July 2014 - Categories - categories.org.au

Partial functor
by David Leduc 17 Mar '15

17 Mar '15

Hi, A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: cleavages and choice
by Jean Bénabou 05 Aug '14

05 Aug '14

Dear Thomas, I am a bit surprised that you, of all people, should defend cleavages, i.e. indexed categories. As far as I remember, there not many of them in the notes you wrote on Fibered categories a la Benabou. I remind you that these notes were written not only after lectures I gave, but after long conversations we had, many times, in my flat, and also a few days you spent in my house in the south of France, where, for at least 10 hours everyday I explained in detail to you my work on fibered categories and corrected many mistakes you made in first drafts of that paper. Nevertheless, for the sake of the people on the category list,to whom this message is also addressed, I shall answer your questions an remarks. Let p: X --> S be a surjective group homomorphism. It is a fibration, and a cleavage is a section of p (in Set). This was explicitly noted by Grothendieck more than 50 years ago! Does such a p come equipped with a cleavage? Take for p the morphim : R --> R/Z of the reals on the circle. Suppose I were to teach periodic functions, i.e functions with domain R which factor through R/Z. Wouldn't it be ridiculous to use a section of p ? Which one by the way? Take the theorem : The composite of two fibrations is a fibration. Does it need cleavages, i.e. AC for classes, to be proved? Of course, if you are cleavage fan, as you seem to be, you can add that, given cleavages of p and q one gets an associated cleavage of pq. Let's look at an important example,namely categories S with pull backs, not with choice of pull backs mind you. This is a first order notion, saying that : for every cospan of S there exists a universal span making the obvious square commutative. Then, without AC, you can prove that the functor Codom: S^2 --> S is a fibration. And again, if you are cleavage happy, add that a cleavage of this fibration, if it exists, (I'm not assuming AC) is a choice of pullbacks in S. I could multiply the examples. But let's look at an important question. Suppose you prove an intrinsic result about fibrations, using cleavages, in principle you'd have to see what happens when you change cleavages. And don't wave your hands and tell me that, for formal 2-categorical reasons, the result is obvious. I'll believe you only when you write a precise metatheorem which covers ALL the cases. You are convinced, and I am convinced, and everybody is convinced that such a metatheorem is not necessary. But that is NOT A PROOF ! And why are you convinced? Because, even if you say the contrary, deep in your mind you KNOW that intrinsic properties of fibrations AND cartesian functors should not refer to cleavages. Let me insist on the fact that the mythic metatheorem should also cover cartesian functors F: X --> X' where you change the cleavages of both X and X' Of course the theorem I mentioned in my mail on pre foliations, applies to fibrations and gives new results in that case. But this theorem is true for F: X --> X' where X is a prefoliation hence, even with AC, has no cleavage, and X' is an arbitrary category over S, i.e. has even less cleavages than X. In order not to make this mail too long I have not, but I should have, mentioned internalization where cleavages are even more problematic. Best to all, Jean Le 30 juil. 2014 à 17:06, Thomas Streicher a écrit : > Dear Jean, > > of course, you are right when emphasizing that one need choice for > classes to endow an "anonymous" fibration with a cleavage. > But that applies also to catgeories with say binary products. One > needs choice for classes in order to choose a product cone for every > pair of objects. > In many instances, however, categories come together with a choice of > products and fibrations come together with a choice of a cleavage. > > For example Set comes with a choice of a cleavage. Fibrations arising > from internal categories are even split. Many constructions on fibrations > allow one to choose a cleavage given cleavages for the arguments. > Do you know of any construction on fibrations which is not "cleavage > preserving" in this sense? > > Of course, one should not require cartesian functors to preserve > cleavages just as one should not require functors to preserve chosen > products. > > Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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A brief survey of cartesian functors
by Jean Bénabou 01 Aug '14

