RE: Finitely presentable presheaves

RE: Finitely presentable presheaves

[S.J.Vickers]

> 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.

Finitely presentable presheaves

[Marco Grandis]

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?

Best regards

Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: grandis@dima.unige.it
tel: +39.010.353 6805   fax: +39.010.353 6752

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