Prostrong monad

Last updated May 2, 2024

A prostrong monad over a promonoidal category $𝕍$ is a monad $(T,μ,η)$ endowed with two strengths $$st_l﹕ 𝕍(X,Y;Z) → 𝕍(TX,Y;TZ) \mbox{ and } st_r﹕ 𝕍(X,Y;Z) → 𝕍(X,TY;TZ).$$ These must satisfy the following axioms.

1. Unitality, $T(λ(u|v)) = λ(u|st_l(v))$ and $T(ρ(u \mid v)) = ρ(u \mid st_r(v))$.
2. Associativity,$$α(st_l(v) \mid st_l(w)) = \left{ \begin{aligned} & let\\ α(v|w) → (w’ \mid v’)\\ in\\ \\ & (w’ \mid st_l(v’)) \end{aligned} \right};$$
3. Neutrality, $$η > st_l(p) = p < η\mbox{ and }η > st_r(p) = p < η\\ ;$$
4. Multiplicativity, $$μ > st_l(p) = st_l(st_l(p)) < μ\mbox{ and }μ_r > st_r(p) = st_r(st_r(p)) < μ\\ ;$$
5. Coherence,$$α(st_r(v)\mid st_l(w)) = \left{\begin{aligned} & let\\ α(v \mid w) → (w’ \mid v’)\\ in \\ & (st_r(w’) \mid st_l(v’)) \end{aligned}\right}.$$

• Personal communication, James Hefford.
• Note on Monoidal Monads (Day, 1976). Day provides a definition of promonoidal promonad, a prostrong monad is its weakened version.