Re: Not invariant but good

Re: Not invariant but good

[Thomas Streicher]

Dear Michael,

> Does this clarify what I meant?

Yes, indeed that has been very helpful. 

> Perhaps "not well-behaved" was a poor choice of words since it implies
> a value judgement, but I think it is correct to say that in the
> absence of choice, category theory becomes very unfamiliar unless we
> replace functors with anafunctors.  For instance, if we insist on
> using only functors, then a category with finite products does not
> necessarily become a monoidal category, as there is no "product"
> functor from A×A to A.  Also, without choice the 2-category Cat using
> functors is not even a regular 2-category, let alone a 2-topos.  Since
> the defining characteristics of Set include that it is a well-pointed
> 1-topos, it seems unlikely to me that one will be able to get much of
> anywhere with a version of Cat that is not a 2-topos.

Well, but then we can work with categories with a chosen structure, e.g.
chosen products; that's what is recommended in the Elephant and it looks to
me as close to practice; it's very unlikely to have a nonconstructive proof
of existence of products.

Thomas



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Re: Not invariant but good

[Michael Shulman]

On Fri, Oct 1, 2010 at 2:24 AM, Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
> Well, but then we can work with categories with a chosen structure, e.g.
> chosen products; that's what is recommended in the Elephant and it looks to
> me as close to practice; it's very unlikely to have a nonconstructive proof
> of existence of products.

Unlikely, but it happens occasionally.  For instance, if the morphisms
of a category are equivalence classes, then a "construction" of
equalizers or pullbacks might require choosing representatives; this
actually happens at one point in the Elephant.  The "small complete
categories" in realizability topoi are also generally "weakly
complete," in the sense that "every small diagram has a limit" is true
in the internal logic, but not "strongly complete" in the sense that
there exist internal limit-assigning (non-ana) functors.  The property
of "strong completeness" is also not in general preserved or reflected
by weak equivalence functors.

One can of course develop a theory which distinguishes between weak
and strong equivalence and weak and strong completeness, but I think
it's reasonable to call it "unfamiliar" to most category theorists.
It feels to me like trying to do work constructively with topological
spaces and therefore having to talk about [0,1] being complete and
totally bounded but not compact, instead of realizing that when
working constructively, one should really replace topological spaces
with locales.  Just as in set theory, no axiom of choice is necessary
to define a function whose values are individually uniquely
determined, it seems to me that no axiom of choice should be necessary
in category theory to define a functor whose values are individually
uniquely determined up to unique isomorphism.  But obviously this is a
subjective judgement.

Mike


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Re: Not invariant but good

[Michael Shulman]

Dear Thomas,

I did not intend to "generalize" anything, only to restate the
definition.  There are at least four equivalent definitions of
fibration in a 2-category that I know of:

- the "representable" one which I gave,
- having an adjoint one-sided inverse to a certain morphism between
comma objects
- being an algebra for a certain monad on a slice 2-category
- the one in the Elephant

All the definitions have two versions: a "strict" one a la
Grothendieck (which only makes sense in a strict 2-category) and a
"weak" one a la Street (which makes sense in any bicategory).  All the
strict notions are equivalent to each other, and all the weak notions
are equivalent to each other.  The idea of the equivalence of the
first three is essentially contained in Richard's remark, while a
proof of the equivalence of the Elephant's version with the
representable one can be found in Peter Johnstone's article
"Fibrations and partial products in a 2-category."

Weak fibrations are closed under composition with equivalences, while
strict ones are not.  In Cat and probably other well-behaved
2-categories, being a weak fibration is the same as being the
composite of a strict fibration and an equivalence, and so it ought to
surprise no one that in such cases it is sufficient to consider strict
fibrations.  It's generally only in the bicategorical world that weak
fibrations become important.

