From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6224 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: Not invariant but good Date: Sat, 25 Sep 2010 00:01:57 -0400 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1285451633 15593 80.91.229.12 (25 Sep 2010 21:53:53 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 25 Sep 2010 21:53:53 +0000 (UTC) Cc: "Eduardo J. Dubuc" , , , To: "Categories list" Original-X-From: majordomo@mlist.mta.ca Sat Sep 25 23:53:51 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Ozcgn-00021C-4g for gsmc-categories@m.gmane.org; Sat, 25 Sep 2010 23:53:49 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39068) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Ozcfq-0004kt-97; Sat, 25 Sep 2010 18:52:50 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ozcfn-0000NW-Cl for categories-list@mlist.mta.ca; Sat, 25 Sep 2010 18:52:47 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6224 Archived-At: Dear all, Very briefly. Many good things in mathematics are depending on the choice=20 of a representation which is not invariant under equivalences, or under isomorphisms. Modern geometry would not exists=20 without coordinate systems. This is true also of algebra and category theory. Algebraic structures are often described by=20 generators and relations. Homological algebra is using non-canonical projective or injective resolutions. Choosing a base point may help=20 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:=20 the notion of homotopy pullback square in a Quillen model category is=20 invariant under weak equivalences, but its definition depends on=20 the notion of pullback square which is not invariant under weak = equivalences! Part of the art of mathematics is in constructing invariant notions=20 from non-invariant ones. We should recognize the usefulness and importance of the latter. Please, let us not call them "evil"! Best, Andr=E9 PS: We should reserve the word "evil" to name things that really are. ------_=_NextPart_001_01CB5C66.6607A3FA Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Not invariant but good

