Re terminology:

Re terminology:

[Ronnie Brown]

Peter Sellinger writes recently: 

---------------------------------------------------

I think this is a very apt illustration of what happens if a term with
an existing meaning is redefined to mean something else. Henceforth it
is impossible for anybody to use the term (with either meaning)
without first giving a definition.

---------------------------------------------------

I completely agree. My own problem is with term `infinity groupoid' which  is used to describe something which is not even a groupoid, and whose use seems to me to militate against the understanding of what has been achieved with the original and much earlier definition. I once asked Gian-Carl Rota about such change of terminology, in connection with a refereeing job, and he agreed that mathematicians are used to creating confusion in this way. 

There are two easy tendencies: one is to use an old name in a quite different way, and the other is to use a new name for an old idea, so that the  use of the old term looks old fashioned, and a lot of work may be consigned to the dustbin of history, becoming not easy of access  for new students. 

It seems to be an example of these confusions is the way the simplicial singular complex of a space is called an infinity-groupoid, even the `fundamental infinity groupoid', when what seems to be referred to is that it is a Kan complex, i.e. satisfies the Kan extension condition, studied since 1955. The new term sounds like `dressing up' an old idea to look new. My personal objection to this change of terminology (i.e. axe to grind!) is that this distracts from studying the not so simple proofs that strict  higher homotopical structures exist, which mainly are for structured spaces (in particular filtered spaces (Brown/Higgins, Ashley), n-cubes of spaces (Loday), and more recently smooth spaces (Faria Martins/Picken)). The analysis and comparison of these uses should be made. It was certainly a relief to Philip and I that we could do something with filtered spaces which we could not do for the absolute case; the significance of the fact  that these constructions work and lead to specific calculations should be thought about. 

The term `higher dimensional group theory' which was published in a paper  with that title in 1982 was intended to suggest developing higher groupoid theory and its relations to homotopy theory in the spirit of group theory, which meant specific constructions relevant to geometry and calculations, even computer calculations,  of many examples, in which actual numbers arise as a test of and examples of the general theory, and in which some aspects of group theory are sensibly seen as better represented in the higher dimensional theory; and example of this is the nonabelian tensor  product of groups, where group theorists have found lots of pickings. 

I am not sure how these terminological problems will be resolved, and I know the term (\infty,n)-groupoid has been well used recently but the problem of relation to the older ideas, which have had a certain success, should be recognised. 

Ronnie Brown








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Re terminology:

[soloviev]

My personal opinion is that this process is very much influenced
by the pressure of "bibliometry", "impact factors" and other "modern
trends" - people often not very scrupulously invent and reinvent
terminology to be better cited, and, conscious or not, it often very
much smells of imposture.

Sergei Soloviev



> Peter Sellinger writes recently:
>
> ---------------------------------------------------
>
> I think this is a very apt illustration of what happens if a term with
> an existing meaning is redefined to mean something else. Henceforth it
> is impossible for anybody to use the term (with either meaning)
> without first giving a definition.
>
> ---------------------------------------------------
>
> I completely agree. My own problem is with term `infinity groupoid' which
> is used to describe something which is not even a groupoid, and whose use
> seems to me to militate against the understanding of what has been
> achieved with the original and much earlier definition. I once asked
> Gian-Carl Rota about such change of terminology, in connection with a
> refereeing job, and he agreed that mathematicians are used to creating
> confusion in this way.
>
> There are two easy tendencies: one is to use an old name in a quite
> different way, and the other is to use a new name for an old idea, so that
> the  use of the old term looks old fashioned, and a lot of work may be
> consigned to the dustbin of history, becoming not easy of access  for new
> students.
>
> It seems to be an example of these confusions is the way the simplicial
> singular complex of a space is called an infinity-groupoid, even the
> `fundamental infinity groupoid', when what seems to be referred to is that
> it is a Kan complex, i.e. satisfies the Kan extension condition, studied
> since 1955. The new term sounds like `dressing up' an old idea to look
> new. My personal objection to this change of terminology (i.e. axe to
> grind!) is that this distracts from studying the not so simple proofs that
> strict  higher homotopical structures exist, which mainly are for
> structured spaces (in particular filtered spaces (Brown/Higgins, Ashley),
> n-cubes of spaces (Loday), and more recently smooth spaces (Faria
> Martins/Picken)). The analysis and comparison of these uses should be
> made. It was certainly a relief to Philip and I that we could do something
> with filtered spaces which we could not do for the absolute case; the
> significance of the fact  that these constructions work and lead to
> specific calculations should be thought about.
>
> The term `higher dimensional group theory' which was published in a paper
> with that title in 1982 was intended to suggest developing higher groupoid
> theory and its relations to homotopy theory in the spirit of group theory,
> which meant specific constructions relevant to geometry and calculations,
> even computer calculations,  of many examples, in which actual numbers
> arise as a test of and examples of the general theory, and in which some
> aspects of group theory are sensibly seen as better represented in the
> higher dimensional theory; and example of this is the nonabelian tensor
> product of groups, where group theorists have found lots of pickings.
>
> I am not sure how these terminological problems will be resolved, and I
> know the term (\infty,n)-groupoid has been well used recently but the
> problem of relation to the older ideas, which have had a certain success,
> should be recognised.
>
> Ronnie Brown
>

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Re terminology:

[Urs Schreiber]

Dear Ronnie Brown,

you write::

> My own problem is with term `infinity groupoid' which
>  is used to describe something which is not even a groupoid,

It seems to follow the well established terminology in higher category
theory, which proceeds: category, 2-category, 3-category, ....
infinity-category and  groupoid, 2-groupoid, 3-groupoid, ...
infinity-groupoid.

> It seems to be an example of these confusions is the way the simplicial
> singular complex of a space is called an infinity-groupoid, even the
> `fundamental infinity groupoid', when what seems to be referred to is that
> it is a Kan complex, i.e. satisfies the Kan extension condition, studied
> since 1955.

The notion of Kan complex is one model for the notion of
infinity-groupoid. There are other, equivalent models. And there are
models that model stricter subclasses of infinity-groupoids, such as
those you are famous for having studied. Part of the point of saying
"infinity-groupoid" instead of "Kan complex" or else is to amplify the
general notion over its concrete implementation.

> this distracts from studying the not so simple proofs that strict
>  higher homotopical structures exist,

I don't quite see why the term should distract from or otherwise
diminish accomplishments made in the study of strict
infinity-groupoids. On the contrary, to my mind the general theory of
infinity-groupoids puts many of these constructions into the full
perspective of higher category theory and thereby amplifies their
relevance.


> I know the term (\infty,n)-groupoid has been well used recently


Not quite: the term (infinity,n)-category is used to denote an
infinity-category in which all k-morphisms for k greater than n are
equivalences. So an infinity-groupoid is an (infinity,0)-category.

This terminology was not invented in order to hide anybody's previous
accomplishments. On the contrary, I think, this terminology follows
established use in higher category theory and serves to nicely
organize past, present and future insights into higher category theory
in a coherent picture.

I am hoping that eventually we find a constructive way of thinking
about these things, where past insights are seen as fitting into the
beautiful picture of higher category theory that has recently been
emerging, and are not seen to be in conflict with them.

Here is an example I quite like: in the context of
(infinity,1)-category theory Jacob Lurie gave an entirely intrinsic
category-theoretic definition of (infinity,1)-sheaves, also known as
infinity-stacks: these are infinity-groupoid valued presheaves
satisfying a suitable descent condition.

Working backwards from the abstract higher category-theoretic
definition of these, one can work out how this notion matches related
models that were previously investigated.

Among these are two main strands:

1) the homotopical structures/model category structures on categories
of ordinary presheaves with values in simplicial sets, as developed by
Brown, Joyal, Jardine, Dugger and others.

2) The notion of presheaves with values in strict infinity-groupoids /
strict omega-groupoids, as originally conceived by John Roberts and
then formalized by Ross Street and others.

One might worry that both these decade-old developments might not
harmonize with the intrinsic higher-categorical notion of
(infinity,1)-sheaf. But the opposite is true: one finds that they are
neatly subsumed in the abstract definition and conversely provide
concrete workable models for the abstract notion.

For point 1) this is proven in Jacob Lurie's book on higher topos
theory: the Joyal/Jardine model structure models precisely those
(infinity,1)-sheaves which are "hypercomplete". More generally, the
left Bousfiled localization of the model structure on simplicial
presheaves at Cech nerves models (infinity,1)-sheaves.

For point 2) this has been proven by Dominic Verity, following a
conjecture I made: one can show that under mild conditons, under the
inclusion of strict omega-groupoids / strict infinity-groupoids into
all infinity-groupoids, the Roberts-Street notion of descent for such
strict oo-groupoid sheaves does model the abstractly found
(infinity,1)-sheaf condition.

http://ncatlab.org/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves

So this means now not a diminishing but a considerable increase in
value of the old work on presheaves with values in strict
infinity-groupoids / omega-groupoids: since it embeds these
constructions into a powerful abstract theory, we now conversely have
all the abstract tools and insights available to study and use the
former.

I have been using this embeddingg of strict oo-groupoid valued
oo-stacks into all oo-stacks quite a lot in my research, originally
starting with the observation that the BCSS-model of the string
2-group realizes it as a strict 2-groupoid valued stack on Diff, which
is quite useful for some applications. All along I have greatly
benfitted by having your book and nonabelian algebraic topology next
to me, together with Ross Street's articles on descent, and at the
same time having Higher Topos Theory on the table. I find that that
these two aspects interact very nicely, and I was therefore a bit
saddened by hearing what sounded like accusations that new
developments in higher category theory try to intentionally diminish
previous development.

I think math is a win-win game, not a competition: one person's
insight does not dimish the other person's insight, but both increase
each other's value. I am dearly hoping that those who practiced
aspects of higher category theory for so long see the new developments
not as in conflictt with their work, but as a beautiful blossoming of
the theory that they started developing.

Because it is true.


Best regards,
Urs


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Re: terminology

[Eduardo J. Dubuc]

soloviev@irit.fr wrote:
> My personal opinion is that this process is very much influenced
> by the pressure of "bibliometry", "impact factors" and other "modern
> trends" - people often not very scrupulously invent and reinvent
> terminology to be better cited, and, conscious or not, it often very
> much smells of imposture.
>
> Sergei Soloviev


I agree with this. But it should be clear that many times it is not 
conscious, but certainly it often smells of imposture. Other times it 
smells of excessive logic and formalism.

Concerning "injective" I see no problem at all that some times injective
means (1-1) and other times it means the dual of projective.

Where is the problem !!, the context always tells you which meaning it is
being used.

e.d.


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Re terminology:

[Joyal, Andre]

Dear Urs and Ronnie,

Sergei Soloviev wrote:

>My personal opinion is that this process is very much influenced
>by the pressure of "bibliometry", "impact factors" and other "modern
>trends" - people often not very scrupulously invent and reinvent
>terminology to be better cited, and, conscious or not, it often very
>much smells of imposture.

Urs Schreiber wrote:
 
>It seems to follow the well established terminology in higher category
>theory, which proceeds: category, 2-category, 3-category, ....
>infinity-category and  groupoid, 2-groupoid, 3-groupoid, ...
>infinity-groupoid.


I introduced the terminology "quasi-category" as an alternative name 
for weak Kan complexes because I wanted to suggest that the theory of these objects 
was closer to category theory than to the theory of Kan complexes.
For example, the notion of an initial object in a quasi-category 
is very important, like that of initial object in a category.
But only a contractible Kan complex can have an initial object. 
The theory of quasi-categories turns out to be amazingly close to category theory
despite the fact that its natural setting is simplicial homotopy theory.
The name "quasi-category" is for me less frightening than
"infinity-category" which has the name of God into it.
More seriously, why should we attach the prefix "infinity" to an object 
which is no more endless than the set of natural numbers, or the set of rational numbers, 
or the simplicial category Delta? The terminology could be reflecting the
(relative) failure of the algebraic approach to higher categories.
An algebraic description of homotopy type of the 2-sphere is missing
and it could be endless. But the 2-sphere is easy to describe simplicially: 

   S^2= Delta[2]/partial \Delta[2]


Best, 
André

 


-------- Message d'origine--------
De: categories@mta.ca de la part de soloviev@irit.fr
Date: jeu. 20/05/2010 03:58
À: Ronnie Brown
Objet : categories: Re terminology:
 
My personal opinion is that this process is very much influenced
by the pressure of "bibliometry", "impact factors" and other "modern
trends" - people often not very scrupulously invent and reinvent
terminology to be better cited, and, conscious or not, it often very
much smells of imposture.

Sergei Soloviev


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Re terminology:

[Ronnie Brown]

Dear Urs,

Thanks for your friendly and detailed reply.

I should say that I also feel responsible for defending and advertising
the work of my long time collaborator, Philip Higgins, without whom of
course much of the work would not have got done, certainly not so
quickly. His last contribution to maths was in 2005; I helped with the
presentation of his TAC paper, but insisted that it showed `you know the
lion from his claw', as all the ideas were his. He is happily playing
the violin and making string instruments from bare blocks of wood! (That
shows his craftmanship.) He remembers the project  as very hard work but
a lot of fun!

You mention the process from category to infinity-category. Actually
that was why we introduced the term infinity-category in
34.  (with P.J. HIGGINS), ``The equivalence of $\infty$-groupoids and
crossed  complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981)
371-386.
See also:
33.  (with P.J. HIGGINS), ``The equivalence of $\omega$-groupoids and
cubical  $T$-complexes'', {\em Cah. Top. G\'eom. Diff.} 22
(1981) 349-370.

The paper [34] also gives a definition of what is now called (rightly) a
globular set. I am curious to know whether there are earlier definitions
of these terms. Joachim Lambek once asked me  at a category theory
meeting: `Why don't people from xxxx refer to these papers?' What could
I say? People do write about 2-groupoids without referring to analogous
work on crossed complexes.

The points Andre makes are very interesting. In the 1970s we were very
puzzled by the Kan condition, and still are, for the following reason.
The fact that the simplicial singular complex of a space is a Kan
complex is due to a fact about the models, namely a geometric simplex
retracts onto all faces but one. So we can have in effect fillers for
S(X) natural in X, by making choices on the models. Also these fillers
are clearly related to multiple compositions of the remaining faces. The
problem is that there is no unique choice of such retractions, nor is it
clear what might be the relations between iterates of such fillers.
These considerations led Keith Dakin to the notion of T-complex  for his
1976 thesis; somehow `T-complex' has more recently become `complicial
set', but nobody asked me. (Groan! Groan!) So it seems that the notion
of quasi category as a weak Kan complex still has not captured something
about the basic example; but how to repair that is quite unclear.

As I have said before, we found it necessary for certain aspects to work
cubically, as expressing in a manageable way `algebraic inverses to
subdivision', and also to get monoidal closed structures. Again, it is
not clear how to capture axiomatically the properties of the cubical
singular complex, as some kind of weak cubical infinity groupoid. I have
been unable to cope with the complications of multiple compositions in
globular or simplicial terms. Is there an operad view of the cubical case?

I have no wish to hold things up or disparage work developing these
ideas in different ways, just the contrary, and indeed I wrote that I
was thinking of higher dimensional group theory, rather than category
theory. The contrast and relations between such views could, perhaps
should, be illuminating.

We are putting a photo from Macquarie of John Robinson's sculpture
`Journeys' as a frontispiece to the new book.

Best wishes to all for the future of this great adventure.

