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 $F,G\colon\mathcal{C}\rightrightarrows\mathcal{D}$?

  • $\begingroup$ A monoidal functor $\mathcal C \to \mathcal D$ is the same thing as a 2-functor $B\mathcal C \to B \mathcal D$ between the corresponding 1-object 2-categories, and a monoidal natural transformation between such corresponds to a 2-natural transformation. So it should be describable using a $Cat$-enriched end formula. The tricky thing is that everything needs to be suitably "weak" or "derived" rather than "strict", and I'm not sure precisely what things will look like once such details are ironed out. $\endgroup$
    – Tim Campion
    Aug 23, 2021 at 16:12
  • $\begingroup$ @TimCampion I think this might not work: every monoidal natural transformation $F\Rightarrow G$ gives an oplax $2$-natural transformation $\mathbf{B}F\Rightarrow\mathbf{B}G$, but these are special among all oplax $2$-natural transformations (a reference is Prop. 4.3.11 of Johnson–Yau). So while we can compute $\mathsf{OplaxNat}(\mathbf{B}F,\mathbf{B}G)$ via a weak biend, we have $\mathsf{OplaxNat}(\mathbf{B}F,\mathbf{B}G)\ncong\mathrm{Nat}^{\otimes}(F,G)$ with the former strictly larger than the latter :/ $\endgroup$
    – Emily
    Aug 23, 2021 at 21:19
  • $\begingroup$ @TimCampion I think this would work for "icons" though. I wonder if there's an end formula for them 🤔 $\endgroup$
    – Emily
    Aug 23, 2021 at 21:21
  • $\begingroup$ I am pretty sure that the answer is "No". $\endgroup$ Aug 26, 2021 at 14:49
  • $\begingroup$ @MartinBrandenburg I suspect the same. I think maybe a similar thing to try would be to embed monoidal categories into multicategories and consider some appropriate notion of "multiend", though perhaps it runs in the same kind of problem... (Besides, there's approximately nothing (AFAIK) developed about co/limits in multicategories...) $\endgroup$
    – Emily
    Aug 27, 2021 at 3:26

1 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).

  • $\begingroup$ Thanks! Some time ago I actually tried the exact same thing (in a reply to your question about monoidal Kan extensions), even defining "monoidal ends" too, but gave up on it when I noticed that it doesn't work when $F$ and $G$ are only lax monoidal! $\endgroup$
    – Emily
    Aug 27, 2021 at 21:46
  • $\begingroup$ By the way, is it necessary for $\mathcal{D}$ to be Cartesian? I think we can define 1) a lax monoidal dinatural transformation to be a dinatural satisfying these conditions: (a), (b); 2) monoidal ends via bijections of the form $$h_{\int^{\otimes}_{A\in\mathcal{C}}D^A_A}\cong\mathrm{DiNat}^{\otimes,\mathrm{lax}}(\Delta_{X},D),$$ and then 3) lax monoidal wedges as lax monoidal dinaturals from $\Delta_X$ (equipped with the lax monoidal structure uniquely determined by the monoid structure of $X$) to the diagram $D$. $\endgroup$
    – Emily
    Aug 27, 2021 at 21:48
  • $\begingroup$ This means those will satisfy the conditions here, recovering yours when $\mathcal{D}$ is Cartesian, but also working in general! $\endgroup$
    – Emily
    Aug 27, 2021 at 21:50
  • $\begingroup$ How do you formulate the properties 1 and 2 above (sorry I don't want to think about general dinatural transformations and the double index notation is horrible) when D is not Cartesian? Or do you want the monoidal end to be a comonoid object? Then T has a comonoid structure and it works fine. $\endgroup$ Aug 28, 2021 at 6:42
  • $\begingroup$ See the image in my last comment above (i.e. here). If one departs from limits or ends by replacing di/natural transformations with monoidal di/natural transformations, then $T$ should be either a monoid or a comonoid object (or a bimonoid object, but let's not talk about that), making $\Delta_T$ into a lax or an oplax monoidal functor. $\endgroup$
    – Emily
    Aug 28, 2021 at 19:40

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