Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial strength of $T$. A good reference for these notions is the paper Logical relations for monadic types by J. Goubault-Larrecq, S. Lasota, D. Nowak. It is well-known that $\sigma$ corresponds to a natural transformation

$\tau_{A,B} : \underline{\hom}(A,B) \to \underline{\hom}(T(A),T(B))),$

making $T$ a $C$-functor (at least in the case that $C$ is symmetric, see the early papers by Anders Kock). My first question is: Is it also known and written down somewhere that $\sigma'$ corresponds $1:1$ to a natural transformation

$\tau'_{A,B} : T(\underline{\hom}(A,B)) \to \underline{\hom}(A,T(B)),$

such that two certain diagrams commute? In the introduction of Kock's paper Strong functors and monoidal monads this is mentioned without proof.

Recall that $\sigma$ and $\sigma'$ are called compatible (so that $T$ becomes a monoidal monad) if the following diagrams commute.

$$\begin{matrix} A \otimes T(B) \otimes C & \xrightarrow{\sigma} & T(A \otimes B) \otimes C \\\\ \sigma' \downarrow ~ & & ~ \downarrow \sigma' \\\\ A \otimes T(B \otimes C) & \xrightarrow{\sigma} & T(A \otimes B \otimes C)\end{matrix} ~~~~~~ (\heartsuit) $$

$$\begin{matrix} T(A) \otimes T(B) & \xrightarrow{\sigma} & T(T(A) \otimes B) & \xrightarrow{\sigma'} & T^2(A \otimes B) \\\\ \sigma' \downarrow ~ & & & & ~ \downarrow \mu \\\\ T(A \otimes T(B)) & \xrightarrow{\sigma} & T^2(A \otimes B) & \xrightarrow{\mu} & T(A \otimes B) \end{matrix} ~ ~ ~ ~ ~ ~ ~ ~(\lozenge)$$

Question. What are the corresponding compatibility diagrams for $\tau$ and $\tau'$? I guess that $(\lozenge)$ corresponds to the diagram

$$\begin{matrix} T(\underline{\hom}(T(A),T(B))) & \xrightarrow{\tau'} & \underline{\hom}(T(A),T^2(B)) \\\\ \tau \uparrow ~ & & ~ \downarrow \mu \\\\ T(\underline{\hom}(A,B)) & & \underline{\hom}(T(A),T(B)) \\\\ \tau' \downarrow ~ & & ~ \uparrow \mu \\\\ T(\underline{\hom}(A,T(B))) & \xrightarrow{\tau} & \underline{\hom}(T(A),T^2(B)) \end{matrix}$$

But I cannot even guess what $(\heartsuit)$ should correspond to.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.