Ronnie


Urs Schreiber wrote:
> Dear Ronnie Brown,
>
> you write::
>
>
>> My own problem is with term `infinity groupoid' which
>>  is used to describe something which is not even a groupoid,
>>
>
> It seems to follow the well established terminology in higher category
> theory, which proceeds: category, 2-category, 3-category, ....
> infinity-category and  groupoid, 2-groupoid, 3-groupoid, ...
> infinity-groupoid.
>
>
>> It seems to be an example of these confusions is the way the simplicial
>> singular complex of a space is called an infinity-groupoid, even the
>> `fundamental infinity groupoid', when what seems to be referred to is that
>> it is a Kan complex, i.e. satisfies the Kan extension condition, studied
>> since 1955.
>>
>
> The notion of Kan complex is one model for the notion of
> infinity-groupoid. There are other, equivalent models. And there are
> models that model stricter subclasses of infinity-groupoids, such as
> those you are famous for having studied. Part of the point of saying
> "infinity-groupoid" instead of "Kan complex" or else is to amplify the
> general notion over its concrete implementation.
>

....

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Re terminology:

[Urs Schreiber]

Dear Ronnie,

> You mention the process from category to infinity-category. Actually that
> was why we introduced the term infinity-category

This is why I am thinking you could embrace the way the term is used
these days: because it follows precisely your use back then, only
removing the restriction of strictness. And algebraicity can  be
restored. See below...


> The problem is that there is
> no unique choice of such retractions, nor is it clear what might be the
> relations between iterates of such fillers.  These considerations led Keith
> Dakin to the notion of T-complex  for his 1976 thesis; somehow `T-complex'
> has more recently become `complicial set', but nobody asked me. (Groan!
> Groan!) So it seems that the notion of quasi category as a weak Kan complex
> still has not captured something about the basic example; but how to repair
> that is quite unclear.

This has recently been clarified by Thomas Nikolaus in his work on
algebraic Kan complexes (which are essentially simplicial
T-complexes!) and algebraic quasi-categories:

  http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects

He shows that the model category/quasi-category/(oo,1)-category (check
preferred term) of all Kan complexes is equivalent to that of all Kan
complexes with all horn fillers chosen.

And analogously: that the model
category/quasi-category/(oo,1)-category (check preferred term) of all
quasi-categories is equivalent to that of all quasi-categories with
all inner horn fillers chosen.

This says that while a Kan complex or quasi-category is not directly
an algebraic model for an oo-groupoid or (oo,1)-category,
respectively, you can immediately turn it into an algebraic model by
making choices, and up to equivalence, the resulting algebraic model
does not depend on these choices.

Best,
Urs


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Re terminology:

[Joyal, André]

Dear Urs and Ronnie,

As you know, there are important differences between category theory and classical algebra. 
One lies in the fact that equivalent categories are considered to be the "same", 
even if when they are not isomorphic.
In category theory most constructions are yielding an object
which is not unique, but only unique up to some kind of equivalence, 
at best unique up to a unique isomorphism.
The general idea seems to be that an object is well defined
if its different incarnations are connected by a contractible
network of equivalences.


It seems to me that the challenge of higher dimensional algebra 
is to learn how to handle constructions whose output
are not unique but only unique only up to a contractible network.
Of course, we may decide to replace these constructions by ones
producing a truly unique object, but the replacement seems often artificial.
For example, we may decide to choose a representative for
the cartesian product of every pair of objects in a category. 
We are then lead to distinguish between two kinds of product
preserving functors. The functors preserving the 
product strictly are given a role, but this seems artificial to me.

I do not want to be negative about the idea of turning higher
dimensional algebra into ordinary algebra, because we may learn
something in the process. Also, Quillen homotopical algebra
can be regarded as a method for reducing higher categorical
and homotopy algebra to ordinary categorical algebra.
However, there was a real gain in moving from a purely algebraic description
of higher categories to one based on simplicial sets and homotopical algebra.
The category of quasi-categories is cartesian closed, a property which appears
to be false for the category of fibrant objects in the "algebraic" models.
This is also true for the category of n-quasi-category (Rezk).
 

Best,
André



-------- Message d'origine--------
De: categories@mta.ca de la part de Ronnie Brown
Date: ven. 21/05/2010 13:00
À: Urs Schreiber; categories@mta.ca
Objet : categories: Re terminology:
 
Dear Urs,

Thanks for your friendly and detailed reply.

I should say that I also feel responsible for defending and advertising
the work of my long time collaborator, Philip Higgins, without whom of
course much of the work would not have got done, certainly not so
quickly. His last contribution to maths was in 2005; I helped with the
presentation of his TAC paper, but insisted that it showed `you know the
lion from his claw', as all the ideas were his. He is happily playing
the violin and making string instruments from bare blocks of wood! (That
shows his craftmanship.) He remembers the project  as very hard work but
a lot of fun!

You mention the process from category to infinity-category. Actually
that was why we introduced the term infinity-category in
34.  (with P.J. HIGGINS), ``The equivalence of $\infty$-groupoids and
crossed  complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981)
371-386.
See also:
33.  (with P.J. HIGGINS), ``The equivalence of $\omega$-groupoids and
cubical  $T$-complexes'', {\em Cah. Top. G\'eom. Diff.} 22
(1981) 349-370.

........


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Re: terminology

[Ronnie Brown]

Dear André

There seems to me to be a tremendous amount of great work going on 
higher category theory, but when you write

-----------------------------------------------------
One lies in the fact that equivalent categories are considered to be the 
"same", 

even if [or] when they are not isomorphic.
-----------------------------------------

this seems to go against the grain of what I have been doing in groupoids  since I decided they were valuable in about 1965! It sounds like the old  canard `groupoids reduce to groups', so there must be some confusion in my mind on what you are saying. 

One thing that took me a while to realise was that it was not enough to study the fundamental groupoid or a fundamental group but one needed to consider intermediate cases, namely the fundamental groupoid on a set of base points chosen according to the geometry at hand. (`Algebraic topology'  has not understood this it seems.) The vertices of a groupoid give a spatial component to group theory, a kind of geography, and sometimes, even often, that is needed to model the geometry.  So for example it is useful  to replace the trefoil group which has 2 generators x,y and one relation  x^2=y^3 by the trefoil groupoid which is the double mapping cylinder (homotopy pushout) in groupoids of the two maps Z \to Z, given by squaring  and cubing. So we add an extra groupoid generator iota on different vertices which turns x^2 into y^3. This corresponds to the double mapping construction to give a CW-complex. 

So groupoids give the strict algebra of keeping the information which makes things the same. 

In higher dimensions we want not just commutative diagrams but control of  the ways of filling these diagrams. If the diagram is a pentagon (as we all know does happen) I would want a pentagon as part of the geometry, and the only question is how to deal with multiple compositions of various such objects, and that was the aim of David Jones thesis on Polyhedral T-complexes. The point is that the pieces to be composable have to be all faces but one of a general poyhedral `horn', the process of composing them  is the filler of the horn, and the composite of the pieces  is the remaining face of the filler. (It was not attempted to do this in category rather than groupoid terms, and that is still a mystery!) So you can see I have long been very sympathetic to using the Kan condition for describing algebraic or structural objects, but find the simplicial approach too awkward (for me, of course; I found the way Nick Ashley coped with that was amazing). 

I do not want to consider equivalent groupoids the same, as I may want to  use the spatial components to describe how they might be glued together.  It is partly the old tag of not throwing away information till the last possible moment. 

On the other hand, some computations are best done at the strict level, rather than the weak one. I mention here the rotations in my paper:

``Higher dimensional group theory'', in {\em Low dimensional topology},  London Math Soc. Lecture Note Series 48 (ed. R. Brown and T.L.  Thickstun, Cambridge University Press, 1982), pp. 215-238.

(see also a fuller exposition in the new book on Nonabelian algebraic topology), which would seem to be more difficult to write out at the lax level. The fact that the strict calculations imply the existence of certain homotopies is part of the interest. 

So in the work with Higgins a Kan fibration - from the singular filtered complex of a filtered space to the quotient to give a strict structure - ties in the lax and the strict in a necessary way for the theory and calculations. 

I am really searching for points of agreement. 

Best regards

Ronnie


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Re: terminology

[Joyal, André]

Dear Ronnie,

I totally agree with what you wrote.

I wrote 

------------------------------------------------------------
One lies in the fact that equivalent categories are considered 
to be the "same",
------------------------------------------------------------- 


I was careful not to write

------------------------------------------------------------
One lies in the fact that equivalent categories are considered 
to be the same,
------------------------------------------------------------- 

Sorry for not been clear enough.
I hope this settle our apparent disagreement.

Best,
André




-------- Message d'origine--------
De: Ronnie Brown [mailto:ronnie.profbrown@btinternet.com]
Date: sam. 22/05/2010 17:43
À: Joyal, André
Cc: Urs Schreiber; categories@mta.ca
Objet : Re: RE : categories: Re terminology:
 
Dear André

There seems to me to be a tremendous amount of great work going on 
higher category theory, but when you write

-----------------------------------------------------
One lies in the fact that equivalent categories are considered to be the 
"same", 

even if [or] when they are not isomorphic.
-----------------------------------------

this seems to go against the grain of what I have been doing in groupoids since I decided they were valuable in about 1965! It sounds like the old canard `groupoids reduce to groups', so there must be some confusion in my mind on what you are saying. 


....



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Re: terminology

[Colin McLarty]

I have very much appreciated André's subtlety on this issue in conversation

2010/5/22 Joyal, André <joyal.andre@uqam.ca>:
> I wrote
> ------------------------------------------------------------
> One lies in the fact that equivalent categories are considered
> to be the "same",
> -------------------------------------------------------------
> I was careful not to write
> ------------------------------------------------------------
> One lies in the fact that equivalent categories are considered
> to be the same,
> -------------------------------------------------------------

John Baez has written carefully on this point too.

But not everyone is so careful and Ronnie has good reason to be
concerned about a tendency to sweep away distinctions that do need to
be made.

Isomorphic categories too must be distinguished from one another, some
times and for some purposes notably including all currently
articulated versions of categorical foundations.

Grothendieck gave it a fine nuance in Tohoku (p. 125) saying "Aucune
des equivalences de categories qu'on rencontre en pratique n'est un
isomorphisme (none of the equivalences one meets in practice are
isomorphisms)."  He stressed that we must distinguish isomorphisms
from equivalences.  Throughout that and later works he *constructs* a
great many categories up to isomorphism, and not just up to
equivalence.  We do not meet these isomorphisms, we construct them --
and it is quite important that once constructed they are not merely
equivalences.

It is an interesting impulse in higher category theory to avoid
identity in favor of isomorphism on the level of objects, and to avoid
isomorphism in favor of equivalence on the level of categories.   But
so far as I know no one has yet articulated a way to avoid ever using
identity of objects and identity of categories.

Colin






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equivalence terminology

[Paul Taylor]

Colin McLarty said,

> It is an interesting impulse in higher category theory to avoid
> identity in favor of isomorphism on the level of objects, and to avoid
> isomorphism in favor of equivalence on the level of categories.   But
> so far as I know no one has yet articulated a way to avoid ever using
> identity of objects and identity of categories.

I am not going to get involved in higher category theory, but one
setting in which (essentially) the question of identity of objects
arises is in the interpretation of type theories in categories,
where one needs to "choose" binary products, to give the simplest case.

Any type theory has its category of contexts and substitutions
(or classifying category).  This has the categorical structure
that is analogous to the type theoretic connectives, for example
it's a CCC if we started with lambda calculus.

Conversely, any category has its proper language, consisting of names
for its objects and morphisms and various axioms.

Without even having the structure, let alone a choice of it,
the category is embedded in the category of contexts and substitutions
of its proper language.

If the category has choices for the structure then this embedding
is a strong equivalence, ie with a pseudo-inverse,

If it has the structure but not choices for it then it is a weak
embedding - full, faithful and essentially surjective.

The upshot of this is that, by replacing the category with a weakly
equivalent one, you become able to talk about equality of objects,
choices of product, etc.

In other words, using the principle of interchangeability at a
higher categorical level, we get the convenience of working with
equality in the original structure.

This is all explored in my book, "Practical Foundations of Mathematics".

Paul


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we do meet isomorphisms of categories

[Marco Grandis]

On 23 May 2010, at 17:39, Colin McLarty wrote:

> Grothendieck gave it a fine nuance in Tohoku (p. 125) saying "Aucune
> des equivalences de categories qu'on rencontre en pratique n'est un
> isomorphisme (none of the equivalences one meets in practice are
> isomorphisms)."  He stressed that we must distinguish isomorphisms
> from equivalences.  Throughout that and later works he *constructs* a
> great many categories up to isomorphism, and not just up to
> equivalence.  We do not meet these isomorphisms, we construct them --
> and it is quite important that once constructed they are not merely
> equivalences.

We do meet isomorphisms of categories. Only, they are so obvious that
sometimes we do not see them.

For instance:

The category of abelian groups is (canonically) isomorphic to the
category
of Z-modules.

Groups are often defined as semigroups satisfying two conditions; but
they
can also be defined as sets with a zeroary operation, a unary
operation and
a binary operation satisfying certain axioms. Again, we have two
isomorphic
categories. An unbiased definition would give a third isomorphic
category
(and one can form infinitely many intermediate cases between the second
and the third, likely of little interest). Algebras for the free
group monad are
directly linked with the unbiased version, yet not the same.

Lattices (with 0 and 1) can be defined as ordered sets satisfying
some conditions;
or as sets with two binary operations satisfying other conditions;
then one can
add two zeroary operations;...

Best regards

Marco Grandis


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Re: terminology

[Vaughan Pratt]

On 5/23/2010 8:39 AM, Colin McLarty wrote:
> It is an interesting impulse in higher category theory to avoid
> identity in favor of isomorphism on the level of objects, and to avoid
> isomorphism in favor of equivalence on the level of categories.   But
> so far as I know no one has yet articulated a way to avoid ever using
> identity of objects and identity of categories.

Is identity even definable?  I thought it was a kind of received wisdom,
like the natural numbers.  All of us seem to be working with the same
notions of = and N, but what are they, exactly?

That was only intended as a rhetorical question, btw.  We can readily
agree on some properties of = and N, which logicians of various stripes
have gone to the trouble of spelling out, and which number theorists
both analytic and algebraic have expanded on.

Moreover most of us would agree that the proposition "the prime factors
of M = 7^7^7^7 + 5^5^5^5 + 1 (7#4 + 5#4 + 1 where m#n denotes an
exponential stack of n m's) are all greater than 2 billion and there are
more than a thousand distinct such" not only makes perfect sense but is
either true or false.  However fewer might be willing to join me in
insisting that it is certainly true.

Those who question excluded middle for this proposition may have
received different wisdom about N than the rest of us, though if I'm
right then there's a constructive proof of the proposition that can be
checked on any laptop in under an hour, which should then oblige the
intuitionistic objectors to stand down.

(No, I don't currently know a single prime factor of M and I don't
believe anyone else does either.  I do however know the least prime
factors of both M + 1 and M + 958; leaving the former as an exercise,
the latter is 1,985,781,901.  M in decimal is 1755522...1375469 where
the number of omitted digits when itself written in decimal has 695,975
digits, so although M in binary wouldn't fit in the universe let alone a
laptop's random-access memory its length in binary would easily fit in
the latter.  The requisite calculations for all these observations take
only minutes on an ordinary laptop.)

