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.