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,
Andr=E9

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

------_=_NextPart_001_01CB5C66.6607A3FA-- [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6232 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Sun, 26 Sep 2010 11:29:07 +0800 Message-ID: References: Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: dough.gmane.org 1285542162 20170 80.91.229.12 (26 Sep 2010 23:02:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 26 Sep 2010 23:02:42 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Mon Sep 27 01:02:40 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P00Ev-0003Sl-GM for gsmc-categories@m.gmane.org; Mon, 27 Sep 2010 01:02:37 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:48684) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P00EC-0002Gi-8w; Sun, 26 Sep 2010 20:01:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P00E8-0004ls-Jb for categories-list@mlist.mta.ca; Sun, 26 Sep 2010 20:01:48 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6232 Archived-At: 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6236 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Sun, 26 Sep 2010 23:54:44 -0300 (ADT) Message-ID: References: Reply-To: selinger@mathstat.dal.ca (Peter Selinger) NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1285635580 22482 80.91.229.12 (28 Sep 2010 00:59:40 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 28 Sep 2010 00:59:40 +0000 (UTC) Cc: categories@mta.ca (categories) To: baez@math.ucr.edu Original-X-From: majordomo@mlist.mta.ca Tue Sep 28 02:59:39 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P0OXe-0001LP-Td for gsmc-categories@m.gmane.org; Tue, 28 Sep 2010 02:59:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51871) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P0OWo-0005jY-Jn; Mon, 27 Sep 2010 21:58:42 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P0OWj-0004x2-OT for categories-list@mlist.mta.ca; Mon, 27 Sep 2010 21:58:37 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6236 Archived-At: 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. > ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6238 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Mon, 27 Sep 2010 13:36:29 +0800 Message-ID: Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: dough.gmane.org 1285635707 23062 80.91.229.12 (28 Sep 2010 01:01:47 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 28 Sep 2010 01:01:47 +0000 (UTC) To: categories , Peter Selinger Original-X-From: majordomo@mlist.mta.ca Tue Sep 28 03:01:44 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P0OZj-0001ts-9Y for gsmc-categories@m.gmane.org; Tue, 28 Sep 2010 03:01:43 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42744) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P0OYY-0005MC-0a; Mon, 27 Sep 2010 22:00:30 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P0OYQ-0004zR-E0 for categories-list@mlist.mta.ca; Mon, 27 Sep 2010 22:00:22 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6238 Archived-At: 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6250 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Tue, 28 Sep 2010 16:11:06 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1285802089 7812 80.91.229.12 (29 Sep 2010 23:14:49 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 29 Sep 2010 23:14:49 +0000 (UTC) Cc: categories To: John Baez Original-X-From: majordomo@mlist.mta.ca Thu Sep 30 01:14:47 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P15rK-0003b6-PB for gsmc-categories@m.gmane.org; Thu, 30 Sep 2010 01:14:46 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:40484) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P15qY-0000UO-QN; Wed, 29 Sep 2010 20:13:58 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P15qV-0001UJ-Rm for categories-list@mlist.mta.ca; Wed, 29 Sep 2010 20:13:56 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6250 Archived-At: On Sun, Sep 26, 2010 at 10:36 PM, John Baez 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 alon= g > 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, "=C9quivalence naturelle et formules logiques en th=E9orie des cat=E9gories", 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6258 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Wed, 29 Sep 2010 14:25:37 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1285802736 9985 80.91.229.12 (29 Sep 2010 23:25:36 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 29 Sep 2010 23:25:36 +0000 (UTC) Cc: categories@mta.ca To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Thu Sep 30 01:25:35 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P161k-0006LW-Oq for gsmc-categories@m.gmane.org; Thu, 30 Sep 2010 01:25:33 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46647) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P160p-0003Zj-LX; Wed, 29 Sep 2010 20:24:36 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P160j-0001l3-I7 for categories-list@mlist.mta.ca; Wed, 29 Sep 2010 20:24:29 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6258 Archived-At: On Tue, Sep 28, 2010 at 3:18 AM, Thomas Streicher 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6261 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Thu, 30 Sep 2010 13:07:29 +1000 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1285898412 7295 80.91.229.12 (1 Oct 2010 02:00:12 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 1 Oct 2010 02:00:12 +0000 (UTC) Cc: Thomas Streicher , categories@mta.ca To: Michael Shulman Original-X-From: majordomo@mlist.mta.ca Fri Oct 01 04:00:10 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P1Uuv-0006bu-8M for gsmc-categories@m.gmane.org; Fri, 01 Oct 2010 04:00:09 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52501) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P1Uu9-0005cy-2B; Thu, 30 Sep 2010 22:59:21 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P1Uu6-0001ea-BQ for categories-list@mlist.mta.ca; Thu, 30 Sep 2010 22:59:18 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6261 Archived-At: > 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. =A0(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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6264 Path: news.gmane.org!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Thu, 30 Sep 2010 13:11:21 +0200 Message-ID: <20100930111121.GA25969@mathematik.tu-darmstadt.de> References: Reply-To: Thomas Streicher NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1285898773 8408 80.91.229.12 (1 Oct 2010 02:06:13 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 1 Oct 2010 02:06:13 +0000 (UTC) Cc: categories@mta.ca To: Michael Shulman Original-X-From: majordomo@mlist.mta.ca Fri Oct 01 04:06:09 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P1V0i-0008Ca-0x for gsmc-categories@m.gmane.org; Fri, 01 Oct 2010 04:06:08 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:36469) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P1Uzt-00068H-Sc; Thu, 30 Sep 2010 23:05:17 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P1Uzq-0001ly-S6 for categories-list@mlist.mta.ca; Thu, 30 Sep 2010 23:05:15 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6264 Archived-At: 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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6265 Path: news.gmane.org!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Thu, 30 Sep 2010 13:34:21 +0200 Message-ID: <20100930113421.GA23528@mathematik.tu-darmstadt.de> References: Reply-To: Thomas Streicher NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1285898847 8618 80.91.229.12 (1 Oct 2010 02:07:27 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 1 Oct 2010 02:07:27 +0000 (UTC) Cc: categories@mta.ca To: Michael Shulman Original-X-From: majordomo@mlist.mta.ca Fri Oct 01 04:07:25 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P1V1w-0008U2-2y for gsmc-categories@m.gmane.org; Fri, 01 Oct 2010 04:07:24 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:38580) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P1V1A-00087x-K8; Thu, 30 Sep 2010 23:06:36 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P1V16-0001o9-LQ for categories-list@mlist.mta.ca; Thu, 30 Sep 2010 23:06:32 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6265 Archived-At: > 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6268 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Thu, 30 Sep 2010 12:39:00 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1285899123 9580 80.91.229.12 (1 Oct 2010 02:12:03 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 1 Oct 2010 02:12:03 +0000 (UTC) Cc: categories@mta.ca To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Fri Oct 01 04:12:01 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P1V6L-00019s-Uk for gsmc-categories@m.gmane.org; Fri, 01 Oct 2010 04:11:58 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44445) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P1V5W-0006Y3-OY; Thu, 30 Sep 2010 23:11:06 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P1V5R-0001v1-7K for categories-list@mlist.mta.ca; Thu, 30 Sep 2010 23:11:01 -0300 In-Reply-To: <20100930111121.GA25969@mathematik.tu-darmstadt.de> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6268 Archived-At: 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 abse= nce > 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=D7A 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6269 Path: news.gmane.org!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Fri, 1 Oct 2010 14:36:53 +0200 Message-ID: Reply-To: Thomas Streicher NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1286025156 29160 80.91.229.12 (2 Oct 2010 13:12:36 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 2 Oct 2010 13:12:36 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sat Oct 02 15:12:35 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P21tD-0000jE-0A for gsmc-categories@m.gmane.org; Sat, 02 Oct 2010 15:12:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53837) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P21s6-0003iG-6g; Sat, 02 Oct 2010 10:11:26 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P21s1-0006x4-GB for categories-list@mlist.mta.ca; Sat, 02 Oct 2010 10:11:21 -0300 Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6269 Archived-At: Dear Michael, > Does this clarify what I meant? Yes, indeed that has been very helpful.=20 > 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=D7A 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 pro= of of existence of products. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ] From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6279 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Sun, 3 Oct 2010 15:10:29 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1286194086 24763 80.91.229.12 (4 Oct 2010 12:08:06 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 4 Oct 2010 12:08:06 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Mon Oct 04 14:08:05 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P2jps-0008Cz-Rp for gsmc-categories@m.gmane.org; Mon, 04 Oct 2010 14:08:05 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58703) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P2jp4-0004uu-TP; Mon, 04 Oct 2010 09:07:14 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P2jp0-0001fC-5Y for categories-list@mlist.mta.ca; Mon, 04 Oct 2010 09:07:10 -0300 In-Reply-To: <20101001092434.GA9359@mathematik.tu-darmstadt.de> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6279 Archived-At: On Fri, Oct 1, 2010 at 2:24 AM, Thomas Streicher 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]