Without exponentiation in the language, M would not be known to us: with
only the polynomial operations the requisite expression 7*7*...*7 +
5*5*...*5 + 1 would stretch beyond the farthest known galaxies.  This
question about M, which is a question about N, could therefore not have
arisen.  With it, the question becomes part of our understanding, or
lack thereof, of N.

The same can be said of identity.  The richer the language, the more
tools we have to probe our understanding of identity, and the clearer
our lack of complete understanding of it becomes.

Vaughan Pratt

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Equality again

[Joyal, André]

Dear Colin,

You wrote:

>It is an interesting impulse in higher category theory to avoid
>identity in favor of isomorphism on the level of objects, and to avoid
>isomorphism in favor of equivalence on the level of categories.   But
>so far as I know no one has yet articulated a way to avoid ever using
>identity of objects and identity of categories.

I love the equality symbol more than an isomorphism symbol,
and an isomorphism symbol more than an equivalence symbol.
I always try to use the equality symbol whenever possible.
I often use the equality symbol for a canonical isomorphism.
Is there a special symbol for canonical isomorphism? (as oppose
to a plain isomorphism). I would love to write something like

  A times (B times C) =' (A times B) times C

André




-------- Message d'origine--------
De: categories@mta.ca de la part de Colin McLarty
Date: dim. 23/05/2010 11:39
À: categories@mta.ca
Objet : categories: Re: terminology
 
I have very much appreciated André's subtlety on this issue in conversation


...


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Re: terminology

[John Baez]

Colin wrote:

It is an interesting impulse in higher category theory to avoid identity in
> favor of isomorphism on the level of objects, and to avoid isomorphism in
> favor of equivalence on the level of categories.   But
> so far as I know no one has yet articulated a way to avoid ever using
> identity of objects and identity of categories.
>

I think Michael Makkai has done it.  He has formulated a foundational
approach to mathematics based on infinity-categories, in which equality
plays no fundamental role:

http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf

I think some approach along these general lines might ultimately become
quite popular.  However, to think in an easy intuitive way about a
mathematical world without equality, we need new definitions of words such
as "the" and "is". Those who find it unpleasant to change the definition of
words such as "autonomous" may think it absurd to consider a such a radical
shift in basic terminology. However, we can already see these words changing
their meanings as we pass from reasoning within sets - where we say "the"
product 2 x 3 "is" 6 - to reasoning within categories - where we say "the"
product of "the" 2-element set and "the" 3-element set "is" "the" 6-element
set.

For a readable introduction to some of Makkai's ideas, try:

http://www.math.mcgill.ca/makkai/equivalence/equivinpdf/equivalence.pdf

Best,.
jb

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Re: Re: terminology

[Colin McLarty]

As to articulating a way to avoid ever using identity of objects and
identity of categories, John Baez writes

> I think Michael Makkai has done it.  He has formulated a foundational
> approach to mathematics based on infinity-categories, in which equality
> plays no fundamental role:
>
> http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf
>
> I think some approach along these general lines might ultimately become
> quite popular.

But so far as  know, this remains an approach, and not any specific
set of axioms offered as foundation.

In this paper Michael defines "multitopic ω-category" more or less
analogously to how Eilenberg and Mac~Lane defined "category," and he
defines "the (large) multitopic set of all
(small) multitopic ω-categories" using that definition.  If I
understand these correctly (and have not, for example, confused
intuitive motivation with strict definition) they take for granted
such as ideas as the category Set of sets, and Set-valued functors.

While Eilenberg and Mac~Lane saw (and referred to) the foundational
significance of their ideas, they did not offer their definition as a
foundation per se.  And they were right.   Lawvere's foundations ETCS
and CCAF are first-order axiomatizations which suffice to prove the
theorems of mathematics (exactly which theorems depending on exactly
which axioms, but he clearly defined variants suited to classical
analysis and various extensions of that).

Has anyone yet offered a  first-order (or ML-typetheoretic)
axiomatization of mathematics along Makkai's lines?

Popular is another question!   And I am not worried about finding a
final form of such axioms.  But I do not yet know of such axioms.

best, Colin

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Re: Equality again

[Patrik Eklund]

I may have missed some parts of the equality message exchange, but here 
a few lines from general equational programming point of view.

I believe it is always important to note where equality or its genetic 
siblings reside. As far as I understand we do category theory mostly over 
ZFC, so ZFC is a metalanguage for category theory. The equality for "the 
diagram commutes" is in ZFC, but the "equation" t1 = t2 involving two 
terms over a signature is more tricky. You might say it's an ordered pair 
(t1,t2), and that structure is in ZFC. The objective of rewriting is to 
find a substitution (Kleisli morphism) s so that s(t1)=s(t2). More 
precisely, the substitution is a morphism s : X -> TY, so you extend it to
Ts : TX -> TTY, and bring it to mu_Y o Ts : TX -> TY with the mu from the 
term monad. All this is done over Set, i.e. T is a monad over Set, and 
therefore TX and TY are sets in ZFC. So, the equality in mu o Ts(t1) = mu 
o Ts(t2) is the equality in ZFC. Incidently, Set is already here in 
question as Set covers only the one-sorted signatures case. Moving over to 
many-sorted signatures you need more.

However, you can use composed monads instead of T, and you don't have to 
be over Set, or its multisorted cousin. Even more so, is it really only 
about ordered pairs? In the end, we are looking for a substitution 
bringing that "possibly something else than just an ordered pair" to 
something close to a 'singleton', where the notion of 'singleton' then 
should reside mostly in the purely categorical language, rather than in 
only and exclusively in ZFC.

General logics (Meseguer, Goguen, Burstall et al) in a general monadic 
setting both for terms as well as sentences, invites to this thinking, 
even if admittedly the programing examples at this point, for the monadic 
extensions, are rather artificial.

Also note that syntactics has for quite a while been studied with respect 
to categorization, but semantics is mostly seen in the metalanguage of set 
theory. Doesn't have to be so? Cannot be so? We are obviously trying to 
complicate things as much as possible in syntactics, and when it comes to 
semantics, our semantics domains are mostly sets, and equality is like 
the emperor, changing clothes all the time.

Best regards,

Patrik



On Mon, 24 May 2010, Joyal, André wrote:

> Dear Colin,
>
> You wrote:
>
>> It is an interesting impulse in higher category theory to avoid
>> identity in favor of isomorphism on the level of objects, and to avoid
>> isomorphism in favor of equivalence on the level of categories.   But
>> so far as I know no one has yet articulated a way to avoid ever using
>> identity of objects and identity of categories.
>
> I love the equality symbol more than an isomorphism symbol,
> and an isomorphism symbol more than an equivalence symbol.
> I always try to use the equality symbol whenever possible.
> I often use the equality symbol for a canonical isomorphism.
> Is there a special symbol for canonical isomorphism? (as oppose
> to a plain isomorphism). I would love to write something like
>
>  A times (B times C) =' (A times B) times C
>
> André
>
>

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Re: terminology

[Toby Bartels]

Vaughan Pratt wrote in part:

>Moreover most of us would agree that the proposition "the prime factors
>of M = 7^7^7^7 + 5^5^5^5 + 1 (7#4 + 5#4 + 1 where m#n denotes an
>exponential stack of n m's) are all greater than 2 billion and there are
>more than a thousand distinct such" not only makes perfect sense but is
>either true or false.  However fewer might be willing to join me in
>insisting that it is certainly true.

Since I know very little about these issues,
I'm not ready to accept your claim that it is true.
(I know that you sketched a way for me to verify it
by performing some calculations on my laptop,
but it would take a while for me to figure out what to program
and then to convince myself that the output meant what you say.)

However, I am happy to agree that the statement is true or false.

>Those who question excluded middle for this proposition may have
>received different wisdom about N than the rest of us, though if I'm
>right then there's a constructive proof of the proposition that can be
>checked on any laptop in under an hour, which should then oblige the
>intuitionistic objectors to stand down.

Anyone who doubts excluded middle for *this* proposition
is not merely a constructivist, or even an intuitionist.
Excluded middle for this proposition is provable in Heyting arithmetic.
While a straightforward calculation of the factors of M
would not fit into the physical universe, it is still finite.

Those who doubt excluded middle (or meaningfulness) for this proposition
go beyond intuitionism; they have been called "ultra-intuitionists",
although the preferred term these days is "ultra-finitists".
As someone who is quite comfortable with constructivism,
I still find ultra-finitism a very strange way to think.

Ultra-finitists definitely have a different recieved wisdom about N
from what the rest of us have received.

Ob categories:  Does anybody know any work on ultra-finitism
from the perspective of categorial logic? (somewhat in the way
that topos theory can provide a perspective on constructivism).
I doubt that any exists, but I would it would be nice if it did.


--Toby


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Re: terminology

[Reinhard Boerger]

Dear Colin, dear all,

you wrote:

> It is an interesting impulse in higher category theory to avoid
> identity in favor of isomorphism on the level of objects, and to avoid
> isomorphism in favor of equivalence on the level of categories.   But
> so far as I know no one has yet articulated a way to avoid ever using
> identity of objects and identity of categories.

As far as I remember I listened to a talk by a logician called H. Preller in
the seventies. She developed a language for categories, which did not
contain the identity of objects.


Greetings
Reinhard



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Re: terminology

[John Baez]

Colin wrote:


As to articulating a way to avoid ever using identity of objects and
> identity of categories, John Baez writes
>
>> I think Michael Makkai has done it.  He has formulated a foundational
>> approach to mathematics based on infinity-categories, in which equality
>> plays no fundamental role:
>>
>> http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf
>>
>> I think some approach along these general lines might ultimately become
> quite popular.
>
> But so far as  know, this remains an approach, and not any specific set of
> axioms offered as foundation.
>


I should let Michael speak for himself, but I have the impression that he
intends to found all his work on FOLDS - "first-order logic with dependent
sorts".  In this paper:

http://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf

he writes:

"The restriction on the use of equality in FOLDS is a fundamental feature.
FOLDS is to be used in formulating categorical situations in which, for
example, equality of objects of a category is not an admissible primitive.
The absence of term-forming operators, to be interpreted as
functions, is a consequence of the absence of equality; it seems to me that
the notion of "function" is incoherent without equality.

It is convenient to regard FOLDS a logic without equality entirely, and deal
with equality, as much as is needed of it, as extralogical primitives."

Best,
jb


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we do meet isomorphisms of categories

[Toby Bartels]

Marco Grandis wrote in part:

>Colin McLarty wrote:

>>Grothendieck gave it a fine nuance in Tohoku (p. 125) saying "Aucune
>>des equivalences de categories qu'on rencontre en pratique n'est un
>>isomorphisme (none of the equivalences one meets in practice are
>>isomorphisms)."  He stressed that we must distinguish isomorphisms
>>from equivalences.  Throughout that and later works he *constructs* a
>>great many categories up to isomorphism, and not just up to
>>equivalence.  We do not meet these isomorphisms, we construct them --
>>and it is quite important that once constructed they are not merely
>>equivalences.

>We do meet isomorphisms of categories. Only, they are so obvious that
>sometimes we do not see them.

>The category of abelian groups is (canonically) isomorphic to the category
>of Z-modules.

[further examples cut]

In all of these examples (although obviously not all examples of isomorphisms),
this is more than just an isomorphism; it's an isomorphism over Set.
That is, it's an isomorphism in the slice category Cat/Set.
It may seem beside the point, but in fact it is also important
that it's an isomorphism in the full subcategory of Cat/Set
whose objects are only the faithful functors to Set;
call this the category Conc of CONCRETE categories.
(So they are all concrete isomorphisms of concrete categories.)

If you take a strictly speak-no-evil approach to category theory
(perhaps even going so far as to found your mathematics on FOLDS),
then it is impossible to state that two categories are isomorphic,
because you must speak of equality of objects (or functors) to do this.
In this approach, Cat and Cat/Set are bicategories but not categories.

But it IS still possible to state that two concrete categories are isomorphic;
the bicategory Conc is (up to equivalence) a locally posetal bicategory
(so if you ignore the non-invertible transformations, it's a category).
So it is possible (and necessary) to say, even when you speak no evil,
that all of Marco's examples are concrete isomorphisms.

So I agree that it is important that these are not mere equivalences,
but I claim (playing the role of an equality-is-evil partisan)
that what is important is not so much that they are isomorphisms
as that they are concrete.


--Toby


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Re: we do meet isomorphisms of categories

[Prof. Peter Johnstone]

Yes, we do meet isomorphisms of categories; my favourite algebraic
example is the isomorphism between (Boolean algebras) and (Boolean
rings), and another good one is the isomorphism between (finite
T_0-spaces) and (finite partial orders). But there's a sense in which
these isomorphisms are "accidental", arising from the fact that both
categories are based on the same category of sets, and in practice
(so far as I know) one never makes use of the fact that they are
isomorphisms rather than mere equivalences.

An even better example occurs in realizability. Some years ago on this
list I queried the need for the condition "Sxy is defined for all x
and y" in the definition of a partial combinatory algebra, and John
Longley came up with a beautiful proof that, given a "weak pca" A
which fails to satisfy this condition, there is a pca A' which does
satisfy it, such that the category of A-valued assemblies is *identical*
(not just equivalent, or even isomorphic) to the category of A'-valued
assemblies. (Details can be found in Jaap van Oosten's book.) The
accident arises in this case from the fact that A' happens to have the
same underlying set as A. But, once again, I don't know of any use for
the fact that the correspondence between the categories of assemblies is
anything more than an equivalence.

Peter Johnstone
---------------------------
On Mon, 24 May 2010, Marco Grandis wrote:

> We do meet isomorphisms of categories. Only, they are so obvious that
> sometimes we do not see them.
>
> For instance:
>
> The category of abelian groups is (canonically) isomorphic to the
> category
> of Z-modules.
>
> Groups are often defined as semigroups satisfying two conditions; but
> they
> can also be defined as sets with a zeroary operation, a unary
> operation and
> a binary operation satisfying certain axioms. Again, we have two
> isomorphic
> categories. An unbiased definition would give a third isomorphic
> category
> (and one can form infinitely many intermediate cases between the second
> and the third, likely of little interest). Algebras for the free
> group monad are
> directly linked with the unbiased version, yet not the same.
>
> Lattices (with 0 and 1) can be defined as ordered sets satisfying
> some conditions;
> or as sets with two binary operations satisfying other conditions;
> then one can
> add two zeroary operations;...
>
> Best regards
>
> Marco Grandis

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we do meet isomorphisms of categories

[Marco Grandis]

I should have mentioned another quite elementary example, that is
perhaps more intriguing.

Let us write Top for (topological spaces defined by open sets) and
Top' for
the isomorphic category (topological spaces defined by closed sets).

Let (X, L) be a set X equipped with a complete sublattice L of its
lattice of parts.
Viewing it as on object of Top or Top' will interchange an Alexandrov
topology
for X with the opposite one, generally different.

This says that - formally - we cannot think of these two isomorphic
categories as being the same thing. Even if, of course, we do think
that way, informally and in practice.

I am not entirely convinced by a comment of Peter:
"in practice (so far as I know) one never makes use of the fact that
they are
isomorphisms rather than mere equivalences".

I am happy with the fact that, going from Top to Top' and back, we
get the same
space on the nose; this spares a lot of complications.

