From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6642 Path: news.gmane.org!not-for-mail From: mgroth@math.uni-bonn.de Newsgroups: gmane.science.mathematics.categories Subject: universal actions of pseudo-monoids in 'biclosed' monoidal 2-categories Date: Tue, 26 Apr 2011 16:45:18 +0200 (CEST) Message-ID: Reply-To: mgroth@math.uni-bonn.de NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1303862022 11985 80.91.229.12 (26 Apr 2011 23:53:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 26 Apr 2011 23:53:42 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Wed Apr 27 01:53:38 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QEs4X-0006b0-J5 for gsmc-categories@m.gmane.org; Wed, 27 Apr 2011 01:53:37 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58650) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QEs2C-0001BT-TR; Tue, 26 Apr 2011 20:51:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QEs2A-0006hd-Ob for categories-list@mlist.mta.ca; Tue, 26 Apr 2011 20:51:11 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6642 Archived-At: Dear category theorists, I have a question in the context of 'biclosed' monoidal 2-categories. T= o give an analogy, let us begin with one dimension less, i.e., with a close= d monoidal category C. Given an arbitrary object X \in C it is then easy to show that the internal hom-object END(X)=3DHOM(X,X) is canonically a mono= id and, in fact, the terminal example of a monoid acting on $X$. One way to make the latter result precise is as follows. The category Mod(C) of modules in C, where objects are pairs (M,Y) consisting of a monoid M and = a left M-module Y is endowed with a forgetful functor U:Mod(C)-->C. This functor U can also be obtained by first observing that the monoidal unit = S can be endowed with the structure of a monoid making it the initial monoi= d and such that the category S-Mod is isomorphic to C. U is then induced by restriction of scalars along the unique monoid morphisms S-->M and the isomorphism S-Mod\cong C. The statement that the canonical action of END(X) on X is the terminal example of a monoid acting on X is now precisely the statement that it gives us a terminal object of U^{-1}(X). I would now love to have corresponding results in the following 2-categorical situation where there are two 'degenerations': the closedness of the monoidal structure is not expressed by a 2-adjunction but only by a 'biadjunction' and we consider pseudo-monoids instead of monoids. Thus, let us consider a symmetric monoidal 2-category C which is 'biclosed' in the sense that for every X there is a right biadjoint 2-functor to -\otimes X, i.e., we have an internal hom 2-functor HOM(X,-) and natural equivalences of categories Hom(W\otimes X,Y)-->Hom(W,HOM(X,Y)) (where Hom is the enriched hom of C). I would now love to have the following results: i) For an arbitrary object X \in C, the internal hom END(X)=3DHOM(X,X) ca= n be canonically endowed with the structure of a pseudo-monoid. ii) There is a canonical action of the pseudo-monoid END(X) on X induced by the 'biadjunction counit'. iii) This action is 'the' 'biterminal' example of such an action: using a 2-categorical version of the Grothendieck construction one can form the 2-category PsMod(C) of pseudo-modules in C which is again endowed with a projection functor U:PsMod(C)-->C. Given an object X \in C the canonical action of ii) is then a bi-terminal object of U^{-1}(X) in the sense that all hom-categories of morphisms into that object are equivalent to the terminal category. It would be of great help if someone could give me a reference to the literature where such issues are discussed. Just in case that someone of you has too much time I would also like to get something close to iv) The monoidal S unit is 'the' initial example a of pseudo-monoid and w= e have an equivalence of categories S-PsMod \simeq C but i)-iii) are more important to me. Thanks a lot in advance! Best, Moritz Groth. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]