All the definitions can also be described either as "properties"
(such-and-such thing exists) or as "structure" (equipped with
such-and-such thing), since in all cases the such-and-such is unique
up to unique isomorphism when it exists.  This is the situation also
referred to as "property-like structure."  But perhaps you were
originally referring instead to the structure required on the
2-category itself?  It's true that the latter three definitions
require existence of some limits in the 2-category, while the
representable version does not.

Finally, in Cat with choice, the strict notions are all equivalent to
the usual Grothendieck fibrations, while the weak ones are equivalent
to Street's.  (Street actually originally gave his definition in a
general bicategory, and only later specialized it to Cat.)

Does this clarify what I meant?

> Nothing against anafunctors but it is an exaggeration to say that in absence
> of choice the usual notion of functor is not well-behaved.

Perhaps "not well-behaved" was a poor choice of words since it implies
a value judgement, but I think it is correct to say that in the
absence of choice, category theory becomes very unfamiliar unless we
replace functors with anafunctors.  For instance, if we insist on
using only functors, then a category with finite products does not
necessarily become a monoidal category, as there is no "product"
functor from A×A to A.  Also, without choice the 2-category Cat using
functors is not even a regular 2-category, let alone a 2-topos.  Since
the defining characteristics of Set include that it is a well-pointed
1-topos, it seems unlikely to me that one will be able to get much of
anywhere with a version of Cat that is not a 2-topos.

Mike


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Re: Not invariant but good

[Thomas Streicher]

> However, in the absence of the axiom of choice, the naive definition
> of "functor" is not very well-behaved; it's better to use "anafunctors,"

Nothing against anafunctors but it is an exaggeration to say that in absence
of choice the usual notion of functor is not well-behaved. One just loses that
full and faithful and essential surjective entails equivalence. That's like
abandoning the notion of surjective map in case we can't split them all.

Thomas


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Re: Not invariant but good

[Thomas Streicher]

Dear Mike,

> I'm not sure what you mean here.  The notion of fibration in a
> 2-category can be defined as a property if you like: a morphism E -->
> B in a 2-category K is a fibration if all the induced functors K(X,E)
> --> K(X,B) are fibrations and all commutative squares induced by
> morphisms X --> X' are morphisms of fibrations (preserve cartesian
> arrows).  This is equivalent to giving some structure on E --> B, but
> that structure is unique up to unique isomorphism when it exists.

The definition you give entails that a "generalised" fibration is actually
a Grothendieck fibration (since Cat(1,E) is isomorphic to E). This way you
don't get closure under precomposition by equivalences. I also don't see why
cartesiannness of the functors induced by X --> X' should amount to a choice
of structure (cartesiannness of a functor is a property and not additional
structure). Moreover, this requirement is a property of Grothendieck fibrations
which can be established when having strong choice available.

I was rather alluding to the notion of fibration in 2-cats as can be found
in part B of the Elephant where a fibration is defined as a 1-arrow together
with additional structure.

The definition you gave above (which is not more general) is the obvious thing
to do in case K is not wellpointed enough (as Cat is).

Moreover, your definition of fibration in a 2-category is based on Grothendieck
fibrations and thus employs equality of 1-cells. Since 1-cells are objects of
a category it should be "evil" to speak about their equality. Not that it were
a problem to me...

Thomas

PS  Your definition of fibration in a 2-cat looks much simpler than what I
could find in the papers by Street and Johnstone. That's nothing to complain
about but where is it from? It seems to me the appropriate one when generalising
from Cat to more general 2-cats.


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Re: Not invariant but good

[Richard Garner]

> It is true that if one has a Grothendieck fibration between categories
> in the ordinary sense, then it only becomes an internal fibration in
> the naively defined 2-category Cat if we have the axiom of choice,
> since the latter amounts to saying that we can simultaneously choose
> cartesian liftings for any families of objects of E and morphisms of B
> we might want to pull them back along.  (I don't think any "global
> choice" is necessary, since the definition doesn't require us to make
> such a choice simultaneously for every possible family--only that for
> any particular family, we -could- make such a choice.)

