In Kelly's book on enriched category theory, he seems to claim that it would be easy to prove that if K is a symmetric monoidal closed category in which all small diagrams have limits and colimits, then in the category of small K-categories it is also true that all small diagrams have limits and colimits, but he does not actually prove this. Does anyone know a reference for a proof of this fact? John Baez
Date: Tue, 31 May 94 22:31:33 PDT From: john baez <baez@ucrmath.UCR.EDU> In Kelly's book on enriched category theory, he seems to claim that it would be easy to prove that if K is a symmetric monoidal closed category in which all small diagrams have limits and colimits, then in the category of small K-categories it is also true that all small diagrams have limits and colimits, but he does not actually prove this. Does anyone know a reference for a proof of this fact? John Baez I wrote out the proof for limits in my dissertation. (Ross Casley. On the Specification of Concurrent Systems. Department of Computer Science, Stanford University. Feb 1991. Technical Report STAN-CS-91-1355.) The existence of colimits is shown in a somewhat more general setting in the following: @Article( BCSW83, Author="Betti, R. and Carboni, A. and Street, R. and Walters, R.", Title="Variation Through Enrichment", Journal="Journal of Pure and Applied Algebra", Volume=29, Pages="109-127", Year=1983) -Ross Casley
Date: Thu, 02 Jun 94 08:50:03 PDT From: Ross Casley <casley@ca.merl.com> From: john baez <baez@ucrmath.UCR.EDU> if K is a symmetric monoidal closed category in which all small diagrams have limits and colimits, then all small diagrams [in K-Cat] have limits and colimits. Does anyone know a reference for a proof of this fact? I wrote out the proof for limits in my dissertation. (Ross Casley. On the Specification of Concurrent Systems. Department of Computer Science, Stanford University. Feb 1991. Technical Report STAN-CS-91-1355.) Ross is too modest, the theorem he proves there is a tad stronger: if K has all J-limits for a small diagram J then so does K-Cat. K should be symmetric monoidal but need not be closed and needs no limits or colimits beyond those implied by the existence of all J-limits. Vaughan Pratt
I would have thought that the standard reference to the cocompleteness of V-Cat was Harvey Wolff, V-cat and V-graph, J. Pure and Applied Alg, 4(1974) 123-135. He shows that if V is cocomplete, then so is V-cat, by showing that V-cat is monadic over V-graph, that V-graph is cocomplete and then quoting a result of Linton in F. E. J. Linton, Coequalizers in categories of algebras, in Lecture Notes in Math. 80, Springer-Verlag 1969, 75-90. His main application is the existence of localizations in the V case. Wolff assumes that V is a symmetric monoidal closed category. The existence of the free functor from V-graph to V-cat requires that tensor commute with coproducts, and the existence of coequalizers in V-cat seems to require that tensor commute with coequalizers, so closedness apparently is essential, at least for this form of the argument. He doesn't say anything about the existence of limits in V-cat. John W. Gray
I see that Casley has answered the query of Baez on the completeness and cocompleteness of V-Cat by giving the reference to the Betti, Carboni, Strret, Walters paper "Variation through enrichment" in JPAA 29 (1983), 109-127. That is a fine reference, and everyone should read that paper; but it was of course not my reason for believing the result at the time of writing my book, published in 1982 but available earlier as a preprint. The result that V-Cat is complete and cocomplete when V is so is much older, and is due to Harvey Wolff, J.Pure Appl. Algebra 4 (1974), 123-135. The paper of Betti et al. does indeed mention this reference - but some y younger colleagues may be unaware of the details of Wolff's contribution. Max Kelly.
participants (5)
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john baez -
John Gray -
Max Kelly -
pratt＠CS.Stanford.EDU -
Ross Casley