01 Aug '14

Dear Ross, Dear all, In a recent mail I asked Ross if pseudo cartesian functors between pseudo fibrations had been studied. There are many generalizations of fibrations. Pseudo fibrations are only one of them. But there are also prefibrations, defined by Grothendieck, but almost never considered, and pre foliations, which I define here, which generalise greatly pre fibrations. For such pre foliatons, I define cartesian functors and show that they have striking properties, most of which are not known, even in the very special case of fibrations. I thought this brief survey might interest you, in case you decide to study seriously the properties of pseudo cartesian functors. Best regards to all, Jean 1) PRE FOLIATIONS AND FOLIATIONS. 1.1. Notations and first definitions. If P: X -> S is a functor, I denote by V(P), abbreviated by V, the set of vertical maps for P . For every object s of S I denote by X_s the fiber of X over s. In order to deal, not only with fibrations but also with pre fibrations, as defined by Grothendieck, and even with more general notions such as pre foliations and foliations that I shall define, I adopt Grothendieck's definition of cartesian maps, namely: A map k: y -> x of X is cartesian iff for every map f: z ->x such Pf = Pk there exits a unique vertical map v: z ->y such that f = kv. I denote by K(P) , abbreviated by K , the set of these maps. I call hyper cartesian the maps which in the english texts are called cartesian and I denote by H(P), abbreviated by H, the set of these maps. They will play very little role in this brief survey. 1.2. DEFINITION. A functor P: X --> S is a pre foliation iff every map f of X can be factored as f = kv with k in K and v in V. If moreover K is stable by composition, I say that P is a foliation. 1.3. Remarks. (a) pre foliations and foliations are first order notions and can be internalized. (b) with universes and AC Grothendieck showed that his construction worked also for lax functors into Cat and gave pre fibrations. But there is no reindexing, even lax, for pre foliations. (c) Even when K is stable by composition, in many examples H will be strictly contained in K. (d) Foliations need not even be Giraud fibrations (sometimes called Conduché fibrations). (e) There are many significant examples of (pre) foliations which are not (pre) fibrations, but I cannot give them in such a brief survey. 2) CARTESIAN FUNCTORS Let P: X --> S, P': X' --> S and F: X --> X' be functors such that P = P'F. For every object s of S ,I denote by F_s : X_s --> X'_s the functor induced by F on the fibers. I have a general definition of F being cartesian, without any assumption on P and P' and without any reference to cartesian maps, but it uses distributors in an essential manner. I shall not need it in this survey and shall give the definition only when P is a pre foliation, but without any assumption on P'. 2.1. DEFINITION. If P is a pre foliation and P'F = P, I say that F is cartesian iff it satisfies the following two conditions: (i) It preserves cartesian maps, i.e. k in K(P) => Fk in K(P'). (ii) For every f': y' --> F(x) in X' , with y' in X' and x in X, there exist f: y -->x in X, and v': y' --> F(y) in V(P') such that f' = F(f)v'. 2.2. Remarks: (a) Condition (i) goes without saying but (ii) may seem surprising. However if P is a pre fibration, without any assumption on P', (i) => (ii). Moreover this implication characterizes pre fibrations among pre foliations. In particular if both P and P' are pre fibrations, our definition coincides with Grothendieck's. (b) In the literature cartesian functors have been considered mostly when both P and P' are fibrations, and, even in that case, not much has been said about their properties. Compare with the following: 2.3. THEOREM. If P is a pre foliation, P' arbitrary, and F is cartesian, then: (1) F is faithful iff every F_s is. (2) F is full iff every F_s is. (3) F is essentially surjective iff every F_s is. (4) F is final iff every F_s is. (5) F is flat iff every F_s is. (6) F has a left adjoint iff every F_s has. If moreover P is a foliation, then (7) F is conservative iff every F_s is. 2.3. Remarks: I would like to insist on the fact that I assume nothing on P' in the theorem. Most of these results are not known even in the classical case where both P and P' are fibrations.(See e.g. the Elephant) I had proved all these results, in that case, already in 1983, more than 30 years ago, and I intended to add them, with many other things, to the Roisin notes in the book I was writing on fibered categories. But by that time the notes had been circulated, and their content was used with very little, if any, reference to me. I'm glad I kept these results to myself for two reasons: (a) As many of the results in the Roisin notes, they would be now in the Elephant, of course uglily re-indexed, and of course without any reference to me. If anyone doubts that, let me recall that my paper on distributors is not mentioned in the pharaonic bibliography of the Elephant, and neither is my joint note with Roubaud on descent, although both are used in the book! I have personally addressed my last 3 mails on fibrations to Peter Johnstone and have had no reaction so far. I hope this one will be more successful. (b) By a careful and repeated analysis of the proofs, over many years, trying to understand what made them tick, I ended up with the notion of (pre) foliation which generalizes greatly (pre) fibrations and has a lot of significant examples. Of course the previous theorem is a very small sample of what can be said about (pre) foliations. I have made this mail public. I hope it will not have the same fate as the Roisin notes, and if some of it is used full credit will be given to me. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Stephen H. Schanuel
by wlawvere 01 Aug '14

