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Stream: learning: questions

Topic: Understanding the definition of Lax Monoidal Functor

Ruby Khondaker (she/her) (May 29 2026 at 13:20):

Recently I have been trying to spread my wings a little by working outside ordinary 1-category theory, hence I'm attempting to understand monoidal categories and coherence. After doing some calculations I think I understand MacLane's proof of coherence for monoidal categories better! There's a section of Eugenia Cheng's "The Joy of Abstraction" which served as a useful guiding principle:

In this sense, one can view the coherence theorem for monoidal categories as saying we want the broad structure of "all formal diagrams commute", and then one can show that the triangle and pentagon identities suffice. Of course, for other structures like braided monoidal categories, one has to weaken this to "all formal diagrams with the same underlying braid commute".

Looking into whether there were coherence theorems for lax monoidal functors, I came across Geoffrey Lewis' PhD Thesis "Coherence for a closed functor", where he gives an example of a diagram obtained from a lax monoidal functor that does not generally commute:

Here $\alpha$ is the functor and $I, I'$ are the monoidal units. Lewis then goes onto define another invariant he calls $\Delta$ which does distinguish the two composites for this diagram and "explains" why they don't commute.

Of course, the commutativity of such a diagram is not part of the lax monoidal functor definition, and indeed this would exclude many common examples of such functors. So then my question is - where do the coherence diagrams for such functors come from?

• It doesn't seem to be the case that one first decides a global coherence that such functors need to obey, and then deduces what diagrams suffice to produce this.
• But it also doesn't seem to be the case that one simply produces all 'small' diagrams that can be made from the coherence transformations and asks them to commute, since then commutativity of the above diagram would be enforced.

Ruby Khondaker (she/her) (May 29 2026 at 13:23):

For me I suppose I've had this experience many times when trying to learn enriched category theory - there comes a point where a wealth of coherence identities appear in the definition of enriched category, enriched functor, enriched natural transformation, but there's not an explanation for why these diagrams are the correct ones. Nor is there usually some characterisation of what the corresponding "global" coherence theorem generated from these identities is. Is this something I should just black-box, then?

Nathanael Arkor (May 29 2026 at 14:57):

There are various ways to arrive at the definition of lax monoidal functor, but a particularly pertinent way is to see them as monoids in a certain multicategory. If $\mathbf A$ and $\mathbf B$ are monoidal categories, then $[\mathbf A, \mathbf B]$ has a natural (convolution) multicategory structure, where the multimorphisms $F_1, \ldots, F_n \to G$ are families $F_1(A_1) \otimes \cdots \otimes F_n(A_n) \to G(A_1 \otimes \cdots \otimes A_n)$, natural in each variable. A monoid in this multicategory is precisely a lax monoidal functor from $\mathbf A$ to $\mathbf B$.

Nathanael Arkor (May 29 2026 at 14:58):

Why do I say this is pertinent? Well, enriched categories can also be seen as monoids in a certain monoidal category (namely, of $\mathscr V$-matrices), which explains where the laws for enriched categories come from.

Nathanael Arkor (May 29 2026 at 14:59):

More generally, enriched categories, enriched functors, and enriched natural transformations are precisely monads, monad morphisms, and monad transformations in the (virtual) double category of $\mathscr V$-matrices. (This is a principle that applies generally for "category-like structures".)

Nathanael Arkor (May 29 2026 at 15:00):

Any time you seem to have unitality and associativity constraints, there's a good chance what you're looking at is a monoid in some monoidal category or multicategory.

Ruby Khondaker (she/her) (May 29 2026 at 15:03):

Oh, that's a little mind-bending to think about. So lax-monoidal functors, which I think of as structure-preserving maps, can also be thought of as monoids, which I think of as structures on their own?

What's this category of V-matrices, if you don't mind expanding?

And when you say this is a principle that applies generally for category-like structures - would a special case of this be that categories are monads in the bicategory of spans?

Nathanael Arkor (May 29 2026 at 16:42):

Indeed: while it is natural to think of lax monoidal functors as "weakly structure-preserving morphisms", you can also think of them as structures in their own right, which is often a helpful perspective!

Nathanael Arkor (May 29 2026 at 16:42):

Ruby Khondaker (she/her) said:

And when you say this is a principle that applies generally for category-like structures - would a special case of this be that categories are monads in the bicategory of spans?

Yes, exactly!

Nathanael Arkor (May 29 2026 at 16:43):

Ruby Khondaker (she/her) said:

What's this category of V-matrices, if you don't mind expanding?

Perhaps one nice introduction is @Dan Marsden's blog post Bicategories, monoids, internal and enriched categories.

Mike Shulman (Jul 30 2026 at 02:22):

Coming to this a little late, but I find that I get more intuition for the notion of lax monoidal functor from the observation that a lax monoidal functor between two monoidal categories is precisely a functor between their underlying multicategories. Here the notion of "functor of multicategories" is the obvious/unavoidable thing that maps objects to objects and multimorphisms to multimorphisms, preserving domains, codomains, identities, and composition.

Mike Shulman (Jul 30 2026 at 06:27):

Another way to explain or justify the correctness of the notion of lax monoidal functor is that it is a lax morphism of algebras for the 2-monad whose algebras are monoidal categories. Of course then you have to justify the latter instead, but I think this is easier because (1) you only have to do it once, rather than separately justifying a notion of "lax morphism" for all the different kinds of structured category in the world, and (2) in the abstract setting it's more obviously correct: there's only one square you can write down that obviously needs a 2-cell in it, and then there's only one cube you can write down involving that square that should obviously commute.

