From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9410 Path: news.gmane.org!.POSTED!not-for-mail From: Uwe Egbert Wolter Newsgroups: gmane.science.mathematics.categories Subject: Kleisli categories for monads on presheaves Date: Tue, 07 Nov 2017 09:39:05 +0100 Message-ID: Reply-To: Uwe Egbert Wolter NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit X-Trace: blaine.gmane.org 1510094945 25051 195.159.176.226 (7 Nov 2017 22:49:05 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 7 Nov 2017 22:49:05 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Nov 07 23:49:01 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1eCCg0-0006CE-2V for gsmc-categories@m.gmane.org; Tue, 07 Nov 2017 23:49:00 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58831) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eCCgx-0002rP-Fk; Tue, 07 Nov 2017 18:49:59 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eCCfW-00005x-NG for categories-list@mlist.mta.ca; Tue, 07 Nov 2017 18:48:30 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9410 Archived-At: Dear all, many thanks for the very useful replies concerning my question about Grothendieck-Yoneda-Colimits. Now another question on top of it: I'm more on the "applied side" and interested in syntactic representation of things. For a many-sorted algebraic signature \Sigma with a finite set (discrete category) S of sorts the construction of \Sigma-terms gives us a monad T_\Sigma:Set^S -> Set^S. The syntactic category with S^* as set of objects, finite tuples of terms as morphisms and "composition by substitution" (Lawvere) can be seen as a subcategory of the Kleisli category of this monad. We generalized recently the concept of algebraic signatures and algebras to graphs: input and out put arities of operations are graphs as well as the carriers of algebras are graphs. We describe the construction of "graph terms" and get a monad on Set^B with B the category given by two parallel arrows s,t:E->V. What we would like to have is a nice generalization of the construction of syntactic Lawvere categories to this case. I learned now that "the category [C^op,Set] is the free colimit completion of C". My question is, if there are similar results for the Kleisli category of a monad on [C^op,Set]? Best regards Uwe [For admin and other information see: http://www.mta.ca/~cat-dist/ ]