From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6499 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: Fibrations in a 2-Category Date: Sat, 29 Jan 2011 12:45:10 -0500 Message-ID: References: <20110122220701.C8B538626@mailscan1.ncs.mcgill.ca> Reply-To: Marta Bunge 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 1296427781 10494 80.91.229.12 (30 Jan 2011 22:49:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 30 Jan 2011 22:49:41 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Sun Jan 30 23:49:36 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 1Pjg5O-0004lv-Ig for gsmc-categories@m.gmane.org; Sun, 30 Jan 2011 23:49:34 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:44042) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Pjg4w-0004ng-9b; Sun, 30 Jan 2011 18:49:06 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Pjg4s-0000XQ-NL for categories-list@mlist.mta.ca; Sun, 30 Jan 2011 18:49:02 -0400 In-Reply-To: <20110122220701.C8B538626@mailscan1.ncs.mcgill.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6499 Archived-At: Dear Jean (and Mike)=2C Your guess/proof that anafunctors and representable distributors are equiva= lent notions=2C and by an equivalence which extends to=A0an equivalence of = categories between Rep(A=2CB) and Cat_ana(A=2CB)=2C is not only correct but= also fully discussed in: http://ncatlab.org/nlab/show/anafunctor In that article=2C it is furthermore pointed out that each version has its = advantages over the other=2C and that therefore both are of interest for ca= tegory theory in a topos S without AC=2C where they generalize ordinary fun= ctors.=A0But=2C even in the presence of AC=2C distributors (profunctors) ge= neralize ordinary functors=2C a fact that I have known for 45 years=2C wher= eas anafunctors do not. This ought to be pointed out by the authors of the = article mentioned above.=A0 In my thesis ("Categories of Set-Valued Functors"=2C University of Pennsylv= ania=2C 1966)=2C inspired by a monograph of Michel Andre ("Categories of Fu= nctors and Adjoint Functors"=2C Batelle Report=2C Geneve=2C 1964)=2C I disc= uss what you later called "distributors"=2C including their composition as = a generalized matrix product. It was Bill Lawvere who pointed out to me the= importance of =A0"distributors"=2C well before you introduced them. I gave= an expose of a portion of my thesis at the Oberwolfach meeting in 1966. My= main interest therein was the equivalence between the category of profunct= ors between two small categories and that of those functors between the cor= responding set-valued functor categories that have an adjoint or a coadjoin= t. In turn=2C this led to Morita equivalence theorems. The results are vali= d for an arbitrary (co)complete topos S but=2C even in the case of Set=2C which satisfie= s AC=2C this shows that profunctors generalize ordinary functors.=A0 With best regards to all=2C Marta =A0=A0 > To: mshulman@ucsd.edu=3B categories@mta.ca > From: jean.benabou@wanadoo.fr > Subject: categories: Re: Fibrations in a 2-Category > Date: Sat=2C 22 Jan 2011 11:25:55 +0100 >=20 > ANAFUNCTORS VERSUS DISTRIBUTORS >=20 > Dear Mike=2C >=20 > (I apologize for using in a few places capital letters=2C where =20 > normally I would have used italics=2C but html is not accepted in the =20 > Category List) >=20 > In your mail about fibrations in a 2-category=2C dated Jan.14=2C you say: >=20 > "One way to deal with the difficulty you mention is by using > "anafunctors=2C" which were introduced by Makkai precisely in order to > avoid the use of AC in category theory". >=20 > There is "another way"=2C which I prefer. It is using distributors=2C =20 > which do much more than merely "avoid the use of AC"=2C and apply to =20 > more general situations than the ones you consider. Let me first =20 > give a very simple definition: >=20 > Let M: A -/-> B be a distributor=2C identified with a functor A --> (B=20 > =B0=2C Set) =3D B^. > I say that M is "representable" iff for every object a of A the =20 > presheaf M(a) is. With AC=2C such an M is isomorphic to a functor F: A =20 > --> B=2C which is unique up to a unique isomorphism. But my definition =20 > doesn't need any reference to AC. > I shall denote by Rep(A=2CB)) the full subcategory of Dist(A=2CB) having = =20 > as objects the representable distributors. > "Corepresentable" distributors are defined by the canonical duality =20 > of Dist=2C and I denote by Corep(A=2CB) the corresponding category. >=20 > 1- In your example you say: >=20 > "let P --> 2 be a fibration=2C with fibers B and A. Then there is =20 > (without AC) an anafunctor A --> B=2C where the objects of F are the =20 > cartesian arrows of P over the nonidentity arrow of 2=2C and the =20 > projections assign to such an arrow its domain and codomain" >=20 > What I can say with distibutors is: >=20 > 1' - Let P --> 2 be an ARBITRARY functor with fibers B and A. Then =20 > there is=2C without AC=2C a canonical distributor A -/-> B associated to= =20 > this functor. Moreover the the functor is a fibration iff the =20 > associated distributor is representable=2C and a cofibration (I think =20 > you'd say "op-fibation") iff this distributor is corepresentable. .