Best regards

Marco Grandis


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Re: Equality again

[Prof. Peter Johnstone]

On Mon, 24 May 2010, Joyal, André wrote:

> I love the equality symbol more than an isomorphism symbol,
> and an isomorphism symbol more than an equivalence symbol.
> I always try to use the equality symbol whenever possible.
> I often use the equality symbol for a canonical isomorphism.
> Is there a special symbol for canonical isomorphism? (as oppose
> to a plain isomorphism). I would love to write something like
>
>  A times (B times C) =' (A times B) times C
>
> André
>

TeX provides a command \doteq for an equality sign with a dot over it;
this is used in other areas of mathematics to mean "is approximately
equal to", but as far as I know it hasn't yet been used by 
category-theorists. Perhaps we could use it to mean "is canonically
isomorphic to"?

I'd also like to use it (or something like it) between pairs of
morphisms, meaning that (they are not equal but) they become equal
when composed with the appropriate canonical isomorphisms (to which
I can't be bothered to give names) in order to match up their domains
and codomains. (Of course, this is simply saying that they are
canonically isomorphic as objects of the functor category [2,C],
where C is the category in which they live.)

Peter Johnstone



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Re: terminology

[Toby Bartels]

Colin McLarty wrote in part:

>As to articulating a way to avoid ever using identity of objects and
>identity of categories, John Baez writes

[snip]

>Has anyone yet offered a  first-order (or ML-typetheoretic)
>axiomatization of mathematics along Makkai's lines?

I don't know very much about what's been done along Makkai's lines;
I also would like to see a specific (if not final) set of axioms.

But category theory has been done in Martin-Löof type theory:
http://www.cs.st-andrews.ac.uk/~rd/publications/CTMLTT.pdf
It has also been done in the type-theoretic proof assistant Coq:
http://coq.inria.fr/distrib/v8.2/contribs-20090527/ConCaT.html

In both of these, there *is* a notion of equality (or better, identity)
at all types, hence a notion of identity of objects of any given category,
allowing one to define isomorphism of categories, etc.
However, this logicians' identity does not match mathematicians' equality;
the easiest way to see this is that there are no quotient types.
(This means that already to do set theory, let alone category theory,
you must define a set to be a type equipped with an equivalence relation.
Such a thing is also called "setoid", depending on which author you read.)

You an also use Mike Shulman's SEAR, in the variant without identity.
http://ncatlab.org/nlab/show/SEAR
http://ncatlab.org/nlab/show/SEAR#eqfree
This looks much more like the ordinary language of mathematics.
(Actually, one could modify ETCS in a similar way,
although it would no longer deserve to be called "ETCS".)


--Toby

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we do meet isomorphisms of categories

[Joyal, André]

Dear Marco,

Perhaps I can advocate the importance of isomorphism between categories.
An equivalence between skeletal categories is necessarly an isomorphism.
There are many examples of skeletal categories in mathematics.
The category of matrices over a ring is skeletal.
The category Delta in homotopy theory is also skeletal.

The category Delta(+) of all finite ordinals and order preserving maps
is an interesting example because, as everyones know, it is
freely generated as a monoidal category by a monoid object.
It is not free in the category of strict monoidal functors
but free in the category of (strong) monoidal functors.
There is of course another category, FatDelta(+), which is 
freely generated by a monoid object in the category of strict 
monoidal functors, but it is seldom used.
The two categories FatDelta(+) and Delta(+) are equivalent,
but the category Delta(+) is simpler because it is skeletal.

The category of finite cardinals and all maps is also skeletal.
It is freely generated by one object as a category with finite coproducts.
It is also freely generated by a commutative monoid as a
symmetric monoidal category.

A category C is skeletal iff every equivalence A-->C has a section.
This property characterises minimal models in algebraic topology.
For example, a Kan complex Y is minimal iff every homotopy equivalence
X-->Y, with X a Kan complex, has a section. Minimal models are important
in topology. Sullivan's rational homotopy theory is essentially a technique 
for constructing minimal models of graded commutative algebras. The rational 
homotopy groups of a space can be read directly form its minimal Sullivan model. 

Minimal models exists in higher category theory too.
Every quasi-category has a minimal  model (should I say skeletal?).
This not a property shared by all types of (infty,1)-categories.
Some are better than others. For example, simplicial categories 
do not admit minimal models (in general).

Strict monoidal categories do not admit minimal models either.
This is because strict monoidal structures cannot be transported 
(in general) along equivalence of categories.
Of course, non-strict monoidal structures can.
There is an obstruction for transfering a strict monoidal structure 
to its skeletal model. It is represented by a cohomology class 
of degree 3 when the category is groupoidal.
It is a small miracle of nature that the category Delta(+) is both
strict monoidal and skeletal. 
Similarly for the category of finite cardinals and maps. 

Best wishes,
André


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Re: terminology

[Thorsten Altenkirch]

> But category theory has been done in Martin-Löof type theory:
> http://www.cs.st-andrews.ac.uk/~rd/publications/CTMLTT.pdf
> It has also been done in the type-theoretic proof assistant Coq:
> http://coq.inria.fr/distrib/v8.2/contribs-20090527/ConCaT.html
>
> In both of these, there *is* a notion of equality (or better,  
> identity)
> at all types, hence a notion of identity of objects of any given  
> category,
> allowing one to define isomorphism of categories, etc.
> However, this logicians' identity does not match mathematicians'  
> equality;
> the easiest way to see this is that there are no quotient types.
> (This means that already to do set theory, let alone category theory,
> you must define a set to be a type equipped with an equivalence  
> relation.
> Such a thing is also called "setoid", depending on which author you  
> read.)

I don't know any reasonable formalisation in Intensional Type Theory.  
People usually assume that hom sets are a setoid but objects aren't.  
This means that constructions like arrow categories are not available.  
To avoid this one would have to formalize explicitely what is a family  
of setoids indexed over a setoid. After this it is hard to see the  
category theory...

Cheers,
Thorsten

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Re: terminology

[Toby Bartels]

Thorsten Altenkirch wrote:

>Toby Bartels wrote:

[I suggested formalising category theory without equality of objects
in intensional Martin-Löf type theory, Coq, or SEAR without equality]

>I don't know any reasonable formalisation in Intensional Type Theory.
>People usually assume that hom sets are a setoid but objects aren't.
>This means that constructions like arrow categories are not available.
>To avoid this one would have to formalize explicitely what is a family
>of setoids indexed over a setoid. After this it is hard to see the
>category theory...

Why do you say that one cannot construct arrow categories?
We need only dependent sums, which Martin-Löf has in his type theory,
to form the type of all morphisms of a given category C.
We cannot compare these for equality, nor do we want to.
What we do need to compare for equality are commutative squares
(which are the morphisms in the arrow category) with given corners
(actually, even with two given parallel sides), and this we can do.

To be explicit, let Ob be the type of objects of the category C;
given x, y: Ob, let x -> y be the type of morphisms from x to y.
Given further two morphisms f, g: x -> y, we have a proposition f = g.
(Martin-Löf identifies propositions with types, but Coq does not,
so I say "proposition" so you can interpret it in either system.)
Then of course, there are operations and axioms that I will skip,
except to introduce ; as notation for composition in diagrammatic order:
f: x -> y, g: y -> z |- f;g: x -> z.

Then the type of objects of the arrow category of C
is sum_{x:Ob} sum_{y:Ob} x -> y, a dependent sum of dependent sums;
a typical element of this type is (x,y,f), where x,y: Ob and f: x -> y.
Given two objects (x,y,f) and (u,v,g) of the arrow category,
the type of morphisms from (x,y,f) to (u,v,g) is
sum_{a:x->u} sig_{b:y->v} f;b = a;g. (*)
(Again, I say "sig" to keep things correct in Coq,
since then f;b = a;g is a proposition rather than a type;
this is the same as a sum to Martin-Löf.)
A typical element of this type is (a,b,p),
where p is a proof of the relevant equality.

A set theorist might well write (*) above as
{ (a,b) | a: x -> u, b: y -> v, f;b = a;g };
they do not refer to p, since set theorists accept proof irrelevance.
They would then be finished, but as type theorists,
we still need to define when parallel morphisms are equal.
We do this by imposing proof irrelevance in the definition;
that is, the definition of equality makes no reference to p.
Specifically, given parallel morphisms (a,b,p) and (c,d,q),
both from (x,y,f) to (u,v,g) in the arrow category,
we define the proposition that they are equal
to be the conjunction of a = c and b = d.
Notice that this makes sense, since a,c: x -> u and b,d: y -> v.
So we can define that equality which we need in the arrow category.

To sum up:  An object in the arrow category of C is (x,y,f),
where x and y are objects of C and f: x -> y is a morphism of C.
A morphism from (x,y,f) to (u,v,g) in the arrow category of C
is (a,b,p), where a: x -> u, b: y -> v, and p: a;g = f;b.
Finally, two such morphisms (a,b,p) and (c,d,q) are equal
if and only if a = c and b = d.  (I leave it as an exercise
for the reader to define the operations and prove the axioms
that define the arrow category of C as a category.)

If you find the summary above a bit too wordy, say
  A morphism from (x,y,f) to (u,v,g) in the arrow category of C
  is (a,b), where a: x -> u and b: y -> v such that a;g = f;b.
That we do not give a name to the proof that a;g = f;b
makes it obvious what the definition of equality of morphisms should be,
so we leave it out as an abuse of language, or a convention of definition.
If it still seems odd that it is even possible to give a proof a name,
well, that is a feature of Martin-Löf type theory and Coq
that you can ignore (just as I ignore the feature of ZFC
that it is possible to ask whether two arbitrary sets are equal),
but you can also use SEAR without equality to avoid even that.


--Toby


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Re: Equality again

[Marco Grandis]

On 27 May 2010, at 13:30, Prof. Peter Johnstone wrote:

>
> TeX provides a command \doteq for an equality sign with a dot over it;
> this is used in other areas of mathematics to mean "is approximately
> equal to", but as far as I know it hasn't yet been used by category-
> theorists. Perhaps we could use it to mean "is canonically
> isomorphic to"?
>
> I'd also like to use it (or something like it) between pairs of
> morphisms, meaning that (they are not equal but) they become equal
> when composed with the appropriate canonical isomorphisms (to which
> I can't be bothered to give names) in order to match up their domains
> and codomains. (Of course, this is simply saying that they are
> canonically isomorphic as objects of the functor category [2,C],
> where C is the category in which they live.)
>
> Peter Johnstone

Dear Peter,

Isn't this very dangerous?

1. First, I think you are referring to some (specified) *coherent*
(contractible) system of isomorphisms,
otherwise you can easily prove that 1 = - 1 (see an example below).

2. Even in that case, we know that coherence can be a delicate thing.
Let us take the cartesian product in Set (or the tensor product in a
symmetric monoidal category).
Would you write XxY =. YxX for the symmetry isomorphism s?
Then by XxX =. XxX do you mean s or the identity?
For XxXxX =. XxXxX we have six permutations of variables, generated
by sxX and Xxs; and so on.
I hope nobody will suggest some complicated trick to account for this;
transpositions and permutations are already there, known to
everybody; but we have to name them.


3. Coming back to point 1, "canonical" isomorphisms need not be
coherent.
There are a lot of such situations; I like to refer to the induced
isomorphisms in homological algebra,
because much of my early work was linked with that.

A is an abelian group (or an object of an abelian category, or
something more general that we do not need
to consider here); X is a sublattice of the (modular) lattice of
subobjects of A. We consider the subquotients
H/K of A, where H and K belong to X.

Then the canonical isomorphisms between these subquotients (induced
by idA) are coherent if and only if X is distributive.
(This is what I am calling now a "coherence theorem for homological
algebra"; it applies to all the usual systems
that produce spectral sequences, and is the reason "why" one cannot
make errors when using canonical isomorphisms
there.)

An easy example of non-coherence can be built in the group A = ZxZ,
taking for X the whole lattice of subgroups, obviously not distributive.
Then Zx0 is canonically isomorphic to A/diagonal, and the latter is
canonically isomorphic to 0xZ.
Now, Zx0 and 0xZ are not canonically isomorphic, as already remarked
in Mac Lane's "Homology".
But notice that the composite of these isomorphisms is (x, 0) |-->
(0, -x), while when
you go through A/codiagonal, you get the opposite isomorphism, (x, 0)
|--> (0, x).

Best regards

Marco Grandis




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Re: terminology

[Thorsten Altenkirch]

Dear Toby,

Thank you for your reply. Of course I am aware of this construction.

A setoid is the intensional representation of a quotient (ie  
coequalizer) and any construction involving it should respect this  
structure. To use the underlying set of a setoid to construct another  
set seems fundamentally flawed.

My understanding of an arrow category is that it's objects are the  
morphisms of  the underlying category and since this is a setoid  
objects should be represented as a setoid too.

You may say that we are only interested in objects upto isomorphism.  
But what does this mean precisely?

Cheers,
Thorsten


On 30 May 2010, at 21:51, Toby Bartels <toby 
+categories@ugcs.caltech.edu> wrote:

> Thorsten Altenkirch wrote:
>
>> Toby Bartels wrote:
>
> [I suggested formalising category theory without equality of objects
> in intensional Martin-Löf type theory, Coq, or SEAR without equality]
>
>> I don't know any reasonable formalisation in Intensional Type Theory.
>> People usually assume that hom sets are a setoid but objects aren't.
>> This means that constructions like arrow categories are not  
>> available.
>> To avoid this one would have to formalize explicitely what is a  
>> family
>> of setoids indexed over a setoid. After this it is hard to see the
>> category theory...
>

...


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Re: terminology

[Peter LeFanu Lumsdaine]

On Tue, June 1, 2010 03:39, Thorsten Altenkirch wrote:

> A setoid is the intensional representation of a quotient (ie
> coequalizer) and any construction involving it should respect this
> structure. To use the underlying set of a setoid to construct another set
> seems fundamentally flawed.

Indeed; but what we construct is not just a set, it's a category! :-)

In Toby's construction \C |---> arr \C, the setoid structure of the
hom-sets of \C is _not_ respected if you just look at the underlying
objects of arr \C, but it _is_ respected once you look at the whole
resulting category arr \C.

This is surely no worse than the fact that in just about any construction
on setoids X |---> F(X), if you look at the underlying set of F(X), this
will not fully respect the setoid structure of X?

> My understanding of an arrow category is that it's objects are the
> morphisms of  the underlying category and since this is a setoid objects
> should be represented as a setoid too.

The trouble here is that the original setoid structure is not on the whole
arrow-set C_1, but on the individual hom-sets C_1(a,b).  The arrow
category sums this up over all a,b:C_0, and so is no longer a setoid from
this data alone.  (A dependent sum of setoids over a set doesn't have a
natural setoid structure, as far as I can see?)

In our case, of course, the object-sets _are_ also naturally setoids, with
their equalities given by isomorphisms of the categories.  But we don't
want to think of this setoid structure as primary: it's just a coarse
reflection of part of the overall category structure.

> You may say that we are only interested in objects upto isomorphism.
> But what does this mean precisely?

Going on from the above, it's an extension to the statement "we are only
interested in elements of a setoid up to the given equality relation".  So
a more precise statement could go along the lines of:  Any construction
dependent on objects should respect isomorphisms.

Best,
-p.

-- 
Peter LeFanu Lumsdaine
Carnegie Mellon University



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Re: Equality again

[Joyal, André]

Dear Marco,

We could use of the dotted-equality symbol only when the 
canonical isomorphism under consideration is part of
a contractible network of isomorphisms. The network does
not need to be explicitly identified if the context is clear enough. 


