skew-monoidal category?

["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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