From: S.J.Vickers@open.ac.uk
To: categories@mta.ca
Subject: RE: Finitely presentable presheaves
Date: Wed, 17 May 2000 09:04:17 +0100 [thread overview]
Message-ID: <B5A6557CFDF6D211960E0008C7F355850134F3AF@tesla.open.ac.uk> (raw)
> 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.
next reply other threads:[~2000-05-17 8:04 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-05-17 8:04 S.J.Vickers [this message]
-- strict thread matches above, loose matches on Subject: below --
2000-05-16 15:55 Marco Grandis
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