Category without objects

Category without objects

[Uwe Egbert Wolter]

Some years ago (around 30?) I read a book where it was mentioned that
one could define categories without (explicit) objects in the sense that
objects are mimicked by identity morphisms. Unfortunately, I can not
reconstruct what book it was.

I know how this works. I would, however, like to have a reference.

Best

Uwe Wolter


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Re: Category without objects

[Andrew Pitts]

[Note from moderator: Many responses to this; will forward the first 
mentioning each source. Thanks to other posters. ]

On 5 March 2015 at 11:49, Uwe Egbert Wolter <Uwe.Wolter@ii.uib.no> wrote:
> Some years ago (around 30?)

25

> I read a book where it was mentioned that
> one could define categories without (explicit) objects in the sense that
> objects are mimicked by identity morphisms. Unfortunately, I can not
> reconstruct what book it was.
>
> I know how this works. I would, however, like to have a reference.

Peter J Freyd and Andre Scedrov, "Categories, Allegories" (Elsevier,
1990) ISBN 0 444 70368 3

is one such.

Best wishes,

Andy Pitts


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Re: Category without objects

[Jiri Adamek]

See Definition 3.8 in Herrlich & Strecker: Category
Theory (42 years old...).

Cheers
Jiri

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
alternative e-mail address (in case reply key does not work):
J.Adamek@tu-bs.de
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

On Thu, 5 Mar 2015, Uwe Egbert Wolter wrote:

> Some years ago (around 30?) I read a book where it was mentioned that
> one could define categories without (explicit) objects in the sense that
> objects are mimicked by identity morphisms. Unfortunately, I can not
> reconstruct what book it was.
>
> I know how this works. I would, however, like to have a reference.
>
> Best
>
> Uwe Wolter
>

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Re: Category without objects

[René Guitart]

In fact this point is observed even in the original paper of Eilenberg Mac Lane !
Furthermore, I recently proposed a notion of autocategory, a kind of category without object ; this is a little different from categories,  and is based precisely on the distinction between four things : objects, identities, identifiers, domains or codomains (see in CTGD in 2014), and this allows to see with the same "eye" categories, 2-categories, etc.. 
Best,
René Guitart

Le 5 mars 2015 à 12:49, Uwe Egbert Wolter a écrit :

> Some years ago (around 30?) I read a book where it was mentioned that
> one could define categories without (explicit) objects in the sense that
> objects are mimicked by identity morphisms. Unfortunately, I can not
> reconstruct what book it was.
> 
> I know how this works. I would, however, like to have a reference.
> 
> Best
> 
> Uwe Wolter
> 

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Re: Category without objects

[Eduardo J. Dubuc]

Much older, 69-70 years old !

In the original paper:

S. Eilenberg y S. Mac Lane, General Theory of Natural Equivalences,
Trans. Amer. Math. Soc. 58 (1945), 231?294.

They already considered the notion of category without objects, but
choose to use objects. They wrote:

"It is thus clear that the objects play a secondary role, and could be
entirely omitted from the definition of a category. However, the
manipulation of the applications would be slightly less convenient were
this done. [p?g. 238]

On spite of this, Ehresmann was a fan of categories without objects:

Erhesman adopted and pushed forward the notion of categories without
objects, writing many papers and a book where categories did not have
objects.

C. Ehresmann, Cat?gories topologiques et cat?gories diff?rentiables,
Colloque G?om. Diff. Globale (Bruxelles, 1958), Centre Belgue Rech.
Math., Louvain, 1959, 137?150.

