From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8678 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: categories of models of cartesian PROPs Date: Mon, 17 Aug 2015 16:14:53 +0800 Message-ID: Reply-To: John Baez NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1439990017 3125 80.91.229.3 (19 Aug 2015 13:13:37 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 19 Aug 2015 13:13:37 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Wed Aug 19 15:13:27 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.19]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ZS3BG-0005JY-Jf for gsmc-categories@m.gmane.org; Wed, 19 Aug 2015 15:13:26 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54638) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ZS3A0-0000wt-Eq; Wed, 19 Aug 2015 10:12:08 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ZS3A0-00051h-Q0 for categories-list@mlist.mta.ca; Wed, 19 Aug 2015 10:12:08 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8678 Archived-At: Hi - Here are two questions: Suppose you have a category with finite products, say T, and a symmetric monoidal category, say C. Let [T,C] be the category where objects are symmetric monoidal functors from T to C, morphisms are monoidal natural transformations. *1. What structure beyond a mere category does [T,C] automatically get in this sort of situation?* *2. What further structure do we get if C has some particular class of limits or colimits?* I haven't thought about this much. Even if T were just symmetric monoidal, I think [T,C] should get a symmetric monoidal structure due to "pointwise multiplication", just as the set of homomorphisms from one commutative monoid to another becomes a commutative monoid where fg(x) := f(x) g(x) Should [T,C] also have some sort of "comultiplication"? What extra benefits do we get from T being cartesian? Here's why I care: My student Brendan Fong wrote a masters' thesis about Bayesian networks, which he's trying to polish up and publish. In the new improved version, he'll associate to any Bayesian network a category with finite products, say T. This plays the role of a "theory". An assignment of probabilities to random variables consistent with this theory is a symmetric monoidal functor from T to C, where C is some symmetric monoidal category - but not cartesian! - category of probability measure spaces and stochastic maps. So, [T,C] plays the role of the "category of models of T in C". It would be nice to know the properties of [T,C] that follow instantly from what I've said, not reliant on any more detailed information about T and C. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]