For example, the dotted equality

(A times B)times C =. A times (B times C)

is refering to the associativity constraint.
The dotted equality

  A times B =. B times A

is refering to the symmetry constraint. But
the dotted equality

  A times A =. A times A

is ambiguous and should be excluded (actually, it
is not ambiguous, since the identity of A times A
is denoted  A times A = A times A ).

I am proposing a rule of thumb, not a new formalism.
Mathematics is as much an art as it is an exact science.

Best,
André


-------- Message d'origine--------
De: categories@mta.ca de la part de Marco Grandis
Date: mar. 01/06/2010 02:36
À: Prof. Peter Johnstone; categories@mta.ca
Objet : categories: Re: Equality again
 

On 27 May 2010, at 13:30, Prof. Peter Johnstone wrote:

>
> TeX provides a command \doteq for an equality sign with a dot over it;
> this is used in other areas of mathematics to mean "is approximately
> equal to", but as far as I know it hasn't yet been used by category-
> theorists. Perhaps we could use it to mean "is canonically
> isomorphic to"?
>
> I'd also like to use it (or something like it) between pairs of
> morphisms, meaning that (they are not equal but) they become equal
> when composed with the appropriate canonical isomorphisms (to which
> I can't be bothered to give names) in order to match up their domains
> and codomains. (Of course, this is simply saying that they are
> canonically isomorphic as objects of the functor category [2,C],
> where C is the category in which they live.)
>
> Peter Johnstone

Dear Peter,

Isn't this very dangerous?

1. First, I think you are referring to some (specified) *coherent*
(contractible) system of isomorphisms,
otherwise you can easily prove that 1 = - 1 (see an example below).

2. Even in that case, we know that coherence can be a delicate thing.
Let us take the cartesian product in Set (or the tensor product in a
symmetric monoidal category).
Would you write XxY =. YxX for the symmetry isomorphism s?
Then by XxX =. XxX do you mean s or the identity?
For XxXxX =. XxXxX we have six permutations of variables, generated
by sxX and Xxs; and so on.
I hope nobody will suggest some complicated trick to account for this;
transpositions and permutations are already there, known to
everybody; but we have to name them.


3. Coming back to point 1, "canonical" isomorphisms need not be
coherent.
There are a lot of such situations; I like to refer to the induced
isomorphisms in homological algebra,
because much of my early work was linked with that.

...

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Re: terminology

[Toby Bartels]

Thorsten Altenkirch wrote in part:

>A setoid is the intensional representation of a quotient (ie
>coequalizer) and any construction involving it should respect this
>structure. To use the underlying set of a setoid to construct another
>set seems fundamentally flawed.

I would rather not say "underlying set" here, but "underlying type".
The category of types can do what it likes, but the category of sets
should already have coequalisers.  (Some type theorists do say "set" here;
I just think that it is liable to confuse category theorists.)
It is the setoids which behave like the sets that we know.

But what is flawed about using the underlying type of a set(oid)?
Group theorists who found group theory on set theory
are allowed to use the underlying set of a group.
So set theorists who found set theory (as setoid theory) on type theory
should be able to speak of the underlying type of a set(oid),
and category theorists who found category theory on type theory
should be able to speak of the underlying type of their structures.

>My understanding of an arrow category is that it's objects are the
>morphisms of  the underlying category and since this is a setoid objects
>should be represented as a setoid too.

I'm not sure what you mean by "this is a setoid".
If you mean that, given a category C (as formalised in type theory),
the morphisms of C form a setoid, then this is not true.
Given a category C and two objects x and y of C,
then the morphisms of C from x to y form a setoid, nothing more.

Even if they did form a setoid, what of that?
In Peter May's example of the category of intermediate fields
in a given field extension, the objects do form a setoid.
I call such a category a "strict category":
  http://ncatlab.org/nlab/show/strict+category
Any poset defines a strict category in which isomorphic objects are equal.
More generally, any category in which any two parallel morphisms are equal
may be made into such a strict category by defining equality as isomorphism.
Assuming an appropriate version of the axiom of choice,
any category whatsoever may be made into a strict category
by defining equality as isomorphism and making some choices
to match up hom-sets.

The fact that strict categories exist does not invalidate
the perspective from which categories are not inherently strict,
any more than the existence of monoidal categories
invalidates ordinary category theory.
Even assuming the axiom of choice,
that we can make any category into a strict category is like
our ability to make any monoidal category into a strict monoidal category;
the theory of weak categories and weak monoidal categories stands.
(Incidentally, any strict monoidal category must be a strict category,
while a weak monoidal category need not be.)

>You may say that we are only interested in objects upto isomorphism. But
>what does this mean precisely?

What it means is that, whenever anyone refers to equality of objects,
I interpret it as being a statement in strict category theory,
with all other statements being in ordinary (weak) category theory.
It is an empirical claim that the basic results of category theory
as it is normally understood do not refer to equality of objects.
It is conjecture in metamathematics that any such statement,
if a theorem, has a proof that never refers to equality of objects.
(Whether this conjecture is true or false can depend
on exactly what your foundations of mathematics are.
You also have to take care to identify defined concepts
that implicitly make reference to equality of objects.)


--Toby


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Re: Terminology

[Jean Benabou]

Dear Steve,

I totally agree with you.
Let me apply  your zoological criteria to Category Theory. You begin  
with the very simple notions of category, functor and natural   
transformation. But then you start piling in subclauses such as  
categories with finite limits, or regular, or abelian, or the glorious  
toposes. For functors you refine the notion to fully faithful ones or  
those who have an adjoint, or are flat, or are fibrations. I could  
give hundreds of examples, and even a meticulous zoologist would  
say:To much is to much!
Obviously Category Theory is very bad and the very idea of putting in   
a same bag groups, topological spaces, locales, and the glorious  
toposes is a misconception.
Serious mathematicians agreed with this. You are too young to remember  
the time when these mathematicians called, with zoological  
justification, this theory :  Abstract general nonsense.

All the best,

Jean




Le 14 févr. 17 à 09:48, Steve Vickers a écrit :

> Dear Fred,
>
> A good answer, but my point was that it was a bad question.
>
> You see this once you start pressing at the details. Are seals and  
> turtles fish? No, but on your definition it depends on whether  
> flippers count as legs or not. What about sea snakes? Obviously not  
> - they're snakes, that just happen to live in the sea. But then eels  
> do seem a bit more fishy.
>
> A meticulous zoologist would start piling on the subclauses to pin  
> it down more precisely, but we know that that does not actually  
> refine our understanding of zoology. It just amplifies the  
> misconceptions underlying the original question.
>
> I'm saying the same can happen in mathematics.
>
> All the best,
>
> Steve.
>
>> On 11 Feb 2017, at 20:42, Fred E.J. Linton <fejlinton@usa.net> wrote:
>>
>> Steve, et al.,
>>
>> If you want
>>
>>> a definition of "fish", but on the understanding that it has to  
>>> include
>> whales
>>
>> let me offer: "legless marine vertebrates" :-) .
>>
>> Cheers, -- tlvp
>>
>



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Re: Terminology

[Steve Vickers]

Dear Fred,

A good answer, but my point was that it was a bad question.

You see this once you start pressing at the details. Are seals and turtles fish? No, but on your definition it depends on whether flippers count as legs  or not. What about sea snakes? Obviously not - they're snakes, that just happen to live in the sea. But then eels do seem a bit more fishy.

A meticulous zoologist would start piling on the subclauses to pin it down more precisely, but we know that that does not actually refine our understanding of zoology. It just amplifies the misconceptions underlying the original  question.

I'm saying the same can happen in mathematics.

All the best,

Steve.

> On 11 Feb 2017, at 20:42, Fred E.J. Linton <fejlinton@usa.net> wrote:
> 
> Steve, et al.,
> 
> If you want 
> 
>> a definition of "fish", but on the understanding that it has to include
> whales
> 
> let me offer: "legless marine vertebrates" :-) .
> 
> Cheers, -- tlvp
> 



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Re: Terminology

[Fred E.J. Linton]

Steve, et al.,

If you want 

> a definition of "fish", but on the understanding that it has to include
whales

let me offer: "legless marine vertebrates" :-) .

Cheers, -- tlvp



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Re: Terminology

[Steve Vickers]

Dear Jean,

My own understanding (superficial and possibly wrong) of the history is that  since Bourbaki there have been definitions of "structure" with the aim of reconciling the algebraic examples (where the homomorphisms preserve structure) with the topological spaces (where the continuous maps have inverse images that preserve structure). Certainly if you look at Joy of Cats, the prime classes of examples are those of topological and algebraic categories.

But, as we know from topos theory, it is not foundationally robust to treat topological spaces as "sets with structure", i.e. point-set topology. In general we have to work point-free, at least if we want to save important parts of topology from going down the drain.

If such an important source of examples, the point-set topological spaces, turned out to be misleading, then, in retrospect, any "precise meaning [of structure] on which the community of mathematicians agree", was probably misguided.

It's like looking for a definition of "fish", but on the understanding that it has to include whales.

All the best,

Steve.

> On 9 Feb 2017, at 16:38, Jean Benabou <jean.benabou@wanadoo.fr> wrote:
> 
> Dear Christopher,
> What I, personally, mean by structure is not the point. This word is used,  very often, in mathematical texts. Sometimes giving the impression that it has a precise meaning on which the community of mathematicians agree. And I was sure there was at least one definition on which the majority of users did  agree
> 
> Then I received 3 answers all referring to: The joy of Cats, but different:
> For Carsten Führman, only faithfulness is required, which obviously is not enough
> Jiri Adamek adds: an isomorphism in S is an identity if its image is. I agree with this; but again not enough.
> Thomas Streicher adds a third condition, with which I would probably agree  if was sure of the precise meaning of isofibration. Could you please, even at the risk of being pedantic say what you mean by that
> 
> Many thanks to all
> 
> 
>> â€ژHi Jean - I don't quite understand this question but  would like to. What do you mean by 'structure'? Thanks
>> 

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Re: Terminology

[George Janelidze]

Dear Jean and Jiri,

As we know there is no such notion accepted by everybody. I would probably
vote for

faithful + amnestic + iso-fibration.

Best regards, George

--------------------------------------------------
From: "Jir? Ad?mek" <j.adamek@tu-bs.de>
Sent: Wednesday, February 8, 2017 6:34 PM
To: "categories net" <categories@mta.ca>
Subject: categories: Re: Terminology

> Dear Jean,
>
> The simplest answer is: faithful. But a better one (in view of
> `everything up to isomorphism') is: faithful and amnestic. The latter
> means that p reflects identity morphisms: an isomorphism in S is an
> identity if its image by p is. See The Joy of Cats (free on the web).
>
> Best, Jiri
>
>> QUESTION
>> Let  p: S --> X  be a functor. What conditions should satisfy p to be
>> called a structure functor, i.e. such that every object  s of S can be
>> thought of as a structure on the object  p(s).
>


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Terminology

[Andrée Ehresmann]

For Charles Ehresmann, the answer to Jean's question was that p be a
"homomorphism functor", a notion he already defined in his 1957 paper
"Gattungen in Lokalen Strukturen", reprinted in

   http://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_I-1_et_I-2.pdf

In modern terms it should correspond to a faithful and amnestic functor.

Cordially
Andree



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Re: Terminology

[Jean Benabou]

Dear Christopher,
What I, personally, mean by structure is not the point. This word is  
used, very often, in mathematical texts. Sometimes giving the  
impression that it has a precise meaning on which the community of  
mathematicians agree. And I was sure there was at least one definition  
on which the majority of users did agree

Then I received 3 answers all referring to: The joy of Cats, but  
different:
For Carsten Führman, only faithfulness is required, which obviously is  
not enough
Jiri Adamek adds: an isomorphism in S is an identity if its image is.  
I agree with this; but again not enough.
Thomas Streicher adds a third condition, with which I would probably  
agree if was sure of the precise meaning of isofibration. Could you  
please, even at the risk of being pedantic say what you mean by that

Many thanks to all


> â€ژHi Jean - I don't quite understand this question but would  
> like to. What do you mean by 'structure'? Thanks
>
> Sent from my BlackBerry 10 smartphone on the O2 network.
>  Original Message
> From: Jean Benabou
> Sent: Wednesday, 8 February 2017 16:18
> To: Categories
> Reply To: Jean Benabou
> Subject: categories: Terminology
>
>
> Dear all,
>
> I'm sure the following question has been answered to. Could anyone
> give me a precise answer and references to this answer. Many thanks.
>
> QUESTION
> Let  p: S --> X  be a functor. What conditions should satisfy p to be
> called a structure functor, i.e. such that every object  s of S can be
> thought of as a structure on the object  p(s).
>

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Re: Terminology

[Thomas Streicher]

Dear Jean,

in Remark 13.18 of their book on "Algebraic Theories" Adamek, Rosicky
and Vitale suggest the following conditions

1) p faithful (what they call "concrete over X")
2) p-vertical isos are identities (what they call "amnestic"))
3) p is an isofibration (what they call "transportable")

These seem to be reasonable conditions validated by most examples.

Does this confirm with your intuition?

Thomas

> I'm sure the following question has been answered to. Could anyone
> give me a precise answer and references to this answer. Many thanks.
>
> QUESTION
> Let  p: S --> X  be a functor. What conditions should satisfy p to be
> called a structure functor, i.e. such that every object  s of S can be
> thought of as a structure on the object  p(s).


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Re: Terminology

[Carsten Führmann]

Dear Jean,

unless there is a technical meaning of "structure" I'm not aware of, the
answer may be "Concrete categories" in the sense of Adámek, Herrlich, and
Strecker: http://katmat.math.uni-bremen.de/acc/acc.pdf. A concrete category
is just a faithful functor, but a remarkable amount of theory can be build
on that notion. In particular, a classification of "algebra-like" and
"space-like" structures is already possible at that level.



On Wed, Feb 8, 2017 at 4:56 PM Jean Benabou <jean.benabou@wanadoo.fr> wrote:

> Dear all,
>
> I'm sure the following question has been answered to. Could anyone
> give me a precise answer and references to this answer. Many thanks.
>
> QUESTION
> Let  p: S --> X  be a functor. What conditions should satisfy p to be
> called a structure functor, i.e. such that every object  s of S can be
> thought of as a structure on the object  p(s).
>

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Re: Terminology

[Jirí Adámek]

Dear Jean,

The simplest answer is: faithful. But a better one (in view of
`everything up to isomorphism') is: faithful and amnestic. The latter
means that p reflects identity morphisms: an isomorphism in S is an
identity if its image by p is. See The Joy of Cats (free on the web).

Best, Jiri

> QUESTION
> Let  p: S --> X  be a functor. What conditions should satisfy p to be
> called a structure functor, i.e. such that every object  s of S can be
> thought of as a structure on the object  p(s).


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Terminology

[Jean Benabou]

Dear all,

I'm sure the following question has been answered to. Could anyone
give me a precise answer and references to this answer. Many thanks.

QUESTION
Let  p: S --> X  be a functor. What conditions should satisfy p to be
called a structure functor, i.e. such that every object  s of S can be
thought of as a structure on the object  p(s).



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Re: Terminology

[Robert Dawson]

On 02/05/2013 12:57 AM, Fred E.J. Linton wrote:
> Thomas Streicher <streicher@mathematik.tu-darmstadt.de> suggested:
>
>> ... I'd call it "essentially subterminal".
>
> Hmm ... hitting a translation engine in a particularly good mood, I found
> "essentially terminal" rendering, in German, as "wesentlich unheilbar".
>
> (Round-tripping from there, you get "fundamentally incurable". Like that?
> Alas, it drew a blank on the actual proposal, "essentially subterminal" :-)

 	Subterminal? Um, that would be "Unterseebootendbahnhof"?

