From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2446 Path: news.gmane.org!not-for-mail From: "Christopher Townsend" Newsgroups: gmane.science.mathematics.categories Subject: Higher Order Yoneda? Date: Mon, 22 Sep 2003 15:03:26 +0000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed X-Trace: ger.gmane.org 1241018668 4196 80.91.229.2 (29 Apr 2009 15:24:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:28 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Sep 23 16:09:24 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Sep 2003 16:09:24 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A1sSl-0002lc-00 for categories-list@mta.ca; Tue, 23 Sep 2003 16:04:39 -0300 X-Originating-IP: [194.66.147.4] X-Originating-Email: [cft71@hotmail.com] X-OriginalArrivalTime: 22 Sep 2003 15:03:26.0861 (UTC) FILETIME=[AD07D3D0:01C3811A] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 20 Original-Lines: 22 Xref: news.gmane.org gmane.science.mathematics.categories:2446 Archived-At: I was looking for a reference (or correction!) to the following observation in indexed category theory. Let E be a cartesian category and H an E-indexed category (that is H is a functor from E^op to CAT, where CAT is some background category of possibly large categories). Then, if C is an internal category in E we have a categorical equivalence Nat[Cat(_,C),H]=H(C_0) where C_0 is the object of objects of C. The objects of Nat[Cat(_,C),H] are the natural transformations and the morphisms are the modifications (see, e.g. definition B1.2.1(c) in Johnstone's Elephant). On objects, this equivalence is just Yoneda's lemma, so surely it has been observed already that it extends to this 2-categorical statement? Best wishes, Christopher Townsend