(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 my paper [3], cited by Robin Houston. 1. Burroni [1] introduced in 1971, a 'pseudocategory', with the 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 'lax category', in a remark after the definition of bicategory. 2. Leinster's book [2] introduces a 'lax bicategory' as an 'unbiased' structure where all multiple compositions are assigned and there are comparison cells from each iterated 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 of comparisons. 3. My paper [3] is about a fundamental 'd-lax 2-category' for a 'directed space'. The arrows are its directed paths, composed by concatenation; a cell is a homotopy class of 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' concatenations of paths. Eg the lazy path 1 * a sleeps half of the time at its beginning, 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 half of the time at its end. The term 'd-lax' is meant to refer to such a direction of comparisons, motivated by directed homotopy. But you can easily define n-ary concatenations of paths, by the obvious n-partition of the standard interval. In [3] there is also a fundamental 'unbiased d-lax 2-category', where 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 categories for a directed space X. In a recent paper [4], I have taken a different way: an (infinite dimensional) fundamental lax cubical category. An n-cube is a map from the standard directed n-cube [0, 1]^n to X; obviously, they have n concatenation laws (but letting symmetries in, we can reduce everything to one of them). Then we need comparisons, with a strict law; these are obtained by reparametrisation of the standard cube, and behave quite differently from those of point 3: - they are invertible for associativity, where you can reparametrise both ways, - they are directed for unitarity, where you can reparametrise a cube a so to make it lazy at the beginning or the end, but you cannot destroy sleeping times once they are there (!), - they are identical for interchange. Best regards Marco Grandis [1] A. Burroni, T-catégories, Cah. Topol. Géom. Différ. 12 (1971), 215-321. [2] T. Leinster, Higher operads, higher categories, Cambridge 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 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 but they seem to treat the units differently.) It has left and right versions, the "right-monoidal" category consists of

a category C, a functor C x C --> C, <M,N> |--> 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 = R) 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 category M_R of right R-modules are (up to approp. isomorphisms on both sides) precisely the right R-bialgebroids.

(The ordinary monoidal structure remains hidden in the special nature of M_R.)

I would thank for any suggestion.

Kornel Szlachanyi

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]