=20 > (again no AC). > Which statement do you prefer? >=20 > 2- A little bit further on you say: >=20 > "More generally=2C if Cat_ana denotes the bicategory of categories and > anafunctors=2C then from any fibration P --> C we can construct (without > AC) a pseudofunctor C^{op} --> Cat_ana." >=20 > With distibutors I can say: >=20 > 2' - Let F: P --> C be an ARBITRARY functor. From F=2C I can construct=2C= =20 > without AC=2C a normalized lax functor D(F) : C^(op) --> Dist . Then we = =20 > have=2C without AC: > (i) F is a Giraud functor (GIF) iff D(F) is a pseudo functor. > (ii) F is a prefibration iff for every map c of C the distributor D(F)=20 > (c) is representable > (iii) F is a fibration iff it satisfies (i) and (ii) > (Iv) F is a cofibration if it is a GIF and the D(F!(c)'s are =20 > corepresentable. >=20 > Which statement do you prefer ? > In (iv) I insist on the fact that it is the same D(F). Is there a =20 > notion of "ana-cofunctor"? > Note moreover that many other important properties of F can be =20 > characterized by very simple properties of D(F)=2C again without AC! >=20 > 3- You also say: >=20 > "An anafunctor is really a simple thing: a morphism in the bicategory > of fractions obtained from Cat by inverting the functors which are > fully faithful and essentially surjective". >=20 > Woaoo=2C you call this a simple thing! Ordinary categories of fractions = =20 > are very complicated=2C unless you have a calculus of right (or left) =20 > fractions. Is there=2C precisely defined=2C and without neglecting the =20 > coherence of canonical isomorphisms=2C such a "calculus" defined. Does =20 > it apply to the "simple thing" of anafunctors. >=20 > 4- In guise of conclusion you say: >=20 > In general=2C it seems to me that there are two overall approaches to > doing category theory without AC (including with internal categories > in a topos): >=20 > 1) Embrace anafunctors as "the right kind of morphism between > categories" in the absence of AC > 2) Insist on using only ordinary functors=2C so that we can work with > the strict 2-category Cat=2C which is simpler and stricter than Cat_ana. > "Personally=2C while there is nothing intrinsically wrong with (2)=2C I > think (1) gives a more satisfactory theory." >=20 > Sorry=2Cbut your approaches 1) and 2) are not the only ones. I opt for =20 > the following one: > 3) Work with distributors. >=20 > I still have to see precise mathematical applications anafunctors.. =20 > Do I have to mention applications of distributors? Do I have to point =20 > out that distributors can=2C not only be internalized=2C but also be =20 > "enriched"? >=20 > 5 - You are a very persuasive person Mike=2C but I'm not "buying" =20 > anafunctors=2C unless you give me convincing examples of what =20 > anafunctors can do=2C which distributors cannot do much better. > And if you want to generalize functors=2C without going all the way to =20 > arbitrary distributors=2C good candidates=2C for me=2C instead of =20 > anafunctors=2C are representable distibutors=2C which are very simple to= =20 > define rigorously and easy to work with. And of course don't use AC.=2C > I have a very strong guess that anafunctors are "the same thing" as =20 > representable distributors. I can even sketch a proof of my guess. > (i) You say that an anafunctor can be represented by a span A <-- F --=20 > > B where the left leg=2C say p=2C is full and faithful and surjective = =20 > on objects and the right leg=2C say q=2C is arbitrary functor. > In Dist you can take the composite: q p*: A -/-> F --> B=2C where p* is = =20 > the distributor right adjoint to the functor p. It is easy to see =20 > that his composite is representable. > Thus we get a map on objects=2C u: Cat_ana(A=2CB) --> Rep(A=2CB) > (ii) Conversely=2C suppose M: A -/-> B is representable. By 1' we get a = =20 > fibration P --> 2 thus by 1 an anafunctor A --> B . > Thus we get a map on objects=2C v: Rep(A=2CB) --> Cat_ana(A=2CB) . > It should be routine that u and v extend to functors U and V and give =20 > an equivalence of categories between Rep(A=2CB) and Cat_ana(A=2CB) > I didn't write a complete proof because=2C in order to do so=2C I'd have = =20 > to know a little more than what you wrote about the category Cat_ana=20 > (A=2CB) and I'm not ready to spend much time on the study of anafunctors. > Is my guess correct? If it isn't=2C where does my "sketch of proof" =20 > break down? > In particular what is the category of anafunctors with domain the =20 > terminal category 1 and codomain a category C? > I'd be very grateful if you could answer these questions=2C and some of = =20 > the ones I asked in 1) and 2). >=20 > I'm sure I didn't convince you. All I hope for=2C is that a few =20 > persons=2C after reading this mail=2C and your future answer of course=2C= =20 > will think twice before they abandon "old fashioned" Category Theory =20 > with its functors=2C AND DISTRIBUTORS=2C and rush to anafunctors=2C with = =20 > the belief that they are unavoidable foundations for the future AC-=20 > free "New Category Theory". >=20 > Regards=2C > Jean=2C >=20 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]