Re: skew-monoidal category?

[Robin Houston <robin.houston@gmail.com>]

I don’t know the full history of this idea; I came across it in the
work of
Marco Grandis on directed homotopy, such as
http://www.dima.unige.it/~grandis/LCat.pdf

Grandis considers a generalisation of bicategories, rather than monoidal
categories, but of course the monoidal version is the special case of a
bicategory with a single object.

Actually Grandis has the maps you call eta and eps going the other way, so
perhaps the precise notion you’re describing is more closely related to
Burroni’s 1971 notion of ‘pseudocategory’.

Anyway I expect some of the more knowledgeable members of this list will be
able to give you a better answer!

All the best,
Robin

2011/12/2 Szlachanyi Kornel <szlach@rmki.kfki.hu>

> 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
>
>

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