From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10141 Path: news.gmane.io!.POSTED.ciao.gmane.io!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: Interpreting category-valued presheaves Date: Tue, 18 Feb 2020 04:23:38 +0000 Message-ID: References: Reply-To: Ross Street Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="ciao.gmane.io:159.69.161.202"; logging-data="12988"; mail-complaints-to="usenet@ciao.gmane.io" Cc: "categories@mta.ca list" To: Redi Haderi Original-X-From: majordomo@rr.mta.ca Wed Feb 19 19:32:24 2020 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.55]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1j4U91-0003Af-Vu for gsmc-categories@m.gmane-mx.org; Wed, 19 Feb 2020 19:32:24 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:44990) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1j4U6E-00021H-Qi; Wed, 19 Feb 2020 14:29:30 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1j4U7n-0002KI-Be for categories-list@rr.mta.ca; Wed, 19 Feb 2020 14:31:07 -0400 In-Reply-To: Accept-Language: en-GB, en-US Content-Language: en-US Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10141 Archived-At: Dear Redi Haderi Max Kelly's book ``Basic concepts of enriched category theory'' generalises this whole Kan business to V-categories for a decent base V. So, for the case V =3D Cat, you obtain the result for 2-categories. I say ``the result'', but of course, for 2-categories there are versions fo= r weaker structures (bicategories, pseudofunctors, and so on). The Cat-enriched version is the strict case which you seem to want. Regards, Ross Street On 16 Feb 2020, at 6:11 AM, Redi Haderi > wrote: Given a category C we know that the category of set-valued presheaves on C may be interpreted as a free cocompletion of C. More precisely, if we denote PSh(C) the presheaf category, then colimit-preserving functors from PSh(C) to a cocomplete category D correspond to functors from C to D (via Yoneda extension). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]