 	<grin, duck, & run>



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Re: Terminology

[Fred E.J. Linton]

Thomas Streicher <streicher@mathematik.tu-darmstadt.de> suggested:

> ... I'd call it "essentially subterminal".

Hmm ... hitting a translation engine in a particularly good mood, I found  
"essentially terminal" rendering, in German, as "wesentlich unheilbar".

(Round-tripping from there, you get "fundamentally incurable". Like that?  
Alas, it drew a blank on the actual proposal, "essentially subterminal" :-)
.)

Cheers, -- Fred



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Re: Terminology

[Fred E.J. Linton]

Thomas Streicher <streicher@mathematik.tu-darmstadt.de> suggested:

> ... I'd call it "essentially subterminal".

Hmm ... hitting a translation engine in a particularly good mood, I found  
"essentially terminal" rendering, in German, as "wesentlich unheilbar".

(Round-tripping from there, you get "fundamentally incurable". Like that?  
Alas, it drew a blank on the actual proposal, "essentially subterminal" :-)
.)

Cheers, -- Fred



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Re: Terminology

[Fred E.J. Linton]

Forgive my repeating, perhaps unnecessarily, the obvious, but
without doing that I fear I may just get inextricably lost as I
try, once again, to sort my way through this question to more of
an answer than I was able to access the last times I tried.

If we pay momentary attention to the "underlying point-set" functor,
from the category of Topological Spaces to that of Sets, we see that
it "has" both a left adjoint, assigning to each set that self-same set
in its discrete topology, and a right adjoint, assigning to each set
that self-same set in its indiscrete topology.

That said, let me turn instead to the "underlying set of objects"
functor from that category of all small categories to that of sets.
It, too, has both a left adjoint, assigning to each set "the" category
having that self-same set as its set of objects, but admitting no 
morphisms between any two objects other than identity maps where 
identity maps are absolutely required -- what's known as the discrete
category on that set of objects -- and a right adjoint, assigning to
each set "the" category having that self-same set as its set of objects,
with the peculiar feature that each of its hom-sets has cardinality 1
-- category that, by analogy with the topological right adjoint, one
might (as Toby Bartels so deftly reminds us) choose to call indiscrete.

And if these categories are nothing more nor less than those that Jean 
Bénabou envisages, with functor to the terminal category 1 fully faithful,
then I guess "indiscrete" would be my answer, too, to his question,
"what would you call" such a category? But for me the indiscreteness
is not in any way a reflection of that full fidelity -- rather, it is
a reflection of the parallel between the fact that such a category "is" 
the value of the right adjoint to the "underlying set of objects" functor
and that an indiscrete space serves as value of the right adjoint to the 
"underlying point-set" functor.

"Setoïd"? "essentially subterminal"? Come on, folks, give us a break :-) !

Cheers, -- Fred

------ Original Message ------
Received: Mon, 29 Apr 2013 07:53:37 PM EDT
From: Toby Bartels <categories@TobyBartels.name>
To: Categories <categories@mta.ca>
Subject: categories: Re: Terminology

> Thomas Streicher wrote:
> 
>>Jean Bénabou wrote:
> 
>>>What would you call a category X such that the functor X --> 1 is
>>>full and faithful? Please don't tell me what they are, I  know that.
> 
>>Sticking to the pattern I suggested I'd call it "essentially subterminal".
> 
> I learnt to call that an "indiscrete category", so I probably would.
> (Another term that I've heard is "chaotic category", which I never liked.)
> Of course, I could also call it a "truth value",
> but only in a context where I would expect this to be understood
> (and being "non-evil", that is working up to equivalence,
> is not actually sufficient for that).  Thus the nLab has
> http://ncatlab.org/nlab/show/indiscrete+category as its own page.
> 
>>>Non evil is essentially evil.
>>>I rather like this conclusion, don't you?
> 
> It is beautiful, but is it accurate?
> 
>>I'd expect the people abhoring evilness would
>>say that full and faithful and essentially surjective is an "evil" notion
>>of equivalence as opposed to the "good" one of adjoint pair where unit  and
>>counit are isos. The latter makes sense in any 2-category whereas the
former
>>doesn't. However, often you just get the "evil" version when not having
>>a strong form of AC (for classes) available.
> 
> On the contrary, an ff and eso functor between two categories
> is enough for the people who abhor evil, as far as I know,
> to decide that the categories are equivalent (and so essentially the same).
> Yet at the same time, these people tend to abhor AC!  How can this be?
> It works if one works in a 2-category whose 1-morphisms are anafunctors.
> Then it is a theorem requiring no choice (and true internal to any topos)
> that any ff and eso functor can be enriched to an adjoint equivalence
> (and in an essentially unique way).
> 
> Of course, "abhor" here should really be read as "consider optional".
> It is possible to work with strict categories, or to work with AC,
> but the main principles and results of category theory do not require
either.
> 
> 
> --Toby
> 


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Re: Terminology

[Peter May]

I apologize for poor taste, but I do like chaotic: one might substitute
indiscrete
in the title ``Chaotic categories and equivariant classifying spaces''
(posted at
http://front.math.ucdavis.edu/1201.5178), but surely not essentially
subterminal.
The comment I'd really like to make is that such categories can be
seriously
interesting.

Peter


On 4/29/13 3:05 PM, Toby Bartels wrote:
> Thomas Streicher wrote:
>
>> Jean B?nabou wrote:
>>> What would you call a category X such that the functor X --> 1 is
>>> full and faithful? Please don't tell me what they are, I  know that.
>> Sticking to the pattern I suggested I'd call it "essentially subterminal".
> I learnt to call that an "indiscrete category", so I probably would.
> (Another term that I've heard is "chaotic category", which I never liked.)
> Of course, I could also call it a "truth value",
> but only in a context where I would expect this to be understood
> (and being "non-evil", that is working up to equivalence,
> is not actually sufficient for that).  Thus the nLab has
> http://ncatlab.org/nlab/show/indiscrete+category as its own page.
>
>>> Non evil is essentially evil.
>>> I rather like this conclusion, don't you?
> It is beautiful, but is it accurate?
>
>> I'd expect the people abhoring evilness would
>> say that full and faithful and essentially surjective is an "evil" notion
>> of equivalence as opposed to the "good" one of adjoint pair where unit and
>> counit are isos. The latter makes sense in any 2-category whereas the former
>> doesn't. However, often you just get the "evil" version when not having
>> a strong form of AC (for classes) available.
> On the contrary, an ff and eso functor between two categories
> is enough for the people who abhor evil, as far as I know,
> to decide that the categories are equivalent (and so essentially the same).
> Yet at the same time, these people tend to abhor AC!  How can this be?
> It works if one works in a 2-category whose 1-morphisms are anafunctors.
> Then it is a theorem requiring no choice (and true internal to any topos)
> that any ff and eso functor can be enriched to an adjoint equivalence
> (and in an essentially unique way).
>
> Of course, "abhor" here should really be read as "consider optional".
> It is possible to work with strict categories, or to work with AC,
> but the main principles and results of category theory do not require either.
>
>
> --Toby


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Re: Terminology

[Toby Bartels]

Thomas Streicher wrote:

>Jean B?nabou wrote:

>>What would you call a category X such that the functor X --> 1 is
>>full and faithful? Please don't tell me what they are, I  know that.

>Sticking to the pattern I suggested I'd call it "essentially subterminal".

I learnt to call that an "indiscrete category", so I probably would.
(Another term that I've heard is "chaotic category", which I never liked.)
Of course, I could also call it a "truth value",
but only in a context where I would expect this to be understood
(and being "non-evil", that is working up to equivalence,
is not actually sufficient for that).  Thus the nLab has
http://ncatlab.org/nlab/show/indiscrete+category as its own page.

>>Non evil is essentially evil.
>>I rather like this conclusion, don't you?

It is beautiful, but is it accurate?

>I'd expect the people abhoring evilness would
>say that full and faithful and essentially surjective is an "evil" notion
>of equivalence as opposed to the "good" one of adjoint pair where unit and
>counit are isos. The latter makes sense in any 2-category whereas the former
>doesn't. However, often you just get the "evil" version when not having
>a strong form of AC (for classes) available.

On the contrary, an ff and eso functor between two categories
is enough for the people who abhor evil, as far as I know,
to decide that the categories are equivalent (and so essentially the same).
Yet at the same time, these people tend to abhor AC!  How can this be?
It works if one works in a 2-category whose 1-morphisms are anafunctors.
Then it is a theorem requiring no choice (and true internal to any topos)
that any ff and eso functor can be enriched to an adjoint equivalence
(and in an essentially unique way).

Of course, "abhor" here should really be read as "consider optional".
It is possible to work with strict categories, or to work with AC,
but the main principles and results of category theory do not require either.


--Toby


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Re: Terminology

[Olivier Gerard]

On Sun, Apr 28, 2013 at 5:49 AM, Jean Bénabou <jean.benabou@wanadoo.fr>
  wrote:

I don't like very much "setoids", and I am very tempted by "essentially
> discrete" as Thomas suggested.


For these ones, I would suggest "catégories timides" or "catégories
réservées", as a play on "discrete", something you could translate as "shy"
or "bashful" or "shrinking categories".

What would you call a category X such that the functor X --> 1 is full and
> faithful? Please don't tell me what they are, I  know that. I'm not even
> asking if there is a we'll established name for them. I don't think there
> is one. What I ask is: Could you suggest one? Preferably a name which would
> be suitable when we work with categories internal to a Topos E where
> supports don't split.


If you are in a playful mood, one could call them "catégories unspirées".
Another suggestion is "catégories modestes".  This would make a good trio
with "catégories discrètes".

Olivier Gérard

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Re: Terminology

[Thomas Streicher]

Dear Jean,

> What would you call a category X such that the functor X --> 1 is
> full and faithful? Please don't tell me what they are, I  know that.

Sticking to the pattern I suggested I'd call it "essentially subterminal".

> Non evil is essentially evil.
> I rather like this conclusion, don't you?

Of course, that's brilliant dialectics! I'd expect the people abhoring evilness
would say that full and faithful and essentially surjective is an "evil" notion
of equivalence as opposed to the "good" one of adjoint pair where unit and
counit are isos. The latter makes sense in any 2-category whereas the former
doesn't. However, often you just get the "evil" version when not having
a strong form of AC (for classes) available. That's why your dialectics
definitely applies!

Best regards, Thomas



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Re: Terminology

[Jean Bénabou]

Dear Thomas, Dear all,

My definition of poset is: "preordered set". I don't know if there is a general agreement, since some answers seemed to suppose that I meant "partially ordered set". It is because I feared this confusion that i specified by adding: equivalent to a discrete category. 
Of course I knew that they were "equivalence relations", and had also many other simple characterizations. One which I like and need is: X is equivalent to a discrete category iff the functor X --> 1 is faithful  and conservative (i.e. reflects isos) because it has the following generalization:

Let P: X --> S be a prefibration. The following are equivalent:
(i) P is equivalent to a discrete fibration.
(ii) P is faithful and consevative.
(iii) each fiber of P is equivalent to a discrete category.

Thus my question  was not:  what are such categories, for which I knew perfectly many answers, but : is there a well established name for them? 
Suggestions such as "setoids" or "essentially discrete" show that this is not the case.

I don't like very much "setoids", and I am very tempted by "essentially discrete" as Thomas suggested. 
But I shall make my question a bit more difficult. 
What would you call a category X such that the functor X --> 1 is full and faithful? Please don't tell me what they are, I  know that. I'm not even asking if there is a we'll established name for them. I don't think there is one. What I ask is: Could you suggest one? Preferably a name which would be suitable when we work with categories internal to a Topos E where supports don't split.

As a side remark, let me say that I don't care very much for the distinction between "evil" and "non evil". Apart  from obvious moral or philosophical reasons, for the following purely mathematical one: Non-evilness depends on the notion of equivalence of categories. And this in turn may heavily depend on which notion of equivalence you chose. And some of these notions depend on the axiom of choice, which I might be tempted to call "evil". Thus we'd reach the following conclusion: 
Non evil is essentially evil. 
I rather like this conclusion, don't you?

Best regards,
Jean

Le 27 avr. 2013 à 15:08, Thomas Streicher a écrit :

> Dear Jean,
> 
a
> 
>> As many of you I presume, I have for ages, and very often, had to deal with categories which are both groupoïds and posets, or again which are equivalent to a discrete category. Is there a well established name for them?
> 
> What about "essentially discrete" like in "essentially small" or
> "essentially surjective". Generally, for any property P of categories
> I would say a category is "essentially P" if it is equivalent to a
> category with property P.
> So "essentially" is a kind of magic word transforming "evil" properties
> into "non-evil" ones. (I don't think one should always do this!)
> 
> Thomas



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Re: Terminology

[Thomas Streicher]

Dear Jean,

> As many of you I presume, I have for ages, and very often, had to deal with categories which are both groupo?ds and posets, or again which are equivalent to a discrete category. Is there a well established name for them?

What about "essentially discrete" like in "essentially small" or
"essentially surjective". Generally, for any property P of categories
I would say a category is "essentially P" if it is equivalent to a
category with property P.
So "essentially" is a kind of magic word transforming "evil" properties
into "non-evil" ones. (I don't think one should always do this!)

Thomas


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Re: Terminology

[David Roberts]

One option is "setoid".

Best regards,
David Roberts
On Apr 25, 2013 7:38 AM, "Jean Bénabou" <jean.benabou@wanadoo.fr> wrote:

> Dear all,
>
> As many of you I presume, I have for ages, and very often, had to deal
> with categories which are both groupoïds and posets, or again which are
> equivalent to a discrete category. Is there a well established name for
> them?
>
> Best regards,
> Jean

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Terminology

[Jean Bénabou]

Dear all,

As many of you I presume, I have for ages, and very often, had to deal with categories which are both groupoïds and posets, or again which are equivalent to a discrete category. Is there a well established name for them?

Best regards,
Jean

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Re: terminology

[Todd Trimble]

In reference to Eduardo's recent comment

>I feel the need to clarify some of my postings.
>
> Due to some public and private mails I realized that most people though
> that I
> was talking about the nLab.
> Well, all the time, when referring to "ghetto" or "subculture" I was
> aiming to
> the WHOLE of the category community within mathematics, not at the nLab
> within
> the category community.
>

I for one didn't get the impression you were referring to the
nLab.  My own comment was in response to Andre Joyal who
wrote "The 'evil' terminology is promoted by a small group of
peoples active in the nLab. It does not reflect a commun usage
in themathematical community."

I thought this could lead to a misunderstanding about the nLab,
hence my comment.

> Actually, I was not even aware of the existence of the nLab.
Due to this
> controversy, I visit the nLab and at first sight I essentially (not fully)
> agree with Andre's comments about the nLab in his recent posting.
>
> I say, go ahead !, nice work !
>
> I can add that I liked the lack of solemnity and the freedom to write down
> your understanding without fear to be wrong, and the freedom of the reader
> to
> insert comments and ask questions. The whole thing is very useful to all
> interested in the subjects being written about, and should not to be taken
> as
> a book in final form, which is not intended to be. Encyclopedia (18
> century)
> and Bourbaki are very important, but some fresh air is also important.
>

Thank you for the nice words (and I'm glad that -- even if nothing else
gets resolved -- at least this discussion has heightened awareness of the
existence of this project!).