Though we can always make such a simultaneous choice as soon as we
have it in one particular case. Given the Grothendieck fibration p:
E->B in Cat, and letting X denote the comma object (B,p), it is enough
to choose a lifting for the X-indexed family of morphisms of B
corresponding to the projection X --> B^2 at the X-indexed family of
objects of E corresponding to the projection X --> E; for then to give
a Y-indexed family of morphisms in B and a Y-indexed family of objects
of E over their codomains is to give a morphism Y -> X, and so the
chosen lifting for the latter induces a chosen lifting for the former.

Richard


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Re: Not invariant but good

[Michael Shulman]

On Tue, Sep 28, 2010 at 3:18 AM, Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
> The notion of Grothendieck fibration is a property
> of functors and not an additional structure. However, the notion of fibration
> in a(n abstract) 2-category can be formulated only postulating a certain kind
> of structure which, however, is unique up to canonical isomorphism. But this
> amounts to defining Grothendieck fibrations in terms of cleavages (which
> certainly are all canonically isomorphic). But choosing cleavages amounts
> to accepting very strong choice principles which is maybe no real problem
> but at least aesthetically moderately pleasing.

I'm not sure what you mean here.  The notion of fibration in a
2-category can be defined as a property if you like: a morphism E -->
B in a 2-category K is a fibration if all the induced functors K(X,E)
--> K(X,B) are fibrations and all commutative squares induced by
morphisms X --> X' are morphisms of fibrations (preserve cartesian
arrows).  This is equivalent to giving some structure on E --> B, but
that structure is unique up to unique isomorphism when it exists.

(One can argue that both ordinary fibrations and fibrations in a
2-category are actually "property-like structures," or "properties
that are not necessarily preserved by morphisms," since their
forgetful functors are pseudomonic but not full.  But that applies
equally to both.)

It is true that if one has a Grothendieck fibration between categories
in the ordinary sense, then it only becomes an internal fibration in
the naively defined 2-category Cat if we have the axiom of choice,
since the latter amounts to saying that we can simultaneously choose
cartesian liftings for any families of objects of E and morphisms of B
we might want to pull them back along.  (I don't think any "global
choice" is necessary, since the definition doesn't require us to make
such a choice simultaneously for every possible family--only that for
any particular family, we -could- make such a choice.)

However, in the absence of the axiom of choice, the naive definition
of "functor" is not very well-behaved; it's better to use
"anafunctors," or equivalently to invert the weak equivalences
(fully-faithful and essentially surjective functors) in the naively
defined 2-category Cat.  In the resulting bicategory, I think any
ordinary Grothendieck fibration will indeed be an internal fibration,
without any need for choice.  (Of course, "internal fibration" should
probably now be interpreted in the looser sense of Street, as
appropriate when working in a non-strict 2-category.)

Mike


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Re: Not invariant but good

[Michael Shulman]

On Sun, Sep 26, 2010 at 10:36 PM, John Baez <baez@math.ucr.edu> wrote:
> Can every property of categories that is invariant under equivalence be
> expressed in some language that doesn't include equations between objects?
> Or conversely? Or what precise conditions are needed to get theorems along
> these lines?

The converse is very easy, and it's something that I and others have
frequently mentioned in these discussions: if we write category theory
in dependent type theory with arrows dependent on their source and
target and no equality predicate on objects, then all formulas and
constructions in this language are easily proven to be invariant under
equivalence and isomorphism.

The forward direction is trickier, but essentially the answer is yes:
I believe theorems along these lines can be found in:

1) Peter Freyd, "Properties invariant within equivalence types of
categories", 1976

2) Georges Blanc, "Équivalence naturelle et formules logiques en
théorie des catégories",  1979

3) Michael Makkai, "First-order logic with dependent sorts, with
applications to category theory,"
http://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf

(and perhaps others that I'm unaware of).

Mike


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Re: Not invariant but good

[John Baez]

Peter wrote:


> In order to focus on the math and not on the terminology, let me today use
> the word "XXXX" instead of "evil".
>

Good.  I think you're secretly on my side, though, because you're using four
X's.

