From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7098 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: skew-monoidal category? Date: Tue, 6 Dec 2011 10:30:43 +0100 Message-ID: <6CE8DCE6-3790-46E3-863E-82571E4BE2C9@dima.unige.it> References: Reply-To: Marco Grandis NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1323193912 1813 80.91.229.12 (6 Dec 2011 17:51:52 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 6 Dec 2011 17:51:52 +0000 (UTC) To: Kornel SZLACHANYI , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Dec 06 18:51:48 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 1RXzBD-00049h-CY for gsmc-categories@m.gmane.org; Tue, 06 Dec 2011 18:51:47 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:47727) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RXz7x-0007xZ-3i; Tue, 06 Dec 2011 13:48:25 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RXz7u-0006H7-Rn for categories-list@mlist.mta.ca; Tue, 06 Dec 2011 13:48:22 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7098 Archived-At: (Subject: lax 2-categories; lax cubical categories) Sorry for the delay. You can find part of the story (as I know it) in the Introduction of =20 my paper [3], cited by Robin Houston. 1. Burroni [1] introduced in 1971, a 'pseudocategory', with the =20 following directions for the comparison cells: f --> f*1, f --> 1*f, (h*g)*f --> h*(g*f). Borceux, in his text on category theory, mentions a similar notion of =20= 'lax category', in a remark after the definition of bicategory. 2. Leinster's book [2] introduces a 'lax bicategory' as an 'unbiased' =20= structure where all multiple compositions are assigned and there are comparison cells from each iterated =20 composition to the corresponding multiple composition, as in the following examples: (k*h*g)*f --> k*h*g*f, (1*(h*g*1))*f --> h*g*f. This has the advantage of a clear formal criterion for the direction =20 of comparisons. 3. My paper [3] is about a fundamental 'd-lax 2-category' for a =20 'directed space'. The arrows are its directed paths, composed by concatenation; a cell is a homotopy class of =20 homotopies of paths; we have comparison cells with direction: 1 * a --> a --> a * 1, a*(b*c) -> (a*b)*c, because these (directed) homotopies can only move towards 'hastier' =20 concatenations of paths. Eg the lazy path 1 * a sleeps half of the time at its beginning, =20 then runs to reach its end; the original a is hastier (makes at each instant a longer way); but a * 1 is even hastier: it runs twice as fast, then can sleep =20 half of the time at its end. The term 'd-lax' is meant to refer to such a direction of =20 comparisons, motivated by directed homotopy. But you can easily define n-ary concatenations of paths, by the =20 obvious n-partition of the standard interval. In [3] there is also a fundamental 'unbiased d-lax 2-category', where =20= we also have comparison cells: a*(b*c) --> a*b*c --> (a*b)*c, 4. It seems to be difficult to proceed this way to higher fundamental =20= categories for a directed space X. In a recent paper [4], I have taken a different way: an (infinite =20 dimensional) fundamental lax cubical category. An n-cube is a map from the standard directed n-cube [0, 1]^n to =20 X; obviously, they have n concatenation laws (but letting symmetries in, we can reduce =20 everything to one of them). Then we need comparisons, with a strict law; these are obtained by =20 reparametrisation of the standard cube, and behave quite differently from those of point 3: - they are invertible for associativity, where you can reparametrise =20 both ways, - they are directed for unitarity, where you can reparametrise a =20 cube a so to make it lazy at the beginning or the end, but you cannot destroy sleeping times once =20 they are there (!), - they are identical for interchange. Best regards Marco Grandis [1] A. Burroni, T-cat=E9gories, Cah. Topol. G=E9om. Diff=E9r. 12 (1971), = =20 215-321. [2] T. Leinster, Higher operads, higher categories, Cambridge =20 University Press, Cambridge 2004. [3] M. Grandis, Lax 2-categories and directed homotopy, Cah. Topol. Geom. Differ. Categ. 47 (2006), 107-128. http://www.dima.unige.it/~grandis/LCat.pdf [4] M. Grandis, A lax symmetric cubical category associated to a =20 directed space, to appear in Cahiers. http://www.dima.unige.it/~grandis/FndLx.pdf On 2 Dec 2011, at 11:43, Szlachanyi Kornel wrote: > Dear All, > > I wonder if the following notion has already a name and disscussed > somewhere: It is like a monoidal category but the associator and units > are not invertible. (Lax monoidal categories share this property =20 > but they > seem to treat the units differently.) It has left and right =20 > versions, the > "right-monoidal" category consists of > > a category C, > a functor C x C --> C, |--> M*N, > an object R > and natural transformations > gamma_L,M,N: L*(M*N) --> (L*M)*N > eta_M: M --> R*M > eps_M: M*R -->M > > satisfying 5 axioms (1 pentagon, 3 triangles and eps_R o eta_R =3D R) =20= > that > are obtained from the usual monoidal category axioms by expressing > everything in terms of the associator, the right unit (eps), and the > inverse left unit (eta) never using their inverses. > > I find this structure interesting because of the following: > > Thm: Let R be a ring. Closed right-monoidal structures on the =20 > category M_R > of right R-modules are (up to approp. isomorphisms on both sides) =20 > precisely > the right R-bialgebroids. > > (The ordinary monoidal structure remains hidden in the special =20 > nature of > M_R.) > > I would thank for any suggestion. > > Kornel Szlachanyi > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]