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Stream: theory: mathematics

Topic: relating left and right actions

Matteo Capucci (he/him) (Nov 28 2023 at 10:51):

Sometimes an object $X$ is acted upon by another object $M$ both on the left and the right, but instead of forming a bimodule structure (meaning $(mx)n = m(xn)$), these two actions are such that $mx = xm$. This is clearly a trivial concept in sets, since this latter equation implies $X$ is in fact receiving a left action of $M$ (albeit factoring through $M/[M,M]$.

In higher settings, however, this isn't necessarily the case.
The first place I encountered this is [[actegories]], in which one might have non-invertible maps $mx \to xm$ (or invertible but non-trivial ones, chiefly in the case of actions of braided monoidal categories, which can be 'mirrored'). The second place is in _The formal theory of monads II_, where Lack and Street characterize the Kleisli cocompletion of a 2-category as the 2-category with object monads and morphisms 'intertwined modules', i.e. a 1-cell $(a,t) \to (b,s)$ is an object $x$ with a left $t$-action and a right $s$-action and a map $\lambda : xt \to sx$ satisfying some laws.

So, I'm curious if people have seen such structures around, or just have something interesting to say about these.

Ralph Sarkis (Nov 28 2023 at 11:56):

The example that came to my mind is the action of functors on natural transformations:
image.png

Morgan Rogers (he/him) (Nov 29 2023 at 09:46):

Looking at the type in isolation it looks like a distributive law, or a commutativity rewrite rule/lax commutativity arrow.