01 Aug '14

Stephen H. Schanuel Dear colleagues, It is with deep sadness that I report that my best friend, Steve Schanuel, died today. His intellectual generosity and quick mathematical wit, legendary among colleagues and students alike, will continue to inspire many. Bill Lawvere [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: functors defined by well-founded induction
by Michael Shulman 31 Jul '14

31 Jul '14

Dear Jaap, I also overlooked this email somehow. (-: On Thu, Jul 10, 2014 at 5:40 AM, Oosten, J. van <j.vanoosten(a)uu.nl> wrote: > is the following too simple-minded? > > Given a well-founded poset (X,<), a category C and a function F which, > to every functor G from an initial segment of X to C, assigns a cocone > for G. > Then there is a unique functor H:X-->C with the property that for every > x\in X, H(x) is the vertex of the cocone which is F applied to the > restriction of H to {y|y<x}. I think my desired application is probably a bit more complicated than this, but if you have >> a place >> where someone has written out exactly how to construct such a functor for the simpler version, then I'd love to see it too! Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Composition of Fibrations
by Jean Bénabou 31 Jul '14

31 Jul '14

A few weeks ago there has been a discussion about stability by composition of fibrations, bifibrations, and similar notions. Obviously the results depend on how such notions are defined. I would like to make a few comments, in particular about Steve Vickers' mail, since all the other participants to the discussion seemed to accept his approach. (I ) VICKERS' DEFINITION OF FIBRATION if C is a 2-category with comma objects and 2-pullbacks, a one cell p: B -> A is a fibration iff it satisfies the Chevalley condition. Let us test this definition in special cases. If S is a category with finite limits the 2-category Cat(S) of internal categories in S satisfies Vickers' conditions hence we know when an internal functor is a fibration. Let Set be the category of sets, except WE DON'T ASSUME THE AXIOM OF CHOICE (AC). Then Cat(Set), abbreviated by Cat, is the 2-Category of small categories. An easy verification shows that a functor p: B -> A satisfies the Chevalley condition iff it is a fibration which admits a cleavage. Thus Vickers' argument, in that case, gives as result: fibrations WHICH ADMIT A CLEAVAGE are stable by composition. On the other hand, it is easy to show that: Every small fibration has a cleavage is equivalent to AC. This well known fact can be very much strengthened by the following example: If AC does not hold in Set, one can construct in Cat a bifibration p: B -> A with internal products and coproducts where A and B are pre-ordered sets, with pullbacks preserved by p, every map of B is both cartesian and cocartesian, and add each of the following conditions: (i) p has neither a cleavage nor a cocleavage. (ii) A bit surprisingly: p is a split fibration but has no cocleavage. (iii) Dual of (ii): p is a cosplit cofibration but has no cleavage. And of course we don't need AC to show that arbitrary fibrations in Cat are stable by composition. (II) 2-CTEGORICAL FIBRATIONS Other definitions of fibrations in an arbitrary 2-Category C have been proposed. The principal one, based on Yoneda, is: A one cell p: B -> A of C is a fibration iff for every object X of C the obvious functor C(X,B) -> C(X,A) is a fibration in Cat, functorial in X. If C has comma objects and 2-pullbacks, it is easy to see that this is equivalent to Vickers' notion, and we have already seen how it can be inadequate. Of course, I don't refer here to Sreet's notion which describes a totally different kind of fibration, stable by equivalences. (III) BIFIBRATIONS For bifibrations the situation is even more confusing: Ghani defines them, in Cat, by the existence of left adjoints to the reindexing functors, Except that without AC reindexing functors need not exist. Vickers uses two duals of the 2-category C where the fibration lives. However if C has comma objects and 2-pullbacks, there is no reason why these duals have the same properties. Moreover, even