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nCatLab states that a monoid in the Day-convolution monoidal category is equivalent to a lax monoidal functor. In the Haskell community this result is used to explain the construction of the applicative functor. Rivas and Jaskelioff show that a monoid in the endofunctor category $[Set, Set]$ with Day convolution is equivalent to a strong lax monoidal functor. They need strength to get the natural transformation $Id \to F$ from the (lax) preservation of the unit $1 \to F(1)$. I'm trying to understand why they need strength even though the nCatLab result doesn't. Granted, nCatLab deals with enriched functors, and self-enrichment is related to strength, but functors in $Set$ are both strong and self-enriched.

Instead of starting from $[Set Set]$, I'd like to start from a monoidal category $C$ and work with the functor category $[C, Set]$ with Day convolution. A monoid is a functor $F$ in this category, together with two natural transformations. I'm interested in one of them.

The unit under Day convolution is $C(I, -)$ where $I$ is the unit object in $C$. Consider the set of natural transformations from $C(I, -)$ to $F$:

$[C, Set](C(I), -), F) = \int_a C(I, a) \to F(a)$

By Yoneda, this is equal to $F(I)$.

A function from $1$, the singleton set, will pick up a natural transformation in $[C, Set](C(I, -), F)$ and an element of $F(I)$:

$1 \to F(I)$

This is exactly one of the morphisms defining the lax monoidal functor. It seems like, in this formulation, strength is not needed. (It's not needed in the rest of the proof either.) What am I missing?

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  • $\begingroup$ It seems to me that you also answered your own question with "functors in Set are both strong and self-enriched". If Rivas and Jaskelioff used strength when talking about Set, then they didn't need to, since all functors $\mathrm{Set}\to\mathrm{Set}$ are strong. Similarly, the nLab is talking only about enriched functors, hence in the case $C=V$ they are also automatically strong. $\endgroup$ – Mike Shulman Jan 30 '17 at 17:53
  • $\begingroup$ Is it correct to say that all functors $[C, D]$ between monoidal categories are strong as long as $D$ is closed? Closedness let us curry strength: $a \to F(b) \to F(a \otimes b)$ and we can get it by lifting $(a \otimes -)$. $\endgroup$ – Bartosz Milewski Jan 31 '17 at 19:54
  • $\begingroup$ I don't know what it means to talk about a functor between two arbitrary monoidal categories being strong. The basic definition of strong functor is only for an endofunctor of a monoidal category. $\endgroup$ – Mike Shulman Feb 1 '17 at 0:27
  • $\begingroup$ Good point! I guess what I'm trying to say is covered in nCatLab using the language of (self-) enriched categories. There is a hint though that strength could make sense outside of endofunctors: 'More generally, the notion makes sense not just for endofunctors of $V$, but for functors between any categories that are “tensored over $V$.”' $\endgroup$ – Bartosz Milewski Feb 1 '17 at 1:31
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You are correct.

The most straightforward structure that you have on $[C, Set]$ is that of a multicategory (https://ncatlab.org/nlab/show/multicategory). Given functors $F_1, F_2, ..., F_n$ (n can be 0) and $F$ the set of multimorphisms $(F_1, F_2, ..., F_n) \rightarrow F$ is defined as the set $$Nat(F_1(X_1)\otimes ... \otimes F_n(X_n), F(X_1\otimes ... \otimes X_n)).$$ The multicategory structure is obvious.

The notion of monoid makes sense in any multicategory. Particularly, a monoid in the multicategory $[C, Set]$ is exactly a lax monoidal functor (not necessarily strong).

Furthermore, via Day convolution $[C, Set]$ happens to be a representable multicategory. More concretely, pretty much by (one of) the definitions of the Day convolution we have universal natural transformations $$F_1(X_1)\otimes ... \otimes F_n(X_n) \rightarrow (F_1\otimes_{Day}...\otimes_{Day}F_n)(X_1\otimes ... \otimes X_n)$$

which is the property guaranteeing representability (See C. Hermida, Representable multicategories). Specifically, the multicategory structure on $[C, Set]$ is represented by the Day monoidal structure on $[C, Set]$, which of course just means that $$[C, Set]_{Mult}((F_1, F_2, ..., F_n), F) \cong [C, Set](F_1\otimes_{Day}...\otimes_{Day}F_n, F).$$

This implies that a monoid in the multicategory [C, Set], i.e. a lax monoidal functor as noted before, is the same as a monoid in the Day-convolution monoidal category [C, Set]. There is no need of further conditions.

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