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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.