in Cat, Vickers' approach will work only for bifibrations which have both a cleavage and a co-cleavage. Thus the wide generalization asserted by Vickers imposes in the well known situations drastic and unnecessary restrictions. (Compare with the example at the end of (I)) (IV) INTERNAL FIBRATIONS. For a long time I have insisted on the fact that the the theory of fibrations is first order and can be internalized. In particular in Cat(S) where S is a topos. Let me give a very simple example. Suppose A and B are groups of S and p: B -> A is a group morphism. Then p is an internal fibration iff it is an epi of S. It satisfies the Chevalley condition iff p admits a splitting in S.. Without any such splitting, internally, every element of B is an iso, hence it is both cartesian and co-cartesian. Thus p is a bifibration. Moreover, again internally, since A and B are groups, every commutative square of B, or A, is a pullback, thus B and A have pullbacks preserved by p, and in B the pullback of a cocartesian map along a cartesian map exists and is cocartesian. Hence p is a fibration with internal sums, it has also trivially internal products. But it has neither cleavages, cocleavages, nor reindexings. What would the theory of groups, torsors, classifiers etc. in a topos S look like if we were forced to assume that S satisfies AC ? Let me add that the internalization works not only for fibrations, but also for prefibrations and even for the (pre)foliations which I have defined and studied. (V) THE GROTHENDIECK CONSTRUCTION AND INDEXED CATEGORIES. Suppose S is a topos. If S satisfies AC, every fibration in Cat(S) will have a cleavage. However, even if S = Set , the Grothendieck construction doesn't make sense without further assumptions, because: if A is an internal category the very notion of a pseudo functor from A(op) to Cat(S) does not make sense since A is internal and Cat(S) is not an internal 2-category (a notion which can be easily defined) One of the mottos of the Elephant is that fibrations and indexed categories are essentially equivalent, but the second notion doesn't even make sense. it is defined as a pseudo functor from a category, which is a mathematical object, into the meta 2-category Cat. Moreover, to get the 2-category of indexed categories, we are required to collect ALL such pseudo functors and their transformations. Thus I ask the question: What is such a collection, a meta-meta 2-category? Yet another example: If we use Lawvere's Category of categories as a foundation for mathematics, fibrations, or cloven fibrations, make perfect sense, but indexed categories don't, let alone the equivalence between the two notions. In the introduction of the Elephant one can read, I quote: We should make the smallest possible demands on the metatheory within which we interpret the theory of categories (and in particular we shall not assume that it satisfies any form of the axiom of choice ... After reading carefully the chapter on indexed categories and fibrations, I ask Peter Johnstone if the following assertion would not be be more appropriate: We shall make, in particular in Chapter B1, the greatest possible demands on the metatheory and in particular assume that it satisfies the strongest form of the axiom of choice.. Incidentally, that is exactly what Grothendieck does: He uses the axiom of universes, and the tau symbol which is the strongest possible form of AC. But, even under such strong assumptions, I think he would object (and so would many other persons), if only for aesthetic reasons, to the following sentence which can be found many times in the Elephant: Let p be a fibration and C be THE associated indexed category, ... And this of course, according to Johnstone, without ANY form of AC. I'd have many more comments but this mail is already a bit long. I apologize for this length, and also for using capital letters in many places where italics or quotation marks would have been more appropriate. But ... HTML oblige. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: functors defined by well-founded induction
by Michael Shulman 30 Jul '14

