From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/538 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Cocompletions of Categories Date: Wed, 26 Nov 1997 13:42:29 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017023 26181 80.91.229.2 (29 Apr 2009 14:57:03 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:57:03 +0000 (UTC) To: categories Original-X-From: cat-dist Wed Nov 26 13:43:22 1997 Original-Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id NAA13738; Wed, 26 Nov 1997 13:42:30 -0400 (AST) Original-Lines: 78 Xref: news.gmane.org gmane.science.mathematics.categories:538 Archived-At: Date: Wed, 26 Nov 97 10:19:37 +0100 From: Jiri Velebil Dear Colleagues, I wonder whether the following description of a free F-conservative completion of any category under C-colimits (where F and C are classes of small categories such that F is a subclass of C and "F-conservative" means "preserving existing F-colimits"). The description of the cocompletion is as follows: Suppose C is a class of small categories and F is a subclass of C. Let X be any category. Denote by [X^op,Set] the quasicategory of all functors and all natural transformations between them. Denote by F^op-[X^op,Set] the quasicategory of all functors which preserve F^op-limits, i.e. limits of functors d : D -> X^op with D^op in F. Claim 1. F^op-[X^op,Set] is reflective in [X^op,Set] (The proof uses the fact that the above Claim holds for the case when X is small - Korollar 8.14 in Gabriel, Ulmer: Lokal pr"asentierbare Kategorien.) By Claim 1., F^op-[X^op,Set] has all small colimits. Denote by D(X) the closure of X (embedded by Yoneda) in F^op-[X^op,Set] under C-colimits. Then one can prove that D(X) is a legitimate category. The codomain-restriction I: X -> D(X) of the Yoneda embedding fulfills the following: 1. D(X) has C-colimits. 2. I preserves F-colimits. 3. D(X) has the following universal property: for any functor H : X -> Y which preserves F-colimits and the category Y has C-colimits there is a unique (up to an isomorphism) functor H* : D(X) -> Y such that H* preserves C-colimits and H*.I = H. In fact, this gives a 2-adjunction between C-CAT_C : the 2-quasicategory of all categories having C-colimits, all functors preserving C-colimits and all natural transformations and CAT_F : the 2-quasicategory of all categories, all functors preserving F-colimits and all natural transformations. The result also holds for V-categories, instead of a class C of small categories one has to work with a class of small indexing types. Thank you, Jiri Velebil velebil@math.feld.cvut.cz Department of Mathematics FEL CVUT Technicka 2 Praha 6 Czech Republic