June 1992 - Categories - categories.org.au

WANTED: Topos Theory
by Steve Vickers 06 Jul '92

06 Jul '92

Can anyone in Britain sell me a copy of Peter Johnstone's "Topos Theory"? Name your price. Steve Vickers. ==========================================================================

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the 2- (n-?) category of chain complexes
by dyetter＠math.ksu.edu 02 Jul '92

02 Jul '92

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 ==========================================================================

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Re: WANTED: Topos Theory ...
by John C. Mitchell 26 Jun '92

26 Jun '92

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 ==========================================================================

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None
by dpym＠dcs.ed.ac.uk 26 Jun '92

26 Jun '92

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 ==========================================================================

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Re: WANTED: Topos Theory and WANTED Basic Concepts...
by RDAWSON＠HUSKY1.STMARYS.CA 26 Jun '92

26 Jun '92

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 ++++++++++++++++++++++++++

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Definition of regular categories
by barr＠triples.Math.McGill.CA 21 Jun '92

21 Jun '92

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 ==========================================================================

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Horst Reichel
by cfw2＠po.CWRU.Edu 19 Jun '92

19 Jun '92

Does anyone have a current address, email or pmail, for Horst Reichel? Thanks, Charles. ==========================================================================

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H-Spaces
by jds＠rademacher.math.upenn.edu 16 Jun '92

16 Jun '92

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 ...??? ==========================================================================

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Re: Mal'cev relations
by barr＠triples.Math.McGill.CA 15 Jun '92

15 Jun '92

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 ==========================================================================

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Mal'cev relations and dinaturality
by mxh＠dcs.edinburgh.ac.uk 10 Jun '92

10 Jun '92

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 ==========================================================================

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