coMalcev monoidal Monads

Stanisław Szawiel
Monday, July 22, 2013 - from 16:30 to 16:55
Colored species can be thought of as Eilenberg-Moore algebras for the symmetrization monad $\cal S$ on the category (typed) signatures. The monoidal structure on signatures lifts to the category of colored species making it into an Eilenberg-Moore object in the 2-category of monoidal categories and lax monoidal functors. This is an unexpected phenomenon, since the symmetrization monad $\cal S$ on signatures is lax, and not oplax monoidal with respect to the substitution tensor. However, the symmetrization monad $\cal S$ has an additional (lax monoidal) operation $\zeta: {\cal S}\rightarrow{\cal S}^3$ that satisfies conditions dual to those satisfied by the usual Malcev operations: \begin{center} \xext=1000 \yext=600 \begin{picture}(\xext,\yext)(\xoff,\yoff) \settripairparms[-1`-1`-1`-1`-1;500] \putVtrianglepair(0,0)[S^2`S^3`S^2`S;S\mu`\mu_S`S\eta`\zeta`\eta_S] \putmorphism(500,500)(1,0)[\phantom{S^3}`S^2`\mu_S]{500}{1}a \end{picture} \end{center} This is why such an operation will be called {\em coMalcev}. In my talk I will show why a coMalcev monoidal operation on a lax monoidal monad ensures that the usual Eilenberg-Moore object can be equipped with a monoidal structure making it the Eilenberg-Moore object in the 2-category of monoidal categories. If time permits I will sketch how general this phenomenon is and show some other examples of coMalcev monads. (Joint work with Marek Zawadowski.)