Re: idempotent completion

Re: idempotent completion

[Peter Freyd]

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.

RE: idempotent completion

[Marta Bunge]

Dear Jirka,

The result appears in my 1966 thesis ("Categories of Set-valued functors",
U. of Pennsylvania), and in print (with an arbitrary closed category base V
and categories relative to V) in the following 1969 paper, translated into
Russian in 1972.


Marta Bunge, Relative Functor Categories and Categories of Algebras.
J.of Algebra 11 (1969) 64-101.

Russian translation in : Mathematics: Periodical collections of Translations
of
Foreign Articles, Vol.16, Izdat. "Mir", Moscow(1972) 11-46, MR 50, #12532.


Cordially,
Marta



************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics
McGill University
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810
Home: (514) 935-3618
marta.bunge@mcgill.ca
http://www.math.mcgill.ca/~bunge/
************************************************




>From: Jiri Adamek <adamek@iti.cs.tu-bs.de>
>To: categories net <categories@mta.ca>
>Subject: categories: idempotent completion
>Date: Fri, 23 Dec 2005 10:46:30 +0100 (CET)
>
>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. One can find this in a russian paper:
>
>"Morita equivalent categories" by S. V. Polin, Vestnik Mosk. Univ.,
>1974, no.2, 41-45
>
>
>xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>alternative e-mail address (in case reply key does not work):
>J.Adamek@tu-bs.de
>xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
>

idempotent completion

[Jiri Adamek]

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. One can find this in a russian paper:

"Morita equivalent categories" by S. V. Polin, Vestnik Mosk. Univ.,
1974, no.2, 41-45


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
alternative e-mail address (in case reply key does not work):
J.Adamek@tu-bs.de
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx