Subj: A name request Date: Tue, 02 Jun 92 21:20:34 ADT From: dbenson@yoda.eecs.wsu.edu (David B. Benson) Given a pair of arrows with common codomain, the pullback consists of an object and a pair of pullback projections with the requisite universal properties. Now for the question: Regular is to equalizer as X is to pullback projections. What name for a pair of arrows with a common domain should replace X in the above line? Thanks, David ========================================================================== Subj: Re: A name request From: street@macadam.mpce.mq.edu.au (Ross Street) Date: Fri, 5 Jun 92 13:00:10 EST > > Date: Tue, 02 Jun 92 21:20:34 ADT > From: dbenson@yoda.eecs.wsu.edu (David B. Benson) > > Given a pair of arrows with common codomain, the pullback consists of > an object and a pair of pullback projections with the requisite > universal properties. Now for the question: > > Regular is to equalizer as X is to pullback projections. > > What name for a pair of arrows with a common domain should replace X in > the above line? > > Thanks, > David > X = "pullback span" (see March 1972 thesis of Jeanne Meisen supervised by Jim Lambek at McGill) or, in the additive case, X = "relation" (see Peter Hilton's LaJolla paper, 1965). Regards, Ross ========================================================================== Subj: Re: A name request From: Richard Wood Date: Fri, 5 Jun 1992 13:58:40 -0300 I assume that by "regular" you mean "regular mono". If that is so then I think that your X is "difunctional jointly monic pair". Aurelio, Christina and company have recently looked at difunctional relations in their work on Mal'cev matters but I do not have an explicit reference for you. The X above is an ugly mouthfull but it makes sense in that for the most "decent" of categories, toposes, every mono is "regular" suggesting a usualness or normalness about such monos relative to the paradigm, one to one function_but difunctionality is not usual. To be precise, let X<---R--->A be a jointly monic pair in SET then R is a pullback of some cospan if and only if, viewed as a relation, R is "transitive" in the following sense: aRx and bRx and bRy implies aRy . This interpretation is of course possible in much greater generality than SET. Best regards, Richard ========================================================================== Subj: A rose by any other name, Mal'cev relations Date: Mon, 8 Jun 92 15:19:39 EDT From: barr@triples.Math.McGill.CA (Michael Barr) I used the phrase regular--or effective--epimorphic family for the dual notion of pair of arrows that were a pushout. Of course, this is a much wider notion, so regular epi pair would do for that. Since it is almost self-defining and also shorter than some of the alternatives, it is the term I recommend. Richard points out, in effect, that a subset R in A x B is a pullback in this sense iff R o R\op o R is in R. That is true in any topos. It is also true in any abelian category. Is it true in the dual of a topos? In the dual of sets? Whatever, it is an exactness condition (that any such subset be a pullback) that might turn out to be interesting. Conversely, it could turn out to be equivalent (probably with some existence of limits and/or colimits) to effective equivalence relations. Note that it implies that R o R\op and R\op o R are transitive (they are obviously symmetric). Michael ++++++++++++++++++++++++ Date: Tue, 9 Jun 92 10:57:05 EDT From: barr@triples.Math.McGill.CA (Michael Barr) This is a follow up to my post from yesterday. Let me say that a subobject R in A x B is a Mal'cev relation if R o R\op o R is in R. Call a Mal'cev relation effective if there is a pullback R ------> A | | | | | | v v B ------> C Yesterday, I pointed out, in effect, that in toposes and abelian categories, Mal'cev relations are effective and asked if it was true in the dual of sets. The following additional facts have now come to light: 1. Mal'cev relations are not effective in the dual of sets. (Counter-example at end). 2. In a Mal'cev category, every relation is Mal'cev (trivial), but not always effective (see 1). 3. If every Mal'cev relation is effective, then so is every equivalence relation. Since equivalence relations are effective in the opposite of sets, it follows from 1 that the converse is false. Thus effectiveness of Mal'cev relations is strictly stronger than that of ERs. 4. A PER (on a single object A) is a Mal'cev relation, but the converse is not true (trivial counter-example in groups, in which, from 2, every relation is Mal'cev). Thus the condition of effectiveness of Mal'cev relations is an exactness condition that is strictly stronger than that of ERs. Since it is true in any topos, it must have something to do with disjoint universal sums. It wouldn't surprise me if it had something to do with effective unions. Here is the counter-example. If there is a pullback as above, then C can be taken to be the pushout. Thus if sets\op satisfy the condition, so does the opposite of finite sets, which is equivalent to finite boolean algebras. In that category, the following is a pushout and not a pullback (there is no choice in the maps since 2 is initial and 1 is final): 2 ------> 4 | | | | | | v v 1 ------> 1 Since Boolean algebras is a Mal'cev category, all relations are Mal'cev (it is trivial to see that this one is in any case). It follows that effective Mal'cev pair would be another answer to the original question, but not a good one, I think. Michael ========================================================================== Subj: Re: Mal'cev relations Date: Tue, 9 Jun 92 21:37:51 EDT From: barr@triples.Math.McGill.CA (Michael Barr) Please change the line that says that in a Mal'cev category, every relation is Mal'cev to one that says that if an equational category has a Mal'cev operation, then every relation is Mal'cev. Thanks, Mike ========================================================================== Subj: Mal'cev relations and dinaturality From: Date: 10 Jun 92 13:12 +0100 As a followup to the discussion on Mal'cev relations these days I thought it might be a good idea to describe some ideas I had recently concerning these. They seem quite close to Peter Freyd's structors, so I don't claim originality. Let BB be a ccc with finite limits. A pullback square (a:X->A,b:X->B,p:A->Y,q:B->Y) is called a Mal'cev relation between A and B, (by a slight distortion of Michael Barr's definition). Are there any good terms for X and Y? One might call them kernel and image, perhaps. The Mal'cev relations in BB form a ccc with finite limits (morphisms being given by hexagons.). Now given a class of -+ bifunctors on BB we can axiomatise an action on Mal'cev relations as follows: If R:=(a:X->A,b:X->B,p:A->Y,q:B->Y) is a M. rel between A and B and F:BB\op x BB -> BB is in the class of bifunctors, then we assign to this situation a Mal'cev relation F*R between FAA and FBB such that the following axioms are satisfied: 1) (F x G)*R =~= F*R x G*R 2) (F => G)*R =~= F*R => G*R 3) If F_R is the relation between FAA and FBB defined by the span (Fpa:FYX->FAA, Fqb:FYX->FBB) and F^R is the relation defined by the cospan (Fap:FAA->FXY, Fqb: FBB->FXY) then F_R \subset F*R \subset F^R. In 3) the term \subset and "relations are used in an informal way. What 3) precisely means is that FAA FAA /\ \ /\ \ / \ / \ / \/ / \/ W FXY FYX Z \ /\ \ /\ \ / \ / \/ / \/ / FBB FBB commute for W the kernel and Z the image of F*R. => and x denote twisted exponential and pointwise product for bifunctors and exponential and cartesian product for Mal'cev relations. The clauses 1)-3) can easily be generalised to -^n+^n-functors. One can show by induction that the class of bifunctors definable by constants +(if it exists), x and => admit an action on Mal'cev relations, so the definition is consistent. Now given such a class of functors and an action on Mal'cev relations we can define a notion of dinaturality as follows: Def: Let F and G be two -+ bifunctors. A "very strong dinatural transformation" t is then given by a family of maps t_A:FAA to GAA for each object A in C, such that for each Mal'cev relation R between A and B the pair (t_A:FAA->GAA, t_B:FBB->GBB) is a morphism from F*R to G*R in the category of Mal'cev relations. I.e. the "famous hexagon" t_A FAA -------> GAA /\ \ / \ / \ / \/ W Z \ /\ \ / \ / \/ t_B / FBB -------> GBB for W the kernel of F*R and Z the image of G*R. (end of definition) These transformations compose and, in effect, form a ccc (with bifunctors as objects). We may call a polymorphic function parametric if it comes from such a transformation. Remark: By letting R := the graph of some morphism it follows that every "very strong dinatural transfomation" is dinatural in the Yoneda-MacLane-Bainbridge-Freyd-Scedrov-Scott sense. Now we are ready to define the corresponding notion of an "end" as a universal transformation from some constant functor to a bifunctor and prove that in the case of FXY == (TY=>X)=>Y (for T covariant) such an end gives the initial T-algebra for T. Moreover one can show that the initial T-algebra _is_ an end, so that e.g. in the category of sets an end for FXY=(Y=>X)=>(X=>Y) exists (namely the NNO). In this sense the category provides a "bad model for polymorphism" looked for by E.Robinson in that it distinguishes different notions of parametricity. At least as far as ML polymorphism is concerned. As a side effect this provides a characterisation of a strong NNO in a ccc with finite limits by pure equations. Martin Hofmann ========================================================================== Subj: Mal'cev relations Date: Thu, 11 Jun 92 13:34:12 EDT From: barr@triples.Math.McGill.CA (Michael Barr) The observations by Martin Hoffman are interesting. In the interests of clarity, I do object to his calling the category of pullback squares the category of Mal'cev relations. One is an existential structure, while the other builds the thing that exists into the structure. It makes it into an entirely different category. Not less interesting, but different. It is (somewhat) like the difference between the categories of complete lattices and complete semilattices. One interesting (and trivial) observation is that if Mal'cev relations are effective, then a quotient of a subobject of a quotient. This is (like everything I've said, but forget to mention) in the presence of sufficient limits and colimits. Now if one could show that the amalgamation property held, one would have that a pushout of a mono is mono in that case. Michael ========================================================================== Subj: Re: Mal'cev relations Date: Sun, 14 Jun 92 08:08:20 GMT-0400 From: jds@rademacher.math.upenn.edu the difference beteen the two categories triggers memories of discussions with John Moore about the non-category of H-spaces the problem is with the morphisms either an H-map is one that respects the two multiplications up to some unspecified homotopy OR an H-map is a pair consisting of a map say f:X --> Y AND a specific homotopy h_t:X x X --> Y either way there are problems thoughts? jim ========================================================================== Subj: Re: Mal'cev relations Date: Mon, 15 Jun 92 10:05 GMT From: MAS013@BANGOR.AC.UK (This is a reply to Jim's thought on H-spaces, not really to Mike's one on Malcev relations.) There would seem to be a need for a discussion of homotopy everything models of the theory of categories. The objects would have a composition that was homotopy associative up to arbitrary degree, would have homotopy identities, etc. Vogt back in 1974 provided an example of such a thing in the context of homotopy coherent diagams of spaces. The composition morphism needed to be chosen, but once chosen was homotopy associative to infinite degree. (Jean-Marc Cordier and I have examined this in the context of simplicially enriched categories.) It seems that a possible way out is to consider such an infinitely lax category as a simplicial class satisfying a weak Kan condition, that condition giving the composition. Grothendieck in his pursuit of stacks asked for infinitely lax categories or groupoids to model homotopy types, and recent work by Kapranov, Voevodsky, Steiner etc., as well as a mass of material originating "down-under" from Ross, and friends, suggest that the time is approaching when an attack on these problems can be made. Jean-Marc and I have a lot of information on doing homotopy coherence in simplicially enriched settings including homotopy coherent forms of the Yoneda lemma, the interchange law in this setting, and so on. Unfortunately it is not always obvious how to go beyond locally weakly Kan simplicial category, where at least the composition is defined, at least, so as to be able to handle weakly Kan simplicial classes. I would much appreciate peoples reactions to this view. Cheers, Tim. (Sender:Tim Porter: e-mail: mas013@uk.ac.bangor.vaxc School of Maths, University of Wales at Bangor, Dean Street, Bangor, Gwynedd, LL57 1UT. U.K. ========================================================================== Subj: H-Spaces Date: Tue, 16 Jun 92 12:56:09 EDT From: jds@rademacher.math.upenn.edu Jim Stasheff replying to tim Porter's comments on my H-space query: Yes, the analog for H-spaces is fine. If X is not only an H-space but a strongly homotopy associative (sha) space, then it has a classifying space BX and the category of sha H-spaces can be defined so that B is a functor, but at any finite level ...??? ========================================================================== Subj: Horst Reichel Date: Fri, 19 Jun 92 12:54:25 -0400 From: cfw2@po.CWRU.Edu (Charles F. Wells) Does anyone have a current address, email or pmail, for Horst Reichel? Thanks, Charles. ========================================================================== Subj: Definition of regular categories Date: Sun, 21 Jun 92 11:34:13 EDT From: barr@triples.Math.McGill.CA (Michael Barr) Many definitions of regular category have been given. The original one required that every arrow have a kernel pair (it will not do to call it a kernel since that has another, and related, meaning in abelian categories), that the kernel pair of every arrow have a coequalizer and that every coterminous pair of arrows of which one is a regular epi (a coequalizer of a kernel pair) have a pullback in which the arrow opposite the regular epi is regular epi. This definition can be both strengthened and weakened (and has been, including by me), while retaining its essential intent. Strengthenings: You can assume the category has more (finite) limits or more coequalizers. All the way up to all finite limits and finite colimits. Weakenings: You can define a regular epi as I did last fall, in terms of pairs of arrows. You can force it to be stable by requiring that it be a cover in the finest topology for which the representable functors are sheaves. You can then require that every arrow factor as a regular epi followed by a mono. I think that if I were starting from scratch, I would use the latter definition and then strengthen it as