Re: skew-monoidal category?

[Marco Grandis <grandis@dima.unige.it>]

(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/ ]