Strong monad
# Definition
A strong monad over a monoidal category $ℂ$ is a monad $(T,μ,η)$ endowed with two strengths $$\operatorname{st_l} \colon X ⊗ T(Y) → T(X ⊗ Y) \mbox{ and } \operatorname{st_r} \colon T(X) ⊗ Y → T(X ⊗ Y).$$ These strengths must satisfy axioms for
- unitality, $st_l ⨾ T(λ) = λ$ and $st_r ⨾ T(ρ) = ρ$ ;
- associativity, $(id ⊗ st_l) ⨾ st_l ⨾ T(α) = α ⨾ st_l$ and $α ⨾ (st_r ⊗ id) ⨾ st_r = st_r ⨾ α$ ;
- preserving units, $η_{A ⊗ B} = (id_A ⊗ η_B) ⨾ st_l = (η_A ⊗ id_B) ⨾ st_r$;
- preserving multiplication, $st_r ⨾ st_r ⨾ μ = (μ ⊗ id) ⨾ st_r$ and $st_l ⨾ st_l ⨾ μ = (id ⊗ μ) ⨾ st_l$ ;
- and compatibility, $(id_A ⊗ st_r) ⨾ st_l = (st_l ⊗ id_C) ⨾ st_r$.