Re: Reflexive coequalizers

[Jiri Adamek <adamek@iti.cs.tu-bs.de>]

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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?
>
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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.