The nLab (and the companion discussion forum, the nForum) are still
young and small.  It's a wiki, like Wikipedia, so that anyone can edit it.
Therefore, if you or anyone else sees flaws in an nLab article, you have
a warm open invitation to improve it!  It's easy to edit, and we appreciate
your leaving a note at the nForum to mention changes you make, or to
discuss anything you like.

We are a loosely aligned group with perhaps a dozen or so very active
contributors, including Andrew Stacey, Urs Schreiber, Zoran Skoda,
Mike Shulman, Toby Bartels, David Roberts, Tim Porter, David Corfield,
and myself. Perhaps the only things that really unite us are a belief in the
value of category theory and higher category theory, and a belief in the
value of this project.  Of course there is also Andre Joyal's CatLab,
which runs on the same easy-to-use and highly effective software.

Todd



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terminology

[Eduardo J. Dubuc]

I feel the need to clarify some of my postings.

Due to some public and private mails I realized that most people though that I
was talking about the nLab.
Well, all the time, when referring to "ghetto" or "subculture" I was aiming to
the WHOLE of the category community within mathematics, not at the nLab within
the category community.

Actually, I was not even aware of the existence of the nLab. Due to this
controversy, I visit the nLab and at first sight I essentially (not fully)
agree with Andre's comments about the nLab in his recent posting.

I say, go ahead !, nice work !

I can add that I liked the lack of solemnity and the freedom to write down
your understanding without fear to be wrong, and the freedom of the reader to
insert comments and ask questions. The whole thing is very useful to all
interested in the subjects being written about, and should not to be taken as
a book in final form, which is not intended to be. Encyclopedia (18 century)
and Bourbaki are very important, but some fresh air is also important.

I do not appreciate that a controversy about terminology be dismissed by
derision by saying

"thanks for trying to move the discussion away from terminology and back to
actual mathematical matters".

This kind of solemnity makes me shit !!

No need to move away from terminology, nobody is forbidding you to discuss
mathematics by discussing terminology, it is not one thing or the other.

We were talking about terminology, yes !!. Why not !. Terminology is
important, great mathematicians worried about it.

The "evil terminology" is wrong, somebody would even say evil, and it is
important that it does not establish itself.

This is not a fight, to abandon a terminology does not mean to loose a fight,
it just mean to become aware of some sides that were not properly considered
at the beginning. The looser is at the end the winner.

The challenge (not a minor challenge) is to find a good word "x" (or xxxx,
which means the same thing in spite to have four x's, ja!) to mean "invariant
under equivalence", or its negation, once we agree that such a word is
necessary due to the need of brevity justified by frequent use (if this
happens to be the case).

We can discuss the mathematics involved in the presence or lack of invariance
under equivalence, nobody forbids this by talking about the terminology utilized.

e.d.


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Re: terminology

[Colin McLarty]

Yes, Michael has said in several papers that his foundation would be
given in FOLDS.  And perhaps some specific version of the axioms has
been given in some paper that I have missed.  But I do not know of it;
and when Martin-Löf suggested last year that I might want to pursue a
type-theoretic foundation for category theory he did not mention
knowing of one existing yet.  So far as I know it remains a project.

best, Colin



2010/5/26 John Baez <baez@math.ucr.edu>:
> Colin wrote:
>
>
> As to articulating a way to avoid ever using identity of objects and
>> identity of categories, John Baez writes
>>
>>> I think Michael Makkai has done it.  He has formulated a foundational
>>> approach to mathematics based on infinity-categories, in which equality
>>> plays no fundamental role:
>>>
>>> http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf
>>>
>>> I think some approach along these general lines might ultimately become
>> quite popular.
>>
>> But so far as  know, this remains an approach, and not any specific set of
>> axioms offered as foundation.
>>
>
>
> I should let Michael speak for himself, but I have the impression that he
> intends to found all his work on FOLDS - "first-order logic with dependent
> sorts".  In this paper:
>
> http://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf
>
> he writes:
>
> "The restriction on the use of equality in FOLDS is a fundamental feature.
> FOLDS is to be used in formulating categorical situations in which, for
> example, equality of objects of a category is not an admissible primitive.
> The absence of term-forming operators, to be interpreted as
> functions, is a consequence of the absence of equality; it seems to me that
> the notion of "function" is incoherent without equality.
>
> It is convenient to regard FOLDS a logic without equality entirely, and deal
> with equality, as much as is needed of it, as extralogical primitives."
>
> Best,
> jb
>


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Re: terminology

[Peter Selinger]

I had written:
> 
> My last comment is that, unlike what Jeff Egger claimed, "autonomous
> category" is not a special case of "*-autonomous category", because no
> symmetry is assumed in autonomous categories. Unless of course one
> first drops symmetry from the definition of *-autonomous categories,
> as Jeff has also suggested. As it stands, neither of "autonomous" and
> "*-autonomous" implies the other, which is perfectly fine in my
> opinion, since they are two different words.

I would like to clarify that Jeff himself did not say anything false,
because in the context in which he said it, he had in fact assumed the
non-symmetric definition of *-autonomous category (of [Barr 1995]).
Sorry if it sounded like I was accusing him.

My intention was only to point out that the statement "autonomous
categories are a special case of *-autonomous categories" cannot be
quoted out of context, because it is false under the original
definition of *-autonomous category that includes symmetry (of [Barr
1979]). Since it had already been quoted out of context when I wrote
the above, I just wanted to point out how the potential confusion. 

I think this is a very apt illustration of what happens if a term with
an existing meaning is redefined to mean something else. Henceforth it
is impossible for anybody to use the term (with either meaning)
without first giving a definition. That's no problem in a math paper,
where definitions are usually given or cited anyway, and therefore
terminology is in principle arbitrary. But it does tend to hobble
everyday discussion.

-- Peter

P.S.: since I have a demonstrated ability to put my foot in my mouth,
I'd like to clarify that I am not accusing Mike Barr of anything
either. His 1995 paper is clearly entitled "Non-symmetric *-autonomous
categories", and the inside of the paper clearly explains the
distinction. It is only in subsequent use that any confusion arises.
The usual solution, of putting either (non-symmetric) or (symmetric)
in parentheses the first time the term is used, and omitting it for
subsequent uses, is perfectly adequate. I am very happy with the
statement "an autonomous category is a special case of a
(non-symmetric) *-autonomous category".

M. Barr (1979). "*-Autonomous Categories", Lectures Notes in
Mathematics 752. Springer. 

M. Barr (1995). "Non-symmetric *-autonomous categories". 
Theoretical Computer Science 139:115–130.


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Re: terminology

[Toby Bartels]

Thanks for this list, Peter!

I have put much of its content on the nLab at
http://ncatlab.org/nlab/show/category+with+duals
(and Mike has already put more on pages linked from there),
so feel free to speak up again (here or by editing those pages)
if something is wrong.


--Toby


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terminology

[Joyal, André]

Micheal Batanin wrote

>If we follow the principle "foo = 1 foo" and want to agree
>with historical low dimensional terminology we should call categories
>2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.

A few thoughts about terminology.

Categories are tradidionally named according to the nature
of their objects, not the nature of their morphisms. 
We say "the category of sets" not "the category of functions". 
This convention is not respected in the case where the category 
has only one object: we call it a monoid, not because it is a 
mono-object category (maybe we should) but because it has only one 
binary operation in contrast with a ring. 
Like monoids, operads are collections of abstract operations
closed under composition. Classical operads have
only one object, one color. But multi-colored operads
are often called muti-categories, especially when they are big. 

A set is a discrete homotopy type, a 0-type.
This why I like to give the category of sets rank 0.
I like to denote the quasi-category of n-types by U[n].


Best,
André


-------- Message d'origine--------
De: categories@mta.ca de la part de Michael Batanin
Date: jeu. 13/05/2010 19:09
À: Toby Bartels
Objet : categories: Re: bilax_monoidal_functors?=
 

   >> Should we shift the
>> numbers and call category a 3-category?
>
> No, but it seems to me that you are doing something very much like this.

Not at all. It may be was not a good example. A better example would be
categories. If we follow the principle "foo = 1 foo" and want to agree
with historical low dimensional terminology we should call categories
2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.
There are many other examples like stack, gerbes and so on. I agree with
Mike Shulman that this is a byproduct of categorification. But we can
survive with it.


...

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Re: terminology

[wlawvere]

Dear Eduardo and everybody:

In one of your papers you used the term
Nullstellensatz for a special case (in some
sense an "algebraically closed"case). I
propose to use that term in this more
general case.

The parameters for various
traditional cases can be perhaps expessed
by an essential connected morphism of
toposes E->S. That is, a full inclusion of
"relatively discrete" into "relatively
 continuous" which has both left adjoint
("connected components") and right adjoint
("points").

In that context there is a natural map from
points to components; if it is epic, we can
say that the Nullstellensatz holds for
E->S.

If S is just the category of abstract sets,
one could think of E as algebraically closed if
the Nullstellensatz holds.

But as seems implicit in Galois theory, for
algebraic geometry over a non-algebraically
closed K, the appropriate base topos S consists
not of abstract sets, but rather of sheaves
on C = the opposite of the category of finite
extensions of K, with every map covering. If E
is the topos of sheaves on (finitely generated
K-algebras )^op with respect to a topology that
restricts to the above on C, I believe
we have a classical example of both your
formulation and mine.

Bill

PS There are other stronger results that also
could be called  Nullstellensatz, involving
another topos F between E and S, such as
the one generated by algebras that are finite
dimensional as K-vector spaces, or one
suggested by Birkhoff's SDI theorem. What
is the appropriate statement for these results ?


Quoting Eduardo Dubuc <edubuc@dm.uba.ar>:

> hello:
>
> Given a set CC of objects in a topos EE, consider the following
> property:
>
>       " X no= empty  iff  exists C \in CC, hom(C, X) no= empty "
>
> example; CC = a set of generators
>
> Has (this property) already  a name ?
>
> If not, can you suggest one ?
>
> Any answer will be welcome.
>
> (Notice that if CC is a set of points (instead of objects) we say
> that
> there are enough points)
>
> Thanks          Eduardo J. Dubuc
>
>
>
>

terminology

[Eduardo Dubuc]

hello:

Given a set CC of objects in a topos EE, consider the following property:

      " X no= empty  iff  exists C \in CC, hom(C, X) no= empty "

example; CC = a set of generators

Has (this property) already  a name ?

If not, can you suggest one ?

Any answer will be welcome.

(Notice that if CC is a set of points (instead of objects) we say that
there are enough points)

Thanks          Eduardo J. Dubuc

Re: terminology

[Eduardo Dubuc]

I have been asked why I reacted to the intended reeplacement of the names
"cartesian and cocartesian"  by  "prone and supine".  I have given several
reasons, but the one underlying the whole issue is the following:

The reason is that since a long time I have been worried about the ghetto
(in the sense of being isolated from the rest) characteristic of a certain
category theory community (or group of people). And P. May has reacted
concerning "prone and supine" probably because of reasons related to this.

The mathematical community  have been using "cartesian and cocartesian"
since always, and the introduction of "prone and supine" inside this
group will confirm even more the isolation. Examples abound, see M.Barr
introduction of "Molecular topos" to replace Grothendieck's "Locally
connected topos".

No matter how many linguistic points in favor a given name may have (like
prone and supine), to replace a well stablished  name intoduced by a
great mathematician (or school of mathematics) only puts you in
ridiculous.

P. May probably was feeling somehow that this will be extended by the
mathematical community to all category theory practicioners.

I profit by this mail to mention that concerning the concept "final" and
"initial", I am happy (and not surprised) to learn that these words have
been used since a long time to indicate the same categorical concept that
myself, and will certainly refer to the indicated bibliography to further
justify my use of these words.

Re: terminology

[vs27]

On Dec 29 2005, Vaughan Pratt wrote:

> Without taking sides on the prone/supine terminology question, I do have
> a strong reaction to the Benabou/May/Dubuc concern that respect for a
> field is undermined by its adoption of frivolous terminology.
>
Dear Vaughan, as everybody has a say. Just my views.
I prefer some nomenclature that sounds mathematical,
rather than based on the name of a friend or a private joke.
(may be i don't understand all the jokes ?)
Also in any case one should avoid  renaming existing
concepts, that is just not fair.


Good opportunity to wish happy new year to everybody.

Re: terminology

[Eduardo Dubuc]

We should not put everything in the same bag !!

"strange," "charm," "beauty" and even "quark" itself

are beautiful  and poetic names  to refer to objects or concepts which
precisely we do not want to associate any precise meaning in everyday
language, and on the other hand, the objects or concepts are  introduced
whith those names.

"prone/supine"  are all the contrary, they intent to reflect in everyday
language just one aspect of an existing concept which has many, and more
important, they are used in place of a well stablished name.

all this has nothing to do with  "young field" as opposed to "mature
subject"

silly names (if any) in physics would be as bad as in any other subject

do not confuse  things, I found  the  "Scott is sober" an exelent example
of humor that does not undermine respect for the field. Another exelent
example that comes to my mind is M. Barr's "The point of the empty set"

edubuc

>
> Without taking sides on the prone/supine terminology question, I do have
> a strong reaction to the Benabou/May/Dubuc concern that respect for a
> field is undermined by its adoption of frivolous terminology.
>
> This may be a valid concern for a young field like category theory, but
> for a more mature subject such as physics, a more relevant concern is
> the undermining of the ability to poke fun at oneself by the fear of not
> being taken seriously.
>
> Has the adoption of frivolous nomenclature for quarks ("strange,"
> "charm," "beauty" and even "quark" itself) diminished in any way the
> world's respect for quarks and their investigators?
>
> And what of computational topology?  Should we turn a blind eye to
> whether Scott is sober, and substitute a more genteel euphemism for his
> bottom?
>
> Vaughan Pratt
>
>

Re: terminology

[Nikita Danilov]

Vaughan Pratt writes:
 > Without taking sides on the prone/supine terminology question, I do have
 > a strong reaction to the Benabou/May/Dubuc concern that respect for a
 > field is undermined by its adoption of frivolous terminology.
 >
 > This may be a valid concern for a young field like category theory, but
 > for a more mature subject such as physics, a more relevant concern is
 > the undermining of the ability to poke fun at oneself by the fear of not
 > being taken seriously.
 >
 > Has the adoption of frivolous nomenclature for quarks ("strange,"
 > "charm," "beauty" and even "quark" itself) diminished in any way the
 > world's respect for quarks and their investigators?

There indeed are drawbacks whenever scientific terms are contrary to the
centuries old tradition not taken from Greek or Latin languages (that,
thanks to their very regular and flexible system of word formation are
so suitable for taxonomies) shared by many cultures. For one thing,
words of existing languages are not in one to one mapping, and then a
term from contemporary language may be not culturally neutral (consider
silly naming wars for transuranium elements).

On the other hand, I stopped using "co-product" after more than one
person with the background in classical languages read it as
"copro-duct".

 >
 > And what of computational topology?  Should we turn a blind eye to
 > whether Scott is sober, and substitute a more genteel euphemism for his
 > bottom?
 >
 > Vaughan Pratt

Nikita.

Re: terminology

[Vaughan Pratt]

Without taking sides on the prone/supine terminology question, I do have
a strong reaction to the Benabou/May/Dubuc concern that respect for a
field is undermined by its adoption of frivolous terminology.

