March 1992 - Categories - categories.org.au

Pfn
by Lars Birkedal 03 Apr '92

03 Apr '92

Does there exist a comprehensive treatment of the category Pfn, objects are sets and morphisms are partial functions - specifically I`m interested to know if someone has proven the existence of colimits and other such concepts in Pfn. Thanks, Lars Birkedal (birkedal(a)diku.dk) =====================================================================

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Omega completeness of retraction categories.
by cxm7＠pop.cwru.edu 30 Mar '92

30 Mar '92

------------- Asperti and Longo say it is an open question whether there are non-trivial cartesian closed categories with omega complete retraction categories. (Their book, p.249.) Here I show that any category _C_ with binary coproducts has omega complete retraction category iff _C_ is a preorder. The theorem does not even use all binary coproducts. It is constructive so it applies to categories over any topos. To define the terms: a "chain" in _C_ is an infinite sequence of objects, with an initial member, each with a map to the next. A category is "omega complete" if every chain in it has a colimit. Given any category _C_ the "retraction category" is called _C_^Ret and has the same objects as _C_. An arrow in _C_^Ret from A to A' is a retraction pair making A a retract of A'. If _C_ is a preorder then _C_^Ret is a groupoid and trivially omega complete. To connect this with the converse notice that _C_ is a preorder iff every object of _C_ is subinitial (i.e. it has at most one map to any object). Theorem: Suppose a category _C_ has an object A which is not subinitial but which does have a coproduct with every object of _C_. Then _C_^Ret is not omega complete. Proof: Define a sequence of objects An this way: A0=A and for every n we let A(n+1) be the coproduct (An)+A. "The map of A onto the last summand of An" means the coproduct injection of A into A(n-1)+A if n is not 0, and the identity if n=0. Each An is a retract of A(n+1) in a canonical way where the monic is the coproduct injection and the splitting epic from An+A to A acts as the identity on An and maps A onto the last summand of An. These retractions give a chain in _C_^Ret. Suppose the chain has a colimit B. Then there is a map from A to B so B is a retract of B+A. Define a cone over the chain this way: The vertex is B+A. For each An in the chain the monic to B+A is the colimit monic to B followed by the inclusion into B+A. The splitting epic from B+A to An acts as the colimit epic on B and maps A to the last summand in An. But since A is not subinitial the map of A onto the last summand in A(n+1) does not factor through the inclusion of An. Thus no splitting epic from B+A to B to commutes with the cone projections, and so B is NOT a colimit. Our chain has no colimit. Colin McLarty =====================================================================

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Re: Pfn (times 2)
by RDAWSON＠HUSKY1.STMARYS.CA 30 Mar '92

30 Mar '92

Maybe I'm being naive here, but isn't Pfn isomorphic in a rather natural way to Set*, the category of pointed sets? Everything that doesn't get mapped anywhere by a partial function P:X-->Y gets mapped to *(Y) by the corresponding function P*: X* --> Y*: so * acts as a sort of 'wastebasket'. So it would seem that coproducts, etc., would follow from that. -Robert Dawson ++++++++++++++++++++++

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None
by barr＠triples.Math.McGill.CA 27 Mar '92

27 Mar '92

The category of sets and partial functions is equivalent to that of pointed sets and base point preserving morphisms, the algebras for the triple X |---> 1 + X with obvious structural maps, and, as such, is as complete and cocomplete as one could wish. Michael Barr =====================================================================

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multi, poly, stable, etc.
by Paul Taylor 13 Mar '92

