xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx On Mon, 9 Oct 2006, Richard Garner wrote:

I have a proof that the indiscrete category functor Set -> Cat preserves reflexive coequalizers which, although straightfoward, uses the explicit description of colimits in Cat. Is this necessary, or can I deduce the result from general principles?

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx This is a nice example of an algebraically exact functor: for varieties Alg T, where T is an algebraic theory (and Alg T is the category of all finite-product preserving functors in [T, Set]), all theory morphisms F: T -> S induce functors Alg F: Alg S -> Alg T given by precomposing with F; they are called algebraically exact. These are precisely the right adjoints between varieties which preserve sifted colimits- and reflexive coequalizers are special sifted colimits. This all is a part of the duality between varieties and algebraic theories (described by F.W.Lawvere, J. Rosicky and myself, Algebra Universalis 49 (2003), 1-45). Consider the "obvious" algebraic theory S of Set, the dual of finite sets, and the "obvious" theory C of Cat, the dual of finitely presentable cats. The functor F: C -> S which forgets morphisms induces the indiscrete category functor as Alg F.

Ah yes, I see now. I had observed that the underlying diagram of indiscrete graphs was still a coequalizer -- which unfortunately is to truncate one's simplicial sets a little too much! Many thanks for the enlightenment. Richard --On 09 October 2006 20:40 George Janelidze wrote:

Dear Richard,

If we were asking the same question about the category SimplSet of simplicial sets instead of Cat, the answer would be obvious since:

(*) The functor Set ---> Set sending X to X^n preserves reflexive coequalizers for each natural n. This (very simple) fact should be considered as well known since it is involved in one of several well-known proofs of monadicity of varieties of universal algebras over Set.

(**) Since SimplSet is a Set-valued functor category, all colimits in it reduce to colimits in Set.

For those who are familiar with the standard adjunction between SimplSet and Cat, the result for SimplSet will immediately imply the result for Cat (since the fundamental category functor being the left adjoint in that adjunction preserves all colimits and obviously sends "indiscrete simplicial sets" to indiscrete categories).

One could also use various (not too much) truncated simplicial sets instead of the simplicial sets of course (e.g. what I once called "precategories").

I mean, I do not remember any reference, but I would not need the explicit description of colimits in Cat.

George Janelidze

----- Original Message ----- From: "Richard Garner" <rhgg2@hermes.cam.ac.uk> To: <categories@mta.ca> Sent: Monday, October 09, 2006 12:14 PM Subject: categories: Reflexive coequalizers

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Dear categorists,

I have a proof that the indiscrete category functor Set -> Cat preserves reflexive coequalizers which, although straightfoward, uses the explicit description of colimits in Cat. Is this necessary, or can I deduce the result from general principles?

[Also, I'm sure this result must appear somewhere but I can't find a reference for it. If anyone knows of one, I'd be grateful.]

Many thanks,

Richard