Eduardo Dubuc


On 05/03/15 13:49, Jiri Adamek wrote:
> See Definition 3.8 in Herrlich & Strecker: Category
> Theory (42 years old...).
>
> Cheers
> Jiri
>
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> alternative e-mail address (in case reply key does not work):
> J.Adamek@tu-bs.de
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> On Thu, 5 Mar 2015, Uwe Egbert Wolter wrote:
>
>> Some years ago (around 30?) I read a book where it was mentioned that
>> one could define categories without (explicit) objects in the sense that
>> objects are mimicked by identity morphisms. Unfortunately, I can not
>> reconstruct what book it was.
>>
>> I know how this works. I would, however, like to have a reference.
>>
>> Best
>>
>> Uwe Wolter


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Re: Category without objects

[Peter LeFanu Lumsdaine]

We actually had a post-seminar reference-hunt on this in Stockholm quite
recently, and found that the arrows-only definition goes right back to Mac
Lane 1948, “Groups, Categories, and Duality”:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079106/pdf/pnas01707-0037.pdf

This cites two earlier papers only along with the definition (Mac Lane 1942
and Eilenberg–Mac Lane 1945 — the first two papers to mention categories,
right?), but both of those used the objects-and-arrows formulation.  So it
seems that the two-sorted formulation was considered right from the start,
and the arrows-only version either from the start or very soon afterwards.

Of course, the original question has already been well answered, but I
guess the extra history may be of interest to others as well.

Best,
–Peter.

On Fri, Mar 6, 2015 at 1:49 AM, Jiri Adamek <adamek@iti.cs.tu-bs.de> wrote:

> See Definition 3.8 in Herrlich & Strecker: Category
> Theory (42 years old...).
>
> Cheers
> Jiri
>
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> xxxxxxxxxxxxxxx
> alternative e-mail address (in case reply key does not work):
> J.Adamek@tu-bs.de
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> xxxxxxxxxxxxxxx
>
>
> On Thu, 5 Mar 2015, Uwe Egbert Wolter wrote:
>
>  Some years ago (around 30?) I read a book where it was mentioned that
>> one could define categories without (explicit) objects in the sense that
>> objects are mimicked by identity morphisms. Unfortunately, I can not
>> reconstruct what book it was.
>>
>> I know how this works. I would, however, like to have a reference.
>>
>> Best
>>
>> Uwe Wolter
>>

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Re: Category without objects

[Uwe Egbert Wolter]

Many thanks for all the immediate replies and all the interesting
information.

Finally, I could also reconstruct today where I have seen the
arrows-only definition around 30 years ago. There is a four page
introduction into categories in the first chapter of P.M. Cohn's
"Universal Algebra". He outlines that one could do so and gives a
corresponding exercise.

Best

Uwe

On 2015-03-06 00:45, Peter LeFanu Lumsdaine wrote:
> We actually had a post-seminar reference-hunt on this in Stockholm
> quite recently, and found that the arrows-only definition goes right
> back to Mac Lane 1948, ???Groups, Categories, and Duality???:
> http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079106/pdf/pnas01707-0037.pdf
>
>
> This cites two earlier papers only along with the definition (Mac Lane
> 1942 and Eilenberg???Mac Lane 1945 ??? the first two papers to mention
> categories, right?), but both of those used the objects-and-arrows
> formulation.  So it seems that the two-sorted formulation was
> considered right from the start, and the arrows-only version either
> from the start or very soon afterwards.
>
> Of course, the original question has already been well answered, but I
> guess the extra history may be of interest to others as well.
>
> Best,
> ???Peter.
>

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Re: Category without objects

[Ronnie Brown]

I remember Henry Whitehead said that he was very impressed by the axioms
for a category in the Eilenberg-Mac Lane paper.

A curiosity about the definition is that groupoids were defined by
Brandy in 1926, and this definition was used by the Chicago school of
algebra and applied to ring theory.  Bill Cockcroft told me that the
groupoid notion was an influence.  In 1985 I asked Eilenberg about this,
and said no, since if it had been, they would have used it as an
example! I forgot to ask Mac Lane!

Ronnie Brown

On 06/03/2015 14:42, Uwe Egbert Wolter wrote:
> Many thanks for all the immediate replies and all the interesting
> information.
>
> Finally, I could also reconstruct today where I have seen the
> arrows-only definition around 30 years ago. There is a four page
> introduction into categories in the first chapter of P.M. Cohn's
> "Universal Algebra". He outlines that one could do so and gives a
> corresponding exercise.
>
> Best
>
> Uwe
>

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Re: Category without objects

[Eduardo J. Dubuc]

It seems to me:

Eheresmann arrived to the concept of categories as a generalization of
groupoids, and he was dealing with small internal categories (the set of
arrows (or elements) were differential manifolds etc). This explains why
he dismissed objects in his later treatment of abstract categories.

