From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3883 Path: news.gmane.org!not-for-mail From: Michal Przybylek Newsgroups: gmane.science.mathematics.categories Subject: Re: Teaching Category Theory Date: Wed, 05 Sep 2007 15:09:12 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: 7BIT X-Trace: ger.gmane.org 1241019580 10714 80.91.229.2 (29 Apr 2009 15:39:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:40 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Sep 5 20:25:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Sep 2007 20:25:13 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IT4Ay-0003xR-B1 for categories-list@mta.ca; Wed, 05 Sep 2007 20:20:48 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 7 Original-Lines: 39 Xref: news.gmane.org gmane.science.mathematics.categories:3883 Archived-At: "Jeff Egger" wrote: > --- Michael Shulman wrote: [I don't see the original post, so I'm responding here] > Actually, currently my favorite level of generality is something I > call a "monoidal fibration". Roughly, the idea is that you have two > different "base" categories, S and V, such that the object-of-objects > comes from S while the object-of-morphisms comes from V. When S and V > are the same, you get internal categories, and when S=Set, you get > classical enriched categories. This could be regarded as "explaining" > the coincidence of internal and enriched categories for V=Set. Heh... I'm studying the same problem as a part of my Master Thesis (under supervision of prof. Andrzej Tarlecki), but fortunately :-) in a bit different framework. The chief concept of my work is a definition of a category ("elementary category") in a fibred monoidal category (i.e. each fibre is monoidal and reindexing functors preserve the monoidal structure) over a base category with binary products. Than, roughly speaking, for a category C with finite limits, C-enriched categories are just "Fam : Fam(C) -> Set"-elementary categories, and C-internal categories are just "Cod : C^{->} -> C"-elementary categories. It turns out (if I didn't make mistakes :-)), that when C has Set-indexed coproducts, than there is an adjunction between the global section functor C(1, -) : Cod -> Fam and the "coproduct functor" \coprod_{-}(1) : Fam -> Cod. Furthermore, if the coproducts are universal, than these functors are fibred and preserves the monoidal structures, and if additionally all global sections in C are disjoint (i.e. the pullback of two different global section is an initial object) than this adjunction is an equivalence of categories (these results give us approximations C-internal categories ---> C-enriched categories and in the other direction). Best regards, Michal R. Przybylek