13 Mar '92

In reply to Michel Hebert: > ...it seems to me that the existence of a multi-initial object (in the sense > of Diers) does not imply the existence of an initial object in each slice, > unless you assume the existence of equalizers. No, the other way round. The category of algebraically closed fields does have an initial object in each slice, alias initial candidates, alias a polyinitial family. The "poly" initial (etc) family is "multi" iff the relevant functor preserves equalisers. As I have said, all of these things become much simpler if you consider slices instead of these confusing collections (multiple-valued adjoints). > Doesn't this permit to simplify the definition of a locally-finitely > poly-presentable category (as an accessible category with pullbacks)? Indeed it does. I'm afraid my own work on this never got beyond the "notes" stage, but those notes are available by anonymous ftp from theory.doc.ic.ac.uk as /theory/papers/Taylor/LFPP.dvi Probably Pierre Ageron or Francois Lamarche has an account of the same thing. I don't like the term "locally finitely presentable", but given that after twenty years there's no changing it, "locally finitely multi- or poly- presentable" describes the corresponding categories with multi- or poly- (finite) colimits. There is a characterisation of the connections between how "poly" the colimits are in terms of finite limits in the same notes. Paul =====================================================================

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Congratulations
by mxh＠dcs.edinburgh.ac.UK 13 Mar '92

13 Mar '92

...to Gordon Plotkin, on becoming a Fellow of the Royal Society. =====================================================================

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commutative Ext
by James Stasheff 12 Mar '92

12 Mar '92

If A is Hopf, then Ext_A(k,k) is graded commutative what is the earliest or best PUBLISHED proof? =====================================================================

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Hopf algebras
by James Stasheff 12 Mar '92

12 Mar '92

Bill Schmitt asks: >From SCHMITTW(a)hermes.msci.memst.edu Thu Mar 12 16:58:24 1992

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regepis, stablefunctors...
by Michel Hebert 12 Mar '92

12 Mar '92

Ref.: The last message of M. Barr on regepis, etc..., and the one of P. Taylor The claim of Joyal is probably the one mentioned on p.292 of [Cassidy-Hebert- Kelly; J.Austr. Mat.Soc.38 (1985)], where it is assumed that (E,M) = (Strong Epis, Monos) (it is Kelly who reported that). Certainly it is not true for all E: to be more precise, Th. 3.3 and 4.7 of the paper imply that factorization systems (E,M) with E stable under pullbacks (in a finitely complete and well- powered category A) are exactly those couples (E,E*) with E = { f ; r(f) iso }for some left-exact reflection functor r from A and E* obtained by the diagonalproperty (Hence in particular all localisation morphisms a-->r(a) are in such an E, and classical cases give counterexamples). About a previous message about slices, it seems to me that the existence of a multi-initial object (in the sense of Diers) does not imply the existence of an initial object in each slice, unless you assume the existence of equalizers (A counterexample being the algebraically closed fields).(Am I using a different definition of multi-initial object?).In fact, at least for axiomatic categories of finitary models, the existence and preservation of equalizers by the forgetful functor is precisely what makes the difference between multi- adjointness and "local adjointness" (in the sense of [Kaput] and [Adamek- Volger] mentioned by Tholen). About preservation of pullbacks only, note that in axiomatic categories of finitary models, the existence and preservation by the forg.funct. of pullbacks imply the one of wide pullbacks (and filtered colimits) (This follows from results of [Pare, Can. J. Math 42 (1990), 731-746] and Volger: see [Volger, Initial and quasi-initial models of theories, manuscript, 1991]). Doesn't this permit to simplify the definition of a locally-finitely poly-presentable category (as an accessible category with pullbacks)? =====================================================================

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corrections to corrections
by barr＠triples.Math.McGill.CA 11 Mar '92

11 Mar '92

corrections to corrections There is an omission in the corrections to ctcs. The definition \mathrm{Sub} has to be added. A couple of people have written me that it wouldn't tex with their version of the diagram macros. I can suppose only that they have old or small versions of tex. However, it has far fewer macros than the book itself and we did that with a 1989 or so version that was considerably smaller than the one I have today. There may also be a problem with the new font selection scheme that I have not yet adopted. I will see that the latest version of the diagram macros are in the ftp/pub/texmacros directory. Michael =====================================================================

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