Eilemberg and MacLane arrived to the concept of categories as an
abstraction of the large concrete categories of sets with structure and
functions which were considered to be morphisms for the structure.
Objects were essential in this approach.

That the insight of E. M. to do not dismiss objects in the abstract
setting was  wonderful is that to-day we can not conceive groupoids
without objects.


On 7/3/15 11:36, Ronnie Brown wrote:
> I remember Henry Whitehead said that he was very impressed by the axioms
> for a category in the Eilenberg-Mac Lane paper.
>
> A curiosity about the definition is that groupoids were defined by
> Brandy in 1926, and this definition was used by the Chicago school of
> algebra and applied to ring theory.  Bill Cockcroft told me that the
> groupoid notion was an influence.  In 1985 I asked Eilenberg about this,
> and said no, since if it had been, they would have used it as an
> example! I forgot to ask Mac Lane!
>
> Ronnie Brown
>

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Re: Category without objects

[F. William Lawvere]

It is difficult to understand "without objects"  without any definition of "object". Remember that , already before the 21st century, modern mathematics had begun to overcome medieval metaphysics. In fact ,in the late 1950s, Alexander Grothendieck had made explicit the definition of "subobject", which seems relevant here, as does his powerful legacy of relativization in several senses. Now we understand that a category C in a category U  is a truncated simplicial object C0->...->C3 satisfying certain limit conditions. We are free to call C0 'objects" and C1 "maps" and since C0->C1 is a subobject of C1, we could also say that objects "are" maps,but "mimicked by" seems \x10unnecessary (as well as undefined).
(Recall that it is actions of such a C in a topos U that form the topos enveloping, as a full subtopos of sheaves, the typical U-topos E->U).
To give a category "with objects" i\x10n a serious sense would seem to be giving MORE than ju\x10st a category, for example an interpretation as structuresC-> B^A, the (functor category also emphasized by Grothendieck)of structures of shape A in background B. (Where perhaps B is equipped with an internal embedding in U itself)
The case of no structure and featureless background ( which seems to be the   default setting of modern mathematics despite the preference of MacLane'sdear teacher for a vonNeuman-like setting) means in particular that the C0 in a category there consists of "lauter Einsen" in the sense of Cantor.
Those featureless elements X of C0 do obtain a structure by virtue of C1,C2 because taking the latter into account we can see the inside of  X as the "comma" category C/X involving (not only the subobjects of X and their inclusions, but also singular figures and reparameterizations) as very extensively utilized by Grothendieck .
Bill
> Date: Sat, 7 Mar 2015 14:36:36 +0000
> From: ronnie.profbrown@btinternet.com
> To: Uwe.Wolter@ii.uib.no; p.l.lumsdaine@gmail.com
> CC: categories@mta.ca
> Subject: categories: Re: Category without objects
> 
> I remember Henry Whitehead said that he was very impressed by the axioms
> for a category in the Eilenberg-Mac Lane paper.
> 
> A curiosity about the definition is that groupoids were defined by
> Brandy in 1926, and this definition was used by the Chicago school of
> algebra and applied to ring theory.  Bill Cockcroft told me that the
> groupoid notion was an influence.  In 1985 I asked Eilenberg about this,
> and said no, since if it had been, they would have used it as an
> example! I forgot to ask Mac Lane!
> 
> Ronnie Brown
> 

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Re: Category without objects

[Ronnie Brown]

The notion of groupoids having structure in (at least, pace Bill!) 2
dimensions, namely 0 and 1,  was important for me to explain how one
could get a van Kampen type theorem for the fundamental groupoid on a
set of base points which would then determine completely individual
fundamental groups, hidden in the middle. This seemed to be against the
experience in algebraic topology where invariants in adjacent dimensions
are determined by exact sequences which do not determine the result you
want completely. I was well aware of the latter feature since
determining extensions in some track exact sequences by fibration
methods  was the topic of my 1961 DPhil thesis, under Michael Barratt.

