Subject: categories: Re: idempotent completion

From: Peter Freyd <[email-masked]>
Date: Mon, 26 Dec 2005 15:56:42 -0500 (EST)

Jiri asks:

  I would be grateful for getting the earliest reference to the fact
  that for two small categories T and S the corresponding functor-
  categories into Set are equivalent iff T and S have the same
  idempotent (= Cauchy) completion.

The fact that a category and its idempotent completion have equivalent
functor categories was certainly known very early. It does not appear
in the first book on category theory (1964) but the lemma that proves
it, to wit, that idempotent-complete cats form a full reflective
subcategory of the relevant category (COSCANECOF) appears on page 61
(which is 18 pages before any mention of reflective subcats and 48
pages before the first mention of functor categories -- see
www.tac.mta.ca/tac/reprints/articles/3/).

That book was devoted to the additive setting. On page 119 one finds
the additive notion, "amenable", corresponding to the condition of
idempotents splitting. The full subcat of small projectives in the
functor category in the additive setting is dual to the amenable
closure of the domain category -- thus providing an instant proof that
if two cats have equivalent additive functor categories then their
amenable closures are equivalent. The non-additive case is easier: the
full subcat of indecomposable projectives in the category of set-
valued functos is dual to the idempotent completion of the domain
category.