End formulas for sets of monoidal natural transformations

Question

Perhaps the characteristic feature of the theory of ends is that they are extremely useful for computing sets of transformations between two functors. For example, one has the formulas \begin{align*} \mathrm{Nat}(F,G) &\cong \int_{A\in\mathcal{C}}\mathrm{Hom}_{\mathcal{C}}\left(F_{A},G_{A}\right),\\ \mathrm{DiNat}(F,G) &\cong \int_{A\in\mathcal{C}}\mathrm{Hom}_{\mathcal{C}}\left(F^{A}_{A},G^{A}_{A}\right), \end{align*} see Coend Calculus, Theorem 1.4.1 and Example 1.4.4.

Is there a similar end formula for the set $\mathrm{Nat}^\otimes(F,G)$ of monoidal natural transformations between two strong monoidal functors $F,G\colon\mathcal{C}\rightrightarrows\mathcal{D}$?

Answer 1

Monoidal ends

Let $(\mathcal{C},\otimes)$ be a monoidal category and let $(\mathcal{D},\times)$ be a cartesian monoidal category. Let $(X,\eta,\mu) : (\mathcal{C}^{\mathrm{op}},\otimes) \times (\mathcal{C},\otimes) \to (\mathcal{D},\times)$ be a lax monoidal functor. A wedge to $(X,\eta,\mu)$ is a wedge to the underlying functor $X : \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathcal{D}$, thus consisting of an object $T \in \mathcal{D}$ and a family of morphisms $(w_A : T \to X(A,A))_{A \in \mathcal{C}}$, such that the following properties hold:

1. $w_1 : T \to X(1,1)$ is equal to $$T \xrightarrow{\exists!} 1 \xrightarrow{\eta} X(1,1).$$

2. $w_{A \otimes B} : T \to X(A \otimes B, A \otimes B)$ is equal to $$T \xrightarrow{~(w_A,w_B)~} X(A,A) \times X(B,B) \xrightarrow{~~\mu~~} X(A \otimes B, A \otimes B).$$

A universal wedge, i.e. end, is defined as usual. It is easy to see that if $\mathcal{C}$ is small and $\mathcal{D}$ is complete, then any lax monoidal functor has an end. We can denote it by $\int (X,\eta,\mu)$.

Since it is a common practice (sigh) to ignore forgetful functors and just write $X$ both for the functor and the monoidal functor, some people will prefer to call this a "monoidal wedge" and a "monoidal end", the latter then being denoted by something like $\int^{\otimes} X$. I do not know if this concept has appeared elsewhere, I just made it up to answer the question below.

Monoidal natural transformations

Now let $(\mathcal{C},\otimes)$, $(\mathcal{C}',\otimes)$ be two monoidal categories and $(F,\eta_F,\mu_F),(G,\eta_G,\mu_G) : (\mathcal{C},\otimes) \to (\mathcal{C}',\otimes)$ be two strong monoidal functors. Consider the functor $X : \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$ defined on objects by $$X(A,B) := \mathrm{Hom}(F(A),G(B)).$$ We equip it with the following lax monoidal structure: $$\eta_X : 1 \to X(1,1)$$ corresponds to the isomorphism $\eta_G \circ \eta_F^{-1} : F(1) \to 1 \to G(1)$, and $$\mu_X : X(A,A') \times X(B,B') \to X(A \otimes B,A' \otimes B')$$ maps a pair of morphisms $f : F(A) \to G(A')$, $g : F(B) \to G(B')$ to the morphism $$F(A \otimes B) \xrightarrow{\mu_F^{-1}} F(A) \otimes F(B) \xrightarrow{f \otimes g} G(A') \otimes G(B') \xrightarrow{\mu_G} G(A' \otimes B').$$ One needs to check the coherence conditions in the definition of a lax monoidal functor, I will not do this here.

It is straight forward to check that $\int (X,\eta,\mu)$ is the set of morphisms $(F,\eta_F,\mu_F) \to (G,\eta_G,\mu_G)$ (aka monoidal natural transformations).