Mike Shulman (Jul 30 2026 at 06:28):

Ruby Khondaker (she/her) said:

there comes a point where a wealth of coherence identities appear in the definition of enriched category, enriched functor, enriched natural transformation, but there's not an explanation for why these diagrams are the correct ones.

In this specific case, I think the usual justification is that if you pick Set as your enriching category, then the axioms reduce precisely to the usual notions of unenriched category, functor, and natural transformation.

Nathanael Arkor (Jul 30 2026 at 07:37):

Mike Shulman said:

Coming to this a little late, but I find that I get more intuition for the notion of lax monoidal functor from the observation that a lax monoidal functor between two monoidal categories is precisely a functor between their underlying multicategories.

Perhaps it's worth noting that this intuition is recovered by the perspective above, because the convolution multicategory $[\mathbf A, \mathbf B]$ is an exponential in the 2-category of multicategories, whence monoids therein (i.e. functors of multicategories from $\mathbf 1$) correspond to functors of multicategories $\mathbf A \to \mathbf B$ by transposition.

Mike Shulman (Jul 30 2026 at 16:45):

I should remember this, especially since there was a talk about exponentiability of virtual double categories at CT26, but which multicategories are exponentiable? I gather from your remark that the underlying multicategory of any monoidal category is exponentiable.

Sridhar Ramesh (Jul 30 2026 at 16:53):

I believe it's precisely the "promonoidal" multicategories which are exponentiable, which includes every monoidal category.

Kevin Carlson (Jul 30 2026 at 16:53):

The exponentiable multicategories are exactly the promonoidals. This is due to Pisani and Ea and I talk about it in our paper they spoke on at CT as well.

Mike Shulman (Jul 30 2026 at 17:05):

Right, thanks.

Noam Zeilberger (Jul 31 2026 at 07:58):

Also worth mentioning another characterization of exponentiable multicategories (which I remember seeing mentioned in Ea's talk) as so-called "malleable" multicategories. The definition of malleable (found in Mario Román's thesis) may be considered as a special case of the notion of Conduché fibration of multicategories: a multicategory $M$ is malleable just in case the unique functor $M \to \mathbf{1}$ to the terminal multicategory $\mathbf{1}$ (with one object and one operation of every arity) is a Conduché fibration. Since Conduché fibrations of multicategories $E \to B$ are in one-to-one correspondence with pseudofunctors $B \to \mathbf{Prof}$ into the 2-multicategory of small categories and multi-profunctors, every malleable multicategory $M$ gives rise to a pseudofunctor $\mathbf{1} \to \mathbf{Prof}$, hence a (unbiased) pseudomonoid in $\mathbf{Prof}$ = promonoidal category.

Nathanael Arkor (Jul 31 2026 at 08:13):

(I would suggest malleability is really the same characterisation: to even talk of promonoidal categories as multicategories requires characterising the image of ProMonCat -> Multicat. I think this is most easily done by identifying multicategories with normal lax monoids in Dist^op, whence it is clear that promonoidal categories are a special case. The invertibility of the structural cells is precisely the condition Mario calls malleability.)

Nathanael Arkor (Jul 31 2026 at 08:14):

I do think the Grothendieck correspondence perspective is a nice one, and it's a pity no-one has written it up in detail in the literature yet. (It's just mentioned in passing in the arXiv version of @Claudio Pisani's Sequential multicategories.)

Nathanael Arkor (Jul 31 2026 at 08:20):

I think it fits in nicely with the perspective of multicategories as normal lax monoids, given that one should be able to generalise the Grothendieck correspondence from (pseudofunctors into MDist :: exponentiable functors into a multicategory) to (normal lax functors into MDist :: arbitrary functors into a multicategory).

Kevin Carlson (Jul 31 2026 at 17:27):

Ea and I thought about this a bit but I think we didn't pin down the right notion of "2-virtual double category" to receive an indexed form of a virtual double functor, which is the case we were interested in, so let it be.

Claudio Pisani (Aug 01 2026 at 13:45):

Perhaps it is worth mentioning also that in the unbiased approach, wherein a symmetric multicategory is a double functor $p:\mathbb M \to \mathbb P\rm b$, a multicategory is exponentiable iff the loose part of $p$ is an ordinary Conduché fibration.
Indeed, the loose part of $\mathbb M$ is formed by the general operations $\alpha$ of the multicategory (parametrized by the functions $p\alpha$), which are families of elementary operations (those for which $p\alpha$ has a terminal codomain), so that we get the "malleability" condition.

In particular, since Conduché fibrations include fibrations,

1. representable (aka monoidal) symmetric multicategories are exponentiable because they are those for which the loose part of $p$ is an opfibration;
2. cocartesian (aka sequential) multicategories are exponentiable because the loose part of $p$ is a fibration; for instance, if $\cal C_\triangleright$ is the cocartesian multicategory on $\cal C$, then $\cal D_\triangleright ^{\cal C_\triangleright} = (\cal D^{\cal C})_\triangleright$, while $\rm Set^{\cal C_\triangleright}$ has functors $F:\cal C\to \rm Set$ as objects, and arrows $F_i \to G$ are natural transformations $(\Pi_i F_i) \Delta \to G$.