30 Jul '14

Actually, my question is much more basic. On Wed, Jul 9, 2014 at 2:39 AM, Paul Taylor <cats(a)paultaylor.eu> wrote: > The simple answer is that the recursion has to define the functor, > ie the morphisms corresponding to instances of the order relation, > and not just the values at individual ordinals, in order to make sense > of defining the values at limit ordinals as colimits. That's exactly what I said: >> since we have to define the value of the functor on morphisms too, >> and its value at a given object may depend on its value at morphisms >> between previous objects. All I'm looking for is a general theorem of the form "given a well-founded relation < on a set X, and a category C, and such-and-such data, there is an induced functor X -> C." I don't care about set-theoretic issues right now, I'm just looking for a place where someone has written out exactly how to construct such a functor using the well-foundedness of <. It seems like it should be a well-known thing, so that I can just cite it rather than having to write out my own proof. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: A brief survey of cartesian functors
by Jean Bénabou 30 Jul '14

30 Jul '14

Dear George, Thank you for your mail. I see that all my mathematical arguments have not convinced you, and that trying to add more would be useless. I respect your opinion although I totally disagree with it. Best regards, Jean Le 29 juil. 2014 à 21:58, George Janelidze a écrit : > Dear Jean, > > Thank you for your kind words at the beginning of your message, and I apologize if what I said about "factorization" and "cartesian" was unclear. > > I did not mean to say that there is any connection between factorization systems and (pre foliations + cartesian FUNCTORS). What I was trying to say, was only that the following two constructions are essentially the same (up to an isomorphism): > > (a) For a fibration C-->X every morphism f in C factors as f = me, where m is a cartesian ARROW and e is a vertical arrow (with respect to the given fibration). > > (b) For a semi-left-exact reflection C-->X (in the sense of Cassidy--Hebert--Kelly) every morphism f in C factors as f = me, where m is in M, e is in E, E is the class of all morphisms inverted by C-->X, and M is its orthogonal class (M can also be defined as the class of trivial covering morphisms in the sense of Galois theory). > > I know this might sound trivial to you, but I think it is a fundamental connection, which should be widely known. And I believe that instead of > > "indexed categories versus fibrations" > > one should sometimes also consider > > "indexed categories versus fibrations versus semi-left-exact reflections" (this is why I mentioned a "third approach"). > > Let me also add now: according to Cassidy--Hebert--Kelly, the factorization mentioned in (b), where E is as in (b), and M is merely its orthogonal class, also exists under certain assumptions much weaker than semi-left-exactness. > > But again, I never thought that what you do with pre foliations and cartesian functors is a similar kind of factorization and/or that it is contained in the Cassidy--Hebert--Kelly paper! > > And I hope you have never felt from me any disrespect of your opinions and/or of your beautiful ideas and results. > > Best regards, > George > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: New book: Basic Category Theory
by Tom Leinster 30 Jul '14

30 Jul '14

Dear Harley, Thanks! Actually, some of us were just discussing the possibility of writing a second category theory book as a collaborative venture: http://golem.ph.utexas.edu/category/2014/07/basic_category_theory.html#c046… (But personally, I've probably had enough book-writing for a while.) Best wishes, Tom On Tue, 29 Jul 2014, Harley D. Eades III wrote: > Hi, Tom. > > Congrats on the successful completion of your book! > > I really like your idea on a second book. Such a book would be > very helpful for young researchers and students in my opinion. > > Has anyone thought about doing it as a group effort in the same > spirit as the homotopy type theory book was? > > A number of authors would make the writing burden far less per author, > and potentially speed up the writing process. > > Anyway, just an idea. > > I really like how you're are making your new book editable and readable for > free in 18 months. This is a great idea, and a service to the community. > > The second book should do the same. > > Very best, > .\ Harley > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: A brief survey of cartesian functors
by George Janelidze 30 Jul '14