It seemed logical that for a theorem that determined completely
information on 1-type you needed an invariant, in this case a groupoid,
with information in dimensions 0 and 1, to take account of all the
1-type information in the pieces glued together.

This result on groupoids  led to the idea of having invariants with
structure in dimensions 0, ..., n. Though it took 9 years to understand,
it is perhaps not surprisingly that the invariants had to be not of a
bare space but of something which also had structure in dimensions
0,...,n; and so one can use filtered spaces, (with Philip Higgins) and
later, with Loday, n-cubes of spaces.

I mention here that some confusion arises in standard algebraic topology
where people often talk of "the fundamental group of a space";  of
course you can talk only of "the fundamental group of a space with base
point", which is a special case of a space with structure.

This leads me to share with you  some comments of Alexander Grothendieck
on base points, and I hope you enjoy the elegance of the language.   Of
course I entirely agree with his criticism of the limitations of the
concepts considered, but it is also true that even 2-fold groupoids in
general are not well understood, whereas cat^n-groups, i.e. (n+1)-fold
groupoids in which one structure is a group) have this amazing
equivalent format, discovered later by Ellis and Steiner, of crossed
n-cubes of groups.  The more general concept has not, I think, been used
for any specific gluing theorems, but has been studied as a model of
homotopy types Blanc and Paoli).

Comments from Alexander Grothendieck, 12 April, 1983

What you write about Loday's n-Cat-groups makes sense for me and is
quite interesting indeed. When you say they capture truncated homotopy
types, I guess you mean "pointed 0-connected (truncated) homotopy
types". This qualification seems to me an important one - while they are
presumably quite adequate for dealing with a number of situations, it is
kind of clear to me they are not for a "passe partout" description of
homotopy types - both the choice of a base point, and the
0-connectedness assumption, however innocuous they may seem at first
sight, seem to me of a very essential nature. To make an analogy, it
would be just impossible to work at ease with algebraic varieties, say,
if sticking from the outset (as had been customary for a long time) to
varieties which are supposed to be connected. Fixing one point, in this
respect (which wouldn't have occurred in the context of algebraic
geometry) looks still worse, as far as limiting elbow-freedom goes!
Also, expressing a pointed 0-connected homotopy type in terms of a group
object mimicking the loop space (which isn't a group object strictly
speaking), or conversely, interpreting the group object in terms of a
pointed "classifying space", is a very inspiring magic indeed - what
makes it so inspiring it that it relates objects which are definitively
of a very different nature - let's say, "spaces" and "spaces with group
law". The magic shouldn't make us forget though in the end that the
objects thus related are of different nature, and cannot be confused
without causing serious trouble.

(This is taken with thanks from the full edited correspondence available
from
http://webusers.imj-prg.fr/~georges.maltsiniotis/ps.html)

Ronnie









On 08/03/2015 19:53, F. William Lawvere wrote:
> It is difficult to understand "without objects"  without any definition
> of "object". Remember that , already before the 21st century, modern
> mathematics had begun to overcome medieval metaphysics. In fact ,
> in the late 1950s, Alexander Grothendieck had made explicit the definition
>  of "subobject", which seems relevant here, as does his powerful
> legacy of
> relativization in several senses. Now we understand that a category C
> in a category U is a truncated simplicial object C0->...->C3
> satisfying certain
> limit conditions. We are free to call C0 'objects" and C1 "maps" and
> since
> C0->C1 is a subobject of C1, we could also say that objects "are" maps,
> but "mimicked by" seems \x10unnecessary (as well as undefined).
>
> (Recall that it is actions of such a C in a topos U that form the topos
> enveloping, as a full subtopos of sheaves, the typical U-topos E->U).
>
> To give a category "with objects" i\x10n a serious sense would seem to be
> giving MORE than ju\x10st a category, for example an interpretation as
> structures
> C-> B^A, the (functor category also emphasized by Grothendieck)
> of structures of shape A in background B. (Where perhaps B is equipped
> with
> an internal embedding in U itself)
>
> The case of no structure and featureless background ( which seems to
> be the
>  default setting of modern mathematics despite the preference of MacLane's
> dear teacher for a vonNeuman-like setting) means in particular that
> the C0
> in a category there consists of "lauter Einsen" in the sense of Cantor.
>
> Those featureless elements X of C0 do obtain a structure by virtue
> of C1,C2
> because taking the latter into account we can see the inside of  X as
> the "comma" category
> C/X involving (not only the subobjects of X and their inclusions, but
> also singular
> figures and reparameterizations) as very extensively utilized by
> Grothendieck .
>
> Bill
>

