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Stream: learning: questions

Topic: Strictness, 2D TQFTs and Frobenius objects

Marco Vianello (Nov 27 2025 at 03:37):

Monoidal categories and monoidal functors come in many flavors. The former can be weak or strict, while the latter can be lax, strong or even strict themself.

In it's Frobenius Algebras and 2D Topological Quantum Field Theories J. Kock defines a TQFT as a symmetric monoidal functor from the symmetric monoidal category $\mathrm{Bord}(2)$ of $2$ -dimensional bordisms (or its strictification) to the symmetric monoidal category $\mathrm{Vect}_{\mathbb F}$ of vector spaces over the field $\mathbb F$ (or its strictification).

Looking at the book gives the impression that the author deliberately restricts the discussion to the strict case (no pun intended), probably to ease the exposition. What I mean by that is that the relevant categorical toolbox is defined and used mainly in its strict formulation.

So (say) the hexagons equations are not required in the definition of symmetric monoidal category, the associativity and unitality constraints are not required in the definition of (symmetric) monoidal functor (more precisely, monoidal functors are by definition _strict_), and so on. Some blurb here and there introduce the non-strict notions and clarify that we are not always allowed to treat non-strict stuff as if it was strict, but that's it.

In particular, the category $\mathrm{TQFT}(2,\mathbb F)$ of $2$ -dimensional TQFTs is defined in the book as the functor category

$\mathrm{Repr}_{\mathbb F}(\mathrm{Bord}(2)) = \mathrm{SymMonCat}(\mathrm{Bord}(2),\mathrm{Vect}_{\mathbb F})$

of strict monoidal functors between the (strictification of, as he puts it) category of $2$ -dimensional cobordisms and the (strictification of) category of $\mathbb F$ -vector spaces.

A somewhat confusing remark appears as Note 3.2.53 just before the official definition of TQFTs in Chapter 3 and just after having defined the category of symmetric strict (at this point) monoidal functors and monoidal transofrmations between a symmetric (strict?) monoidal category $\mathcal C$ to $\mathrm{Vect}_{\mathbb F}$ (i.e., the linear representations of $\mathcal C$ ). In particular, the author writes that

We consider strict functors. In fact our $\mathrm{Vect}_{\mathbb F}$ is a strictification of the true monoidal category of vector spaces... Otherwise we should consider strong monoidal functors.

Some perplexities of mine:

The Note seems to suggest that by considering the strictification of the target cateogry we are allowed to forget about non-strict functors. I was wondering if this is true, and how would this work in practice. By the way, I suspect it is not.
The main theorem in the book states, in so many words, that for every symmetric (strict?) monoidal category $\mathcal C$ , the (monoidal?) category of (some unspecified kind of) symmetric monoidal functors between the (strictified?) category of $2$ -dimensional cobordisms and $\mathcal C$ is (monoidally?) equivalent to the (monoidal?) category of commutative Frobenius objects of $\mathcal C$ . Do we need so much strictness to make it all work? Or is strictness just an educational choice to not to allow coherence data to hinder the flow of these (quite beautiful I should say!) ideas?

P.S. This question has also been posted to math.stackexchange.com.

John Baez (Nov 27 2025 at 12:54):

@Marco Vianello

The Note seems to suggest that by considering the strictification of the target category we are allowed to forget about non-strict functors. I was wondering if this is true, and how would this work in practice. By the way, I suspect it is not.

Yeah, that sounds totally bogus to me. However, I know a theorem that may apply to this particular case.

The main theorem in the book states, in so many words, that for every symmetric (strict?) monoidal category $\mathcal C$ , the (monoidal?) category of (some unspecified kind of) symmetric monoidal functors between the (strictified?) category of $2$ -dimensional cobordisms and $\mathcal C$ is (monoidally?) equivalent to the (monoidal?) category of commutative Frobenius objects of $\mathcal C$ .

Wow, I've never before met such a nervous-sounding theorem. :laughing:

Do we need so much strictness to make it all work?

No: I know in my bones that it does not. Someone should write a proof of a theorem like this that doesn't assume any strictness.

John Baez (Nov 27 2025 at 14:13):

Okay, here's what I posted to Math Stackexchange:

You can strictify the whole situation here.

A "prop" is a particularly simple sort of strict symmetric monoidal category - see Definition 9 of Props in network theory by Baez, Coya and Rebro.

