From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6491 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Fibrations in a 2-Category Date: Sun, 23 Jan 2011 12:17:45 -0800 Message-ID: References: <43697659-DDA8-44AC-AD7B-077BE1EC3665@wanadoo.fr> 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 1295877438 3723 80.91.229.12 (24 Jan 2011 13:57:18 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 24 Jan 2011 13:57:18 +0000 (UTC) Cc: Categories To: JeanBenabou Original-X-From: majordomo@mlist.mta.ca Mon Jan 24 14:57:05 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PhMug-000575-2P for gsmc-categories@m.gmane.org; Mon, 24 Jan 2011 14:56:58 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:33257) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PhMuY-0003j0-6t; Mon, 24 Jan 2011 09:56:50 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PhMuL-00010t-O8 for categories-list@mlist.mta.ca; Mon, 24 Jan 2011 09:56:38 -0400 In-Reply-To: <43697659-DDA8-44AC-AD7B-077BE1EC3665@wanadoo.fr> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6491 Archived-At: Dear Jean, As you surmise, anafunctors are more or less the same as representable distributors. I apologize for not pointing that out to begin with, and I'm glad you brought it up. Specifically, the bicategory Cat_ana is equivalent to the bicategory Rep of categories and representable distributors, in the way that you sketched. So if by > 3) Work with distributors. you mean to (among other things) regard Rep as a replacement for Cat, then I don't really view that approach as very different from my (1). In particular, it still resolves the issues you raised about internal fibrations: ordinary fibrations should equally well give rise to internal (Street) fibrations, in the representable sense, in Rep. I have no objection to using the language of representable distributors instead of anafunctors. And as you point out, distributors have many other uses -- you don't need to convince me that distributors are useful and important! (I feel obliged to point out that anafunctors can be enriched just as well as distributors can, but distributors can certainly be used for many things that anafunctors cannot.) My experience in mathematics (which is admittedly much shorter than yours) is that usually, when a given concept has two equivalent representations, it is useful to know about both of them and how to go back and forth between them, since each will have its own advantages and be preferred by different people. Thus, my answer to your questions "which do you prefer?" is that I would prefer to have both at my disposal and be able to use either one. Without prejudicing anafunctors over representable distributors, I could rephrase the point I intended to make in my previous email as: by replacing Cat with a bicategory whose morphisms are "functors defined up to isomorphism," we can recover many facts about category theory which classically require AC. Whether that bicategory is Cat_ana or Rep is immaterial to the question of "how to do category theory without AC." Having said that, I suppose I should nevertheless also say a little bit about why I like to have anafunctors at my disposal *in addition to* distributors. One reason is that sometimes it requires a little contortion to put something in the form of a representable distributor. For instance, if a category A has binary products, then there is obviously a product-assigning representable distributor P: A x A -/-> A, defined by P(a,(a_1,a_2)) = Hom_A(a,a_1) x Hom_A(a,a_2) But if A has binary coproducts, then in order to define a coproduct-assigning representable distributor C: A x A -/-> A, then (as far as I know) one needs to say something like C(a,(a_1,a_2)) = the set of triples (a_3,p_1,p_2,f) where p_i: a_i --> a_3 are the injections into a coproduct and f: a --> a_3, modulo an equivalence relation (a_3,p_1,p_2,f) ~ (a_3',p_1',p_2',f') if there exists a (necessarily unique iso)morphism g: a_3 --> a_3' commuting with all the structure maps. Of course, C is more easily defined as a corepresentable distributor. But if you want to define a functor that involves both limits and colimits, like (a,b,c) |--> a x (b + c), then it is not "naturally" represented as either a representable or a corepresentable one -- although it always *can* be so represented, essentially by passing across the equivalence Cat_ana = Rep that you sketched. By contrast, with anafunctors, all of these functors can be represented "naturally" in analogous ways. In the first case, we consider the span AxA <-- P --> A, where P is the category of binary product diagrams in A. In the second case, we consider the span AxA <-- C --> A, where C is the category of binary coproduct diagrams. And in the third case, we consider the span AxAxA <-- D --> A, where D is the category of binary coproduct diagrams together with a product diagram one of whose factors is the vertex of the coproduct diagram. Note that in each case, the middle category is the category of models of a "sketch" in A. Personally, I find the anafunctor way of representing such "functors" a bit cleaner, and sometimes easier to work with. Roughly speaking, I would say that anafunctors are formulated exactly in order to describe "functors defined up to isomorphism." Representable distributors, by contrast, can be described as "functors valued in representable presheaves." "Objects defined up to isomorphism" and "representable presheaves" are *formally* equivalent (without invoking AC), but not every "naturally occurring" object-defined-up-to-isomorphism is "given in nature" by the presheaf it represents. Some are given by the copresheaf they corepresent; others aren't given directly in either of those ways. (But you can guess from the fact that there are lots of quotation marks in this paragraph, none of this is particularly formal. In particular, I don't present it as an argument intended to convince you to prefer anafunctors over representable distributors, but rather as a reason why I or someone else might like to think about anafunctors *in addition to* representable distributors. If you like, you can think of an anafunctor as a particularly convenient "presentation" of a representable distributor.) I also find it illuminating that amongst all the "classical" facts about category theory that naively become false without AC, if you take the single statement "a fully faithful and essentially surjective functor is an equivalence" and "force" it to be true in a universal way, then you end up with a world in which all (or at least most) of the *other* "classical" facts *also* become true again. As far as I know, that fact is easiest to express using anafunctors and calculus of fractions -- although I would be very interested to see a direct proof that Rep is the result of formally inverting the weak equivalences in Cat (i.e. a proof that doesn't essentially go by proving that Rep is equivalent to Cat_ana). In regards to your other questions: > Ordinary categories of fractions are > very complicated, unless you have a calculus of right (or left) fractions. > Is there, precisely defined, and without neglecting the coherence of > canonical isomorphisms, such a "calculus" defined. Does it apply to the > "simple thing" of anafunctors. I wouldn't use the phrase "category of fractions" unless there is a calculus of fractions; otherwise I would probably say "homotopy category" or "localization". Calculi of fractions for bicategories, and the construction of anafunctors thereby, can be found in: Pronk, D. "Etendues and stacks as bicategories of fractions", Compositio Math. 1996 Roberts, D. "Internal categories, anafunctors and localisations", arXiv:1101.2363 > what is the category of anafunctors with domain the terminal category 1 and codomain a category C? It's sometimes called the category of "ana-objects" or "cliques" in C. Its objects are diagrams in C whose domain is a contractible category. In the setting of internal categories in a topos, it is a stackification of C. (Does that count as a precise mathematical application?) Best, Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]