needed. The full embedding theorem is certainly true in that context. As for examples, that's harder. I haven't worked out the details, but I think that finite ordinals and order-preserving maps should be an example. If you stick to non-zero ordinals, then this is the standard simplicial category. All the monics and all the epics are split and hence regular in any definition, but I don't think the epis have kernel pairs, although coequalizers probably exist. The point is that there is no reason to constrict the notion, since you can add whatever you need when you need it. In a regular category, it is certainly equivalent to say that every epi is regular and that every monic epi is an isomorphism. Moreover, it works with the weakest definition, since you need only take an epi and factor it as a regular epi followed by a mono. Then the mono is the second factor of an epi and hence also epic and so if every monic epi is an ismorphism, the original is, up to isomorphism, the regular epi part. The converse is even easier. I thought the definition of pretopos implied regular. Michael ========================================================================== Subj: WANTED: Topos Theory Date: Wed, 24 Jun 92 11:30:02 From: Steve Vickers Can anyone in Britain sell me a copy of Peter Johnstone's "Topos Theory"? Name your price. Steve Vickers. ========================================================================== Subj: Re: WANTED: Topos Theory and WANTED Basic Concepts... Date: Fri, 26 Jun 1992 10:57 ADT From: RDAWSON@HUSKY1.STMARYS.CA This must be the Most Wanted Book In The World... do the publishers realize the demand? To whom it may concern: if anybody brings out an edition at any reasonable price, I will buy one. Paperback would be nice but not essential. I will also recommend that our library purchase a copy,. That's 2. I also know several other people who want copies... Note that I am *not* opening bidding against Steve Vickers; this is intended for use to get the publishers going. But, seriously, somebody must have contacts to pass this on! -Robert Dawson ++++++++++++++++++++++++++ From: dpym@dcs.ed.ac.uk Date: Fri, 26 Jun 92 15:00:36 BST In a vein similar to that of Steve Vickers' note: Will anyone in the U.K. sell me a copy of `Basic Concepts of Enriched Category Theory', by G.M. Kelly ? D.J. Pym ========================================================================== Subj: Re: WANTED: Topos Theory ... From: John C. Mitchell Date: Fri, 26 Jun 92 10:55:21 -0700 I did buy a copy of Topos Theory a couple of years ago direct from the publisher. It turned out to be an "Academic Press Replica Reprint," which means a bound Xerox copy. I think the price was somewhat high, but I don't remember exactly. Despite the poor print quality, I think it is better to own a copy of this sort than none at all. A publisher can certainly do this without any substantial inventory costs, etc. On the other hand, without going into details, there could be other ways to obtain a copy of similar quality ... John Mitchell ========================================================================== Subj: the 2- (n-?) category of chain complexes From: dyetter@math.ksu.edu (David Yetter) Date: Mon, 29 Jun 92 17:28:47 CDT Surely this must be known, but where is there a reference? If one considers any abelian category A, one can from a 2-category as follows: objects: differential graded objects in A (i.e. chain complexes) 1-arrows: chain maps 2-arrows: equivalence classes of chain homotopies where two chain homotopies a,b:F--->G are equivalent if there exits a map x of degree -2 from the common source of F and G to the common target of F and G such that a - b = xd - dx. The two dimensional composition of chain homotopies is addition, the one dimensional is given by if a:F--->G, b:H--->K (source(H)=target(G)) then a *1 b = aH + Gb (composites in diagrammatic order). Also, can one play the game again, and get a 3-category, etc.? A related question: does anyone know of a notion of "weak n-category" in which associativities are strict, but the middle-four-interchange holds only up to higher dimensional data? (That's what happens here if one uses chain homotopies instead of equivalence classes.) --David Yetter ========================================================================== Subj: Re: WANTED: Topos Theory Date: Tue, 30 Jun 92 14:36:15 From: Steve Vickers Having asked for second-hand copies of Johnstone's "Topos Theory" that I could buy, I instead got a couple of commiserational replies saying how terrible it is that the book should have been allowed to go out of print. I intend to write to the publishers, Academic Press, to encourage them to reprint in paperback. If enough people would like to add their "signatures", I'll add a section in the following form: The following people have told me that they will definitely buy a copy if the book is reprinted at not too high a price: name brief address name brief address : : Please email me if you'd like to be included. Steve Vickers. sjv@doc.ic.ac.uk