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
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.
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
participants (3)
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Jiri Adamek -
Marta Bunge -
Peter Freyd