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Re: Category without objects

[Vaughan Pratt]

On 08/03/2015 19:53, F. William Lawvere wrote:
> It is difficult to understand "without objects"  without any definition
> of "object".

One could raise an analogous objection to the notion of a "reflexive
graph without vertices", defined as an M-set for the 3-element monoid M
consisting of the monotone endomorphisms of the poset 0 < 1.  While no
mention is made of vertices in this definition, an equivalent notion
arises in a canonical way by taking the Karoubi envelope of M, yielding
a notion of "vertex".  "Without vertices" then just means economizing by
skipping the step of taking the envelope.

Quine wrote "Word and Object".  Reasoning analogously as above, what a
category theorist would call an object, Quine would call a word.

Vaughan Pratt


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Re: Category without objects

[Tadeusz Litak]

On 11/03/15 05:20, Vaughan Pratt wrote:
>
>
> Quine wrote "Word and Object".  Reasoning analogously as above, what a
> category theorist would call an object, Quine would call a word.


There is actually a (perhaps) more direct reference to Quine in the context of this discussion.

In the entry "Mathematosis" of his "Quiddities: An Intermittently Philosophical Dictionary", he wrote:

> There has been a tendency of late to sacrifice simplicity at the
> altar of model theory. For instance we find a group defined as
> an ordered pair (A, f) where A is a class and f is a FUNCTION,
> q.v., whose arguments and values comprise A and fulfill certain
> axioms that I shall not pause over. This dragging in of a class
> A, and therewith of an ordered pair, is a gratuitous conformity
> to model-theoretic fashion; the function f is enough by itself,
> since A is definable in terms of f as the class of its arguments
> and values.

One can see here an analogy to the redundancy of objects in category theory.

However, the entry "Function" in the same book makes it clear that Quine thinks of functions in entirely set-theoretic
terms: as collections of ordered pairs.

t.


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Re: Category without objects

[Eduardo J. Dubuc]

Redundancy is not the same that lack of simplicity, in fact, most of the
time is the other way around. Unnatural elimination of redundancy makes
things very difficult to understand.

On 11/3/15 21:42, Tadeusz Litak wrote:
>
> On 11/03/15 05:20, Vaughan Pratt wrote:
>>
>>
>> Quine wrote "Word and Object".  Reasoning analogously as above, what a
>> category theorist would call an object, Quine would call a word.
>
>
> There is actually a (perhaps) more direct reference to Quine in the
> context of this discussion.
>
> In the entry "Mathematosis" of his "Quiddities: An Intermittently
> Philosophical Dictionary", he wrote:
>
>> There has been a tendency of late to sacrifice simplicity at the
>> altar of model theory. For instance we find a group defined as
>> an ordered pair (A, f) where A is a class and f is a FUNCTION,
>> q.v., whose arguments and values comprise A and fulfill certain
>> axioms that I shall not pause over. This dragging in of a class
>> A, and therewith of an ordered pair, is a gratuitous conformity
>> to model-theoretic fashion; the function f is enough by itself,
>> since A is definable in terms of f as the class of its arguments
>> and values.
>
> One can see here an analogy to the redundancy of objects in category
> theory.
>
> However, the entry "Function" in the same book makes it clear that Quine
> thinks of functions in entirely set-theoretic
> terms: as collections of ordered pairs.
>
> t.
>



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