The symmetric monoidal category you're calling $\mathsf{Bord}(2)$ is equivalent, as a symmetric monoidal category, to a prop by Proposition 11 in Appendix A of that paper, and so is $\mathsf{FinVect}_\mathbb{F}$ , the symmetric monoidal category of finite-dimensional vector spaces over the field $\mathbb{F}$ . Thus, by Proposition 12 of that paper, every strong symmetric monoidal functor $F\colon \mathsf{Bord}(2) \to \mathsf{FinVect}_\mathbb{F}$ is isomorphic, via a symmetric monoidal natural isomorphism, to a strict symmetric monoidal functor.

You might be annoyed by the limitation to finite-dimensional vector spaces, but it's not really a limitation. Any symmetric monoidal functor $F \colon \mathsf{Bord}(2) \to \mathsf{Vect}_{\mathbb{F}}$ sends the circle to a Frobenius algebra, and every Frobenius algebra must be finite-dimensional - so we don't lose anything by restricting to $\mathsf{FinVect}_\mathbb{F}$ . (Any Frobenius algebra is finite-dimensional because it's isomorphic, as a vector space, to its dual.)

Marco Vianello (Nov 28 2025 at 01:37):

@John Baez The last revision of your paper changed the numbering scheme for theorems, propositions, etc.

Definition 9 became Definition 4.2, and Proposition 11 and 12 became Proposition 4.3 and 4.4 respectively.

With that said, thank you! That was exactly the kind of feedback I was hoping for.

Marco Vianello (Nov 28 2025 at 01:38):

John Baez said:

Wow, I've never before met such a nervous-sounding theorem. :laughing:

Indeed. I'll likely be back in the next few weeks as I try to sort out some "?"s.

Ideally, I'm hoping to find that something along the following lines is true:

Given any monoidal category $\mathcal C$ , there is an equivalence of categories

$\mathrm{MonCat}(\Delta, \mathcal C)\simeq \mathrm{Mon}(\mathcal C)$

where $\mathrm{MonCat}(\Delta, \mathcal C)$ is the category of strong monoidal functors from the (augmented) simplex category $\Delta$ to $\mathcal C$ , and $\mathrm{Mon}(\mathcal C)$ is the category of monoid objects in $\mathcal C$ .

Given any symmetric monoidal category $\mathcal C$ , there is an equivalence of categories

$\mathrm{SymMonCat}(\Phi, \mathcal C)\simeq \mathrm{cMon}(\mathcal C)$

where $\mathrm{SymMonCat}(\Phi, \mathcal C)$ is the category of symmetric strong monoidal functors from the skeleton $\Phi$ of the category of finite sets and functions between them to $\mathcal C$ , and $\mathrm{cMon}(\mathcal C)$ is the category of commutative monoid objects in $\mathcal C$ .

And so on...

These facts are stated on the book I'm currently reading (so the same handwaviness warnings should apply). But I believe it too that (along with the one I mentioned in the OP) they probably hold in their strong glory (with the equivalences in 2. and in its Frobenius counterpart in the OP both being strong monoidal).

I also wonder if these are all specific instances of a more general result. But that's material for another thread.

Nathanael Arkor (Nov 28 2025 at 07:20):

Marco Vianello said:

I also wonder if these are all specific instances of a more general result. But that's material for another thread.

I would recommend taking a look at Weber's Internal algebra classifiers as codescent objects of crossed internal categories.

John Baez (Nov 28 2025 at 09:43):

Marco Vianello said:

Ideally, I'm hoping to find that something along the following lines is true:

Given any monoidal category $\mathcal C$ , there is an equivalence of categories

$\mathrm{MonCat}(\Delta, \mathcal C)\simeq \mathrm{Mon}(\mathcal C)$

where $\mathrm{MonCat}(\Delta, \mathcal C)$ is the category of strong monoidal functors from the (augmented) simplex category $\Delta$ to $\mathcal C$ , and $\mathrm{Mon}(\mathcal C)$ is the category of monoid objects in $\mathcal C$ .

Given any symmetric monoidal category $\mathcal C$ , there is an equivalence of categories

$\mathrm{SymMonCat}(\Phi, \mathcal C)\simeq \mathrm{cMon}(\mathcal C)$

where $\mathrm{SymMonCat}(\Phi, \mathcal C)$ is the category of symmetric strong monoidal functors from the skeleton $\Phi$ of the category of finite sets and functions between them to $\mathcal C$ , and $\mathrm{cMon}(\mathcal C)$ is the category of commutative monoid objects in $\mathcal C$ .

In the special case $\mathcal{C} = \mathsf{FinVect}_\mathbb{F}$ , these results also follow from the propositions on props that I mentioned. But these results must be true in the generality you just stated, and someone should have written up a proof somewhere: it would be a criminal offense if nobody had!

(Thanks for catching the changing in numbering of those propositions - I hadn't updated the version on my website, so it was older than the arXiv version.)