Subject: categories: RE: Finitely presentable presheaves

From: [email-masked]
Date: Wed, 17 May 2000 09:04:17 +0100

> Consider the category of presheaves  C^  on a small category,  C.
> 
> Plainly, a finite colimit  F  of representables presheaves is finitely
> presentable, in the usual sense:  C^(F, -)  preserves 
> filtered colimits.
> But the converse is also true and seemingly well known: every finitely
> presentable presheaf is a finite colimit of representables.
> 
> Is this proved somewhere?

I think this is obvious from the fact that the theory of presheaves over C
is many sorted, essentially algebraic (a finite limit theory), with a sort
for each object of C and a unary operator for each morphism. Then finitely
presentable in the categorical sense is the same as finitely presentable in
the algebraic sense, which is equivalent to being a finite colimit of free
cyclic (i.e. one generator) algebras. In the case of presheaves, Yoneda's
lemma says precisely that the free algebra on a generator of sort X (object
of C) is the representable presheaf for X.

I have exploited some of these facts in a paper with my PhD student Gillian
Hill, "Presheaves as configured specifications". It develops a language for
specifying systems by components with sharing.

Steve Vickers.