On indexed actions

Abstract

We present some laws relating the $\Cat$-indexed categories of left, right and bi-actions: by defining $(A\comp M)x = Mx^{Ax}$ one gets a biclosed monoidal action of $\Set^{X\op}$ on $(\Set^X)\op$, while $\B X$ and $\Cat/X$ act (partially) on their opposites by exponentials; both the inclusions $(\B X,\B X)\to (\Set^{X\op},\Set^X) \to (\Cat/X,\Cat/X)$ preserve the (cartesian) monoidal structures and the actions, and the same holds for substitutions along functors. These strong morphisms of strong indexed monoidal actions have in fact a wider range of applications; in particular, replacing $\Set$ with any (co)complete symmetric monoidal closed category $\V$, we consider the pair of indexed categories $(\V_0^{X\op},\V_0^X ; X\in\Cat)$ with the pair of biclosed indexed monoidal actions of each one on the opposite of the other one and its formal relationships with biactions and constant actions. Some of the resulting laws also hold in a fragment of biclosed bicategory (with an object supporting a symmetric monoidal category) and are taken, in the second part, as the basis for developing some abstract category theory. Finally, we add $\Set^{X\op\tm X}$ to the picture and give a symmetrical version of the comprehension adjunction.