30 Jul '14

Dear Jean, Thank you for your kind words at the beginning of your message, and I apologize if what I said about "factorization" and "cartesian" was unclear. I did not mean to say that there is any connection between factorization systems and (pre foliations + cartesian FUNCTORS). What I was trying to say, was only that the following two constructions are essentially the same (up to an isomorphism): (a) For a fibration C-->X every morphism f in C factors as f = me, where m is a cartesian ARROW and e is a vertical arrow (with respect to the given fibration). (b) For a semi-left-exact reflection C-->X (in the sense of Cassidy--Hebert--Kelly) every morphism f in C factors as f = me, where m is in M, e is in E, E is the class of all morphisms inverted by C-->X, and M is its orthogonal class (M can also be defined as the class of trivial covering morphisms in the sense of Galois theory). I know this might sound trivial to you, but I think it is a fundamental connection, which should be widely known. And I believe that instead of "indexed categories versus fibrations" one should sometimes also consider "indexed categories versus fibrations versus semi-left-exact reflections" (this is why I mentioned a "third approach"). Let me also add now: according to Cassidy--Hebert--Kelly, the factorization mentioned in (b), where E is as in (b), and M is merely its orthogonal class, also exists under certain assumptions much weaker than semi-left-exactness. But again, I never thought that what you do with pre foliations and cartesian functors is a similar kind of factorization and/or that it is contained in the Cassidy--Hebert--Kelly paper! And I hope you have never felt from me any disrespect of your opinions and/or of your beautiful ideas and results. Best regards, George -------------------------------------------------- From: "Jean B?nabou" <jean.benabou(a)wanadoo.fr> Sent: Tuesday, July 29, 2014 11:16 AM To: "George Janelidze" <janelg(a)telkomsa.net> Cc: "Ross Street" <street(a)ics.mq.edu.au>; "Steve Vickers" <s.j.vickers(a)cs.bham.ac.uk>; "Lack Steve" <steve.lack(a)mq.edu.au>; "Peter Johnstone" <P.T.Johnstone(a)dpmms.cam.ac.uk>; "Eduardo Dubuc" <edubuc(a)dm.uba.ar>; "Thomas Streicher" <streicher(a)mathematik.tu-darmstadt.de>; "Robert Par?" <pare(a)mathstat.dal.ca>; "Marta Bunge" <martabunge(a)hotmail.com>; "William Lawvere" <wlawvere(a)hotmail.com>; "Michael Wright" <mpbw1879(a)yahoo.co.uk>; "Categories" <categories(a)mta.ca> Subject: Re: A brief survey of cartesian functors > Dear George, > > I appreciate very much your pioneer work on Galois theories and the > developments you and others have given to that work. > I also believe in the role of analogies in mathematics, and I think > category theory is the ideal place where one can give the DEEP analogies a > mathematical content. > However, in this case, the analogy seems to me totally superficial, > namely: two classes A and B of maps in a category X, and the possibility > to factor every map f of X as ab, with a in A and b in B. > > This won't go very far since you need some axioms on the pair (A,B) to > start proving anything except trivialities. And, I tried to explain in my > previous mail, the properties of pairs (E,M) and (V,K) are so radically > different that a common denominator would be reduced to almost nothing. > > Even more important to me, cartesian functors are a very good notion of > morphism between pairs (V,K) and (v',K') which you can prove non trivial > results, the theorem in my mail is only an example of such results. As far > as I know there is no notion of morphism between pairs (E,M) and (E',M'). > > Let me point out some features of cartesian functors F: X --X' , viewed > abstractly as morphisms (V,K) --> (V',K') where V = V(P), K = K(P), V' > = V(P') and K' = K(P'). > 1) F preserves vertical end cartesian maps. This is harmless, but F also > REFLECTS vertical maps. > 2) We assume that every map of X can be factored as kv, but we make no > such assumption on X' > 3) The very nature of the results: For any important properties, F > satisfies globally the property iff it satisfies it fiberwise. > > If any reasonable notion of morphism of pairs (E,M) was defined someday > would reflection of maps in M be considered? Would one accept that (E',M') > should not be a factorization system even in a very weak sense? > And if non trivial results could be obtained about such notion would some > kind of fibers play a role? > > Sorry George, much as I like unifying notions and theories, I cannot see > any real, non trivial, relation between factorization systems and (pre > folations + cartesian functors) > I insist on the second term of the previous symbolic addition. > There would be a lot more to say about indexed versus fibered, but you > already know my opinion about that. Moreover indexed is totally irrelevant > here becausethere is no reindexing for pre foliations > > Best regards, > Jean > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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