:-)


> I don't think the notion used in your examples is general enough. For
> example, fix some groupoid C, and consider the property of an object
> x: "x is isomorphic to exactly 3 objects of C".


What a fiendishly clever example!


> To me, this is clearly XXXX, because it is not invariant under equivalences
> of
>
C. Yet, according to the definition you used in this email, it extends
> to a functor C -> {F,T}, and therefore is non-XXXX.
>

True.

By the way, in case anyone out there forgets: "Extending to a functor C ->
{F,T}" was merely a pedantic way of saying "being a property of objects of C
that is invariant under isomorphisms" - a pedantic way that lets us easily
generalize this notion to "being a structure on objects of C that is
covariant under isomorphisms".  To generalize, we just replace {F,T} by Set.

It may not be clear to everybody why I like this pedantic approach, so I
should probably explain why.   A (-1)-category is a truth value, a
0-category is a set, and a 1-category is a category.  So, when we replace
{F,T} by Set, we are replacing the 0-category of (-1)-categories by the
1-category of 0-categories.   Since we are just increasing a certain
parameter by 1, it becomes easy to see how to continue this game
indefinitely.

For more details, try this:

http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=12

For a property P of objects x of a category C, "being invariant under
>
  isomorphisms of objects in C" is strictly weaker than "being invariant
> under equivalences of C".


Yes.

Here's my rejoinder:

I had been fixing a groupoid C and asking whether a property of objects of
that category was invariant under isomorphisms.  When you say "x is
isomorphic to exactly 3 objects of C", you are actually treating C not as
fixed but as variable. The more things we let vary, the more invariance
properties we can demand!

In particular, there's a 2-groupoid Cat_* where an object is a "pointed
category" (C,x), that is, a category C with chosen object x.

I can treat "being an object x that is isomorphic to exactly 3 objects in C"
as a property of pointed categories.  And, I would call this property
evil... whoops, I mean XXXX... because it determines a function

Ob(Cat_*)  -> {F,T}

that does not extend to a 2-functor

Cat_* -> {F,T}

Again, this is just a pedantic way of saying what you're saying.  I'm just
trying to point out that I can fit it into my philosophy.


> I tried to give a more general and precise 2-categorical definition on
> the categories list on January 3, 2010, but I am not sure I got it
> quite right.


I remember enjoying that post, but I'll need to reread it to remember what
you said.


> I think it was Mark Weber who also pointed out, around
> the same time, that one person's XXXX concept is another person's
> non-XXXX concept - in a different 2-category.
>

Very much so!  And you've also noticed here that sometimes a property of
objects in a fixed category arises from a property of pointed categories....
so that we can take either a 1-categorical or a 2-categorical approach to
the XXXXness of this property.


> So I don't think it is correct to identify the concept of XXXX with
> "having to talk about equality".
>

I agree.  I take it as a *rule of thumb* that when somebody writes down a
property of categories that involves equations between objects, they're
running the risk that this property is not invariant under equivalence of
categories.   But I don't know the general theorems that make this rule of
thumb precise.

Can every property of categories that is invariant under equivalence be
expressed in some language that doesn't include equations between objects?
Or conversely? Or what precise conditions are needed to get theorems along
these lines?

Best,
jb


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Re: Not invariant but good

[Peter Selinger]

Hi John,

thanks for trying to move the discussion away from terminology and
back to actual mathematical matters. In order to focus on the math and
not on the terminology, let me today use the word "XXXX" instead of
"evil".

I don't think the notion used in your examples is general enough. For
example, fix some groupoid C, and consider the property of an object
x: "x is isomorphic to exactly 3 objects of C".  To me, this is
clearly XXXX, because it is not invariant under equivalences of
C. Yet, according to the definition you used in this email, it extends
to a functor C -> {F,T}, and therefore is non-XXXX.