This may be a valid concern for a young field like category theory, but
for a more mature subject such as physics, a more relevant concern is
the undermining of the ability to poke fun at oneself by the fear of not
being taken seriously.

Has the adoption of frivolous nomenclature for quarks ("strange,"
"charm," "beauty" and even "quark" itself) diminished in any way the
world's respect for quarks and their investigators?

And what of computational topology?  Should we turn a blind eye to
whether Scott is sober, and substitute a more genteel euphemism for his
bottom?

Vaughan Pratt

Re: Terminology

[Eduardo Dubuc]

I very strongly agree with J. Benabou's comments about "prone" and
"supine", and P. May's opinion that "I'd like category theory no longer to
be regarded as nonsense in this country --- it still is in many quarters,
as I could easily prove --- and such terminology is not exactly helpful to
the cause!"

I recall P. Johnstone that he himself named his book  "Elephant Book"
because every body has different version of what a topos is, reflecting
only one of the many aspects of the concept.

Names like "Prone" and "Supine" correspond (with luck) to only one of the
many aspects of the concept of cartesian and its dual (in a sense)
cocartesian.

Also, there is a clear ethical aspect involved when a stablished
terminology that has been historically introduced by particular people
suffers a move to be eliminated and reeplaced by another.

But, coming back to the question above, i am also against the habit to
name a new mathematical concept with words that have a precise meaning in
everyday language (as prone, supine, etc).

Presisely, I do not know what does it mean exactly "Cartesian" (has
something to do with Descartes ...), but I know presisely what it is a
"Cartesian arrow" (in mathematics).

Colorful terminology taken from everyday language is an strong indication
to serious mathematicians that the subject should no be taken seriously
(see for example the claims of  "Catastrofe Theory" as opposed to the
sober "Classification of singularities of C-\infty mappings", and a lot of
similar examples).

As P May points out, "  . . .  such terminology is not exactly helpful to
the cause!".

The meaning of a mathematical concept should be given by the concept
itself, and not by the connotation that its name has in everyday language.

Terminology

[jean benabou]

I have seen in this mail that the suggestion of Taylor and Johnstone to
replace cartesian and cocartesian maps by prone and supine ones begins to
be accepted. When I first saw that suggestion, I was so amazed that I
thought it was a joke, and not such a good one. I still hope it is no more
than that. But, just in case, and before it is too late, I want to say
that I am very strongly opposed to such changes for many reasons:
linguistic, mathematical, and ethical, which I am ready to explain in
detail if I am asked to do so.

Re: terminology

[Marco Grandis]

In reply to Stasheff's question on terminology for homotopy coherent algebras:

>but now what about e.g. 1-homotopy associaitve satisfying a STRICT
>pentagon??
>perhaps strict 1-homotopy

I would say:

"2-strict sha-algebra", as motivated below.
(sha = strongly homotopy associative)

However: after the strict pentagon, this structure has a second coherence
condition for the associativity homotopy
(which disappears for monoidal categories, just because their 2-morphisms
are trivial)

____________

In a paper [*] on strongly homotopy associative (differential) algebras, I
proposed this definition (4.2; pages 38-39).

Notation: a sha-algebra is a graded module  A  with morphisms
(sort of components of a global differential  d of bar coalgebras)

   d_1: A --> A  (degree - 1; the differential)

   d_2: AoA --> A  (degree 0; the product)

   d_3: AoAoA --> A  (degree 1; the associativity 1-homotopy)
   ........
   d_n: A^n  -->  A   (degree  n - 2; the coherence n-homotopy)
   ........

( o = tensor product;  ^n  = tensor power)

under axioms

(1)  d_1.d_1 = 0
(2)  ....

(expressing  dd = 0  for the global differential).

DEF. This is called an *n-strict sha-algebra* if  d_p = 0  for  p > n.

Equivalently,  the morphisms  d_1,..., d_n  have to satisfy the original
axioms (1)  ... (n)
plus n - 1 conditions obtained from the axioms  (n+1) ... (2n - 1),
cancelling the null  d_p's
(the remaining axioms become trivial).

This gives:

1-strict = differential module

2-strict = associative differential algebra

3-strict = 1-homotopy associative differential algebra
   with strict pentagon (from axiom (3)) and axiom (4) reduced to:

   (4)   d3 (1o1od3 + 1od3o1 + d3o1o1) = 0.

_______

So far in that paper.
The name is chosen to make  d_n  the last relevant component, in the
n-strict case.

I might now (more geometrically) prefer a - 1 shift in these names, so that
the last example would be named 2-strict, in accord with the fact that the
last relevant homotopy is a an ordinary ("one-dimensional") homotopy and
everything becomes strict starting with "dimension 2".

_______

Reference:

[*] M. Grandis, On the homotopy structure of strongly homotopy associative
algebras, J. Pure Appl. Algebra 134 (1999), 15-81.

_______

Regards    MG

terminology

[James Stasheff]

In `higher homotopy theory', terminology has not setled down nor is it
transparent

homotopy ___________ algebra can mean a variety of things

letting ______________ = associative
it can mean JUST that there is a homtopy for associaitivity
or
some authors use it to mean A_\infty

which I initially tried to indicate by strongly homtopy associative

_\infty seems to have caught on to mean the presence of higher homtopies
of all orders

in most but not all cases, such algebras have a homtopy invariant
defintion

so I would suggest the following revisionist terminology

1-homotopy associative means JUST that there is a homotopy for
associaitivity

similarly n-homotopy associative would mean homotopies of homotopies
of...

homotopy invariant ___ algebra would mean just what it says

so far so good
but now what about e.g. 1-homotopy associaitve satisfying a STRICT
pentagon??

perhaps strict 1-homotopy

open to suggestions



	Jim Stasheff		jds@math.upenn.edu

		Home page: www.math.unc.edu/Faculty/jds

As of July 1, 2002, I am Professor Emeritus at UNC and
I will be visiting U Penn but for hard copy
        the relevant address is:
        146 Woodland Dr
        Lansdale PA 19446       (215)822-6707

Terminology

[Krzysztof Worytkiewicz]

Dear Categories,

Is it  established terminology  to call *injective* a faithful functor
which is injective (in the usual sense) on objects ?

Cheers, Krzysztof

Re: Terminology

[Max Kelly]

In response to Jean Benabou's question about the terminology for what some
call "cofinal" functors, may I refer him to Section 4.5 of my book "Basic
Concepts of Enriched Category Theory", where such notions are considered
in considerable generality? In so far as we deal with functors - meaning
"V-functors" in the context of V-enriched category theory - the terms I
used, which are those common here at Sydney, are "final functor" and
"initial functor". These notions, however, make sense only when V is
cartesian closed; for a more general symmetric monoidal closed V, what is
said to be initial is a pair (K,x) where K is a V-functor A --> C and x is
a V-natural transformation H --> FK, where H: A --> V and F: C --> V are
V-functors with codomain V, and thus are "weights" for weighted limits.
The 2-cell x expresses F as the left Kan extension of H along K if and
only if, for every V-functor T: C --> B of domain C, the canonical
comparison functor (induced by K and x) between the weighted limits, of
the form

                   (K,x)* : {F,T} ----> {H,TK},
		   
is invertible (either side existing if the other does); the book contains
a third equivalent form making sense whether the limits exist or not. When
these equivalent properties hold, the pair (K,x) is said to be INITIAL.
The point is that, in this case, the F-weighted limit of any T can be
calculated as the H-weighted limit of TK.

When V is cartesian closed, we have for each V-category C the V-functor C
---> V constant at the object 1, limits weighted by which are the CONICAL
limits, which when V = Set are the classical limits. For such a V we can
consider the special case of the situation considered above, where each of
H and F is the functor constant at the object 1, and where x is the unique
2-cell between H and FK; we call the functor K "initial" when this pair
(K,x) is so; equivalently when the canonical lim T ---> lim TK is
invertible for every T (for which one side exists -- or better put in
terms of cones), or equivalently again when

           colim C(K-,c) == 1  for each object c of C.
	  
When V = Set, this is just to say that each comma-category K/c is
connected. When the category C is filtered, a fully-faithful K: A --> C is
final (dual to initial) precisely when each c/K is non-empty.

The book goes on to discuss the Street-Walters factorization of any (ordinary)
functor into an initial one followedby a discrete op-fibration.	

The above being so, it seems that Jean's good taste has led him to suggest
the very same nomenclature that recommended itself to us at Sydney.  I
should have been happier, though, if he had recalled the treatment I gave
lovingly those many years ago. There are many other expositions in the
book that I am equally happy with, and which I am sure Jean would enjoy.
By the way, someone spoke recently on this bulletin board of the book's
being out of print and hard to get; I've been meaning to find the time to
reply to that, and discuss what might be done. The copyright has reverted
to me; but the text does not exist in electronic form - it was written
before TEX existed, and prepared on an IBM typewriter by an excellent
secretary with nine balls.

I suppose I could have some copies - one or more hundreds - printed from
the old master, after correcting the observed typos. But the photocopying
and binding and the postage would cost a bit. I'ld be happy to receive
suggestions, especially from such colleagues as would like to get hold of
a copy. By the way, I sent out preprint copies to about 100 colleagues
back in 1980 or 1981; if any of those are still around, I point out that
they contain the full text. So too do those copies which appeared in the
Hagen Seminarberichte series. Once again, I look forward to any comments,
either in favour of or against making further copies.

Max Kelly.

Re: Terminology

[Dr. P.T. Johnstone]

> I am confronted with problems of "contradictory terminology" which I would
> like to solve and, since english is not my language, I need some
> suggestions.
> Let  F: Y-----> X be a functor such that for every object  x of  X the comma
> category  (x,F) is connected.Such functors, although they are not defined in
> all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's
> handbook (Vol.1-'2.11-p.69) but none of these terms is satisfactory.
>  The "cofinal" name comes obviously from the vocabulary of ordered sets
> which are special cases, but in category theory  "co" is now associated with
> dual notions.

There was some discussion of this point on the categories mailing list
a year or two back. I think there was general consensus that the "co"
in "cofinal" was redundant, and that such functors should simply be
called "final". This is the term used in Mac Lane's book (section IX 3,
p.217) -- I believe Mac Lane was the first to shorten "cofinal" to "final".
For some reason, Borceux chose to use the opposite convention regarding
"initial" and "final" in his book (although, in Exercise 2.17.8 on page 94,
he seems to have reverted to the same convention as Mac Lane).

Peter Johnstone

Re: Terminology

[Steve Lack]

Jean Benabou writes:
 > I am confronted with problems of "contradictory terminology" which I would
 > like to solve and, since english is not my language, I need some
 > suggestions.
 > Let  F: Y-----> X be a functor such that for every object  x of  X the comma
 > category  (x,F) is connected.Such functors, although they are not defined in
 > all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's
 > handbook (Vol.1-§2.11-p.69) but none of these terms is satisfactory.

Mac Lane calls such functors ``final'' in Categories for the Working
Mathematician. I do too.

Steve Lack.

Terminology

[Jean Benabou]

I am confronted with problems of "contradictory terminology" which I would
like to solve and, since english is not my language, I need some
suggestions.
Let  F: Y-----> X be a functor such that for every object  x of  X the comma
category  (x,F) is connected.Such functors, although they are not defined in
all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's
handbook (Vol.1-§2.11-p.69) but none of these terms is satisfactory.
 The "cofinal" name comes obviously from the vocabulary of ordered sets
which are special cases, but in category theory  "co" is now associated with
dual notions.
The "initial" name is even less satisfactory, because:
(i) If  Y=1, F is identified with an object  x of X and F is "initial" iff x
is a terminal object of X  !
(ii) More generally, if Y has a terminal object  t  then F is "initial" iff 
F(t) is terminal !
(iii) Even more generally yet, without assuming the existence of terminal
objects in Y or X :
 Let X^ and Y^ be the categories of presheaves on X and Y, and  F! :X^----->
Y^  the canonical extension of F to these categories.If  T is the terminal
object of Y^ one can easily show that  F has the previous property iff 
F!(T) is terminal in X^.(Which by the way, gives the nicest proof of the
stability under composition of such functors)
I propose to call these functors either "terminal" or better "final" but I
would like to know if this would not conflict with previous terminology.
Thanks for your help. 

re: terminology

[James Stasheff]

So far the clear front runner is `face complex'
Thanks to all the nominators.

Grandis points out why the historical semi-simplicial
won't fly for at least another generation.

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
 \ (    Chapel Hill NC		FAX:(919)-962-2568
  \*)   27599-3250

        http://www.math.unc.edu/Faculty/jds

Re: terminology

[Marco Grandis]

J. Stasheff wrote:

>Has terminology settled down?
>I can recall seeing various terms for
>``simplicial object without degeneracies''


I am afraid it has not.

In my opinion, it should be called 'semi-simplicial object', consistently
with the original terminology in Eilenberg-Zilber (see references below).
Such a term has been adopted in Weibel's text on homological algebra
(1994). But there seems to be some opposition.
___

I hope the following reconstruction of terminology is correct.

1. What is now called a simplicial object was introduced by Eilenberg and
Zilber (1950); they use:

(a) [already existing] 'simplicial complex' = set with distinguished parts;
(b) [new term] 'semi-simplicial complex' = graded set with faces;
(c) [new term] 'complete s.s. complex' = graded set with faces and degeneracies;

2. Later, notion (c) was recognised as more important than (b) and called
'semi-simplicial complex', leaving (b) without any standard name.

3. Since May's book (1967) at least, notion (c) gradually settled down as
'simplicial set', generalised to 'simplicial object' in a category; this is
now standard.

4. It should now be natural to use a similar term, 'semi-simplicial object
(possibly: set)' for (b), i.e. a 'simplicial object without degeneracies'
(as in Weibel 1994). This is consistent with the original use in
Eilenberg-Zilber and gives a non-ambiguous set of terms for the three
notions recalled:

(a) 'simplicial complex' (also: combinatorial complex)
(b) 'semi-simplicial object (set)'
(c) 'simplicial object (set)'

However, I used myself this terminology in a paper published in '97 and had
strong reactions from people attached to the terminology in use between
50's and '60s (point 2 above).

___

References:

S. Eilenberg - J.A. Zilber, Semi-simplicial complexes and singular
homology, Ann. of Math. 51 (1950), 499-513.

J.P. May, Simplicial objects in algebraic topology, Van Nostrand 1967.

C.A. Weibel, An introduction to homological algebra, Cambridge Univ. Press,
Cambridge, 1994.

___

With best regards

Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: grandis@dima.unige.it
tel: +39.010.353 6805   fax: +39.010.353 6752

http://www.dima.unige.it/STAFF/GRANDIS/
ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/

Re: terminology

[Paul Glenn]

How about "face complex"?

James Stasheff wrote:
> 
> Has terminology settled down?
> I can recall seeing various terms for
> ``simplicial object without degeneracies''
> 
> .oooO   Jim Stasheff            jds@math.unc.edu
> (UNC)   Math-UNC                (919)-962-9607
>  \ (    Chapel Hill NC          FAX:(919)-962-2568
>   \*)   27599-3250
> 
>         http://www.math.unc.edu/Faculty/jds

-- 
Paul Glenn
Department of Mathematics
Catholic University of America

terminology

[James Stasheff]

Has terminology settled down?
I can recall seeing various terms for 
``simplicial object without degeneracies''

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
 \ (    Chapel Hill NC		FAX:(919)-962-2568
  \*)   27599-3250

        http://www.math.unc.edu/Faculty/jds