For a property P of objects x of a category C, "being invariant under
isomorphisms of objects in C" is strictly weaker than "being invariant
under equivalences of C". Proof: Clearly, any isomorphism of C can be
mapped to an identity of some category C' by some equivalence of
categories.  Therefore, any property that is invariant under
equivalences of categories is invariant under isomorphisms of
objects. The above example shows that the converse is not true.

I think for XXXXness of structures, a similar refinement is needed.
To me, the intuitive concept of XXXX for structures is "cannot be
transported along equivalences such that the equivalence becomes
structure preserving". To say it more explicitly: if category C has
the structure, and category C' is equivalent to C (as a category),
then C' can be equipped with a structure in such a way that the
equivalence (both directions) is structure preserving.

This is a fairly subtle concept, not least because it depends on the
precise 2-category in question (to fix what "equivalence" and
"structure preserving" means). For example, whether the structure of
being "strictly monoidal" is XXXX or not depends on what one means by
"structure preserving" (e.g., strict monoidal or strong monoidal
functors). The natural transformations need to be specified too, so
that one can define "structure preserving equivalence".

I tried to give a more general and precise 2-categorical definition on
the categories list on January 3, 2010, but I am not sure I got it
quite right. I think it was Mark Weber who also pointed out, around
the same time, that one person's XXXX concept is another person's
non-XXXX concept - in a different 2-category.

The fact that being XXXX depends on an ambient 2-category means that
it is not a moral judgment, and people should not be offended by
it. Some perfectly useful things can be XXXX sometimes, and some
perfectly useless things can be non-XXXX. For example, even a
not-very-natural property like "there are exactly 3 objects isomorphic
to x" can be non-XXXX when viewed in the right 2-category. For
example, this is the case in the 2-category of categories, functors,
and identity natural transformations.

Last comment. Thomas Streicher brought up the example of a fibration
P: XX -> BB as a concept that was XXXX but very useful. But I don't
think this concept is actually XXXX. Certainly if one thinks of the
fibration as a *structure* on BB, then this transports very nicely
along equivalences. Namely, given any equivalence BB <--> BB', one can
find a fibration P' : XX' -> BB' which is equivalent, as a fibration,
to P. Right?

So I don't think it is correct to identify the concept of XXXX with
"having to talk about equality". Rather, it should be defined in some
2-categorical way. See also Mike Shulman's post from January 4, which
discussed this distinction in more depth.

-- Peter


John Baez wrote:
>
> Dear Andre -
>
>> Many good things in mathematics are depending on the choice
>> of a representation which is not invariant under equivalences,
>> or under isomorphisms. Modern geometry would not exists
>> without coordinate systems.
>
> I agree.  I think you're arguing against a position that nobody
> here has espoused.
>
> A coordinate system is a structure, not a property.  In my
> earlier email I said a *property* is evil if it's not invariant under
> equivalences.  But I'd say a *structure* is evil if it's not *covariant*
> under equivalences.  Coordinate systems are covariant under
> equivalences, so they're not evil.
>

...


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Re: Not invariant but good

[John Baez]

Dear Andre -

> Many good things in mathematics are depending on the choice
> of a representation which is not invariant under equivalences,
> or under isomorphisms. Modern geometry would not exists
> without coordinate systems.

I agree.  I think you're arguing against a position that nobody
here has espoused.

A coordinate system is a structure, not a property.  In my
earlier email I said a *property* is evil if it's not invariant under
equivalences.  But I'd say a *structure* is evil if it's not *covariant*
under equivalences.  Coordinate systems are covariant under
equivalences, so they're not evil.

Let me expand on this a bit, first for properties and then for
structures.

Say we have a groupoid C.  A (possibly evil) property of objects
in C is a map

F: Ob(C) -> {F,T}

where Ob(C) is the class of objects of C and {F,T} is the set of
truth values.  I say the property is non-evil if it extends to a
functor

F: C -> {F,T}

where now we regard {F,T} as a discrete groupoid.

For example, suppose C is the groupoid of vector spaces
over your favorite field.

A typical non-evil property is "being 5-dimensional".  A typical
evil property is "having the empty set as its origin".  (In ZF set
theory, we can take any vector space and ask whether
its zero element happens to be the empty set.)

I think most mathematicians would be happy to see a theorem
that begins

Theorem: If V is a vector space that is 5-dimensional...

but somewhat surprised to see a theorem that begins:

Theorem: If V is a vector space with the empty set as its origin...

We instinctively feel that any theorem of the second sort
could have been phrased better: how could it really matter
that the origin is the empty set?

Now, structures.  Again, let C be a groupoid.  A (possibly evil)
structure on objects in C is a map

F: Ob(C) -> Set

The idea is that for any object c in C, F(c) is the set of
structures that can be put on that object.  I say the structure
is non-evil if it extends to a functor

F: C -> Set

If C is the groupoid of vector spaces, a typical non-evil
structure is a basis: here F(c) is the set of bases of the
vector space c.   A typical evil structure would be "a basis,
if the underlying set of c is the real numbers, but two bases
otherwise."

Again, I think most mathematicians would feel happy to work
with the non-evil structure, but somewhat uncomfortable
working with the evil one.  That's the feeling that this concept
of "evil" is trying to formalize.

Note that a non-evil structure on finite sets is what you call a
"species".  You wisely avoided studying the evil ones.

Best,
jb


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Not invariant but good

[Joyal, André]

Dear all,

Very briefly.

Many good things in mathematics are depending on the choice 
of a representation which is not invariant under equivalences,
or under isomorphisms. Modern geometry would not exists 
without coordinate systems. This is true also of algebra
and category theory. Algebraic structures are often described by 
generators and relations. Homological algebra is using non-canonical
projective or injective resolutions. Choosing a base point may help 
computing the fundamental group of a topological space.
Choosing a triangulation may help computing the homology groups.
Invariant notions are often constructed from notions which are not.
For example, the Euler characteristic of a space
is best explaned by using a triangulation.

Another example from homotopy theory: 
the notion of homotopy pullback square in a Quillen model category is 
invariant under weak equivalences, but its definition depends on 
the notion of pullback square which is not invariant under weak equivalences!

Part of the art of mathematics is in constructing invariant notions 
from non-invariant ones. We should recognize the usefulness and
importance of the latter. Please, let us not call them "evil"!

Best,
André

PS: We should reserve the word "evil" to name things that really are.




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<P><FONT SIZE=2>Dear all,<BR>
<BR>
Very briefly.<BR>
<BR>
Many good things in mathematics are depending on the choice<BR>
of a representation which is not invariant under equivalences,<BR>
or under isomorphisms. Modern geometry would not exists<BR>
without coordinate systems. This is true also of algebra<BR>
and category theory. Algebraic structures are often described by<BR>
generators and relations. Homological algebra is using non-canonical<BR>
projective or injective resolutions. Choosing a base point may help<BR>
computing the fundamental group of a topological space.<BR>
Choosing a triangulation may help computing the homology groups.<BR>
Invariant notions are often constructed from notions which are not.<BR>
For example, the Euler characteristic of a space<BR>
is best explaned by using a triangulation.<BR>
<BR>
Another example from homotopy theory:<BR>
the notion of homotopy pullback square in a Quillen model category is<BR>
invariant under weak equivalences, but its definition depends on<BR>
the notion of pullback square which is not invariant under weak equivalences!<BR>
<BR>
Part of the art of mathematics is in constructing invariant notions<BR>
from non-invariant ones. We should recognize the usefulness and<BR>
importance of the latter. Please, let us not call them &quot;evil&quot;!<BR>
<BR>
Best,<BR>
André<BR>
<BR>
PS: We should reserve the word &quot;evil&quot; to name things that really are.<BR>
<BR>
<BR>
<BR>
<BR>
<BR>
</FONT>
</P>

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