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Stream: theory: category theory

Topic: 'semibraidings'

Matteo Capucci (he/him) (Mar 24 2022 at 10:09):

Let $\mathcal M$ be a monoidal category. Suppose I give an identity-on-objects monoidal functor $B : \mathcal M \to \mathcal M^{rev}$, where 'rev' denotes the same monoidal category 'reversed' tensor (as we agreed on #learning: questions > reverse of a monoidal category?).
Let $\epsilon : I = I$ be its unit constraint and $\beta_{m,n} : m \otimes n \to n \otimes m$ its monoidal constraint. The latter has all the looks of a braiding. Indeed, the unit axioms coincide perfectly, but then the axioms for $\beta$ are subtly yet fundamentally different: braidings.png

Matteo Capucci (he/him) (Mar 24 2022 at 10:09):

(The top one is one of the usual axioms of braiding, whereas the bottom one is the associativity axiom for $\beta$)

Matteo Capucci (he/him) (Mar 24 2022 at 10:10):

It's evident that if $\beta$ is indeed a braiding, then it does satisfy the second axiom, i.e. associativity, by coherence. But the converse is not obvious, and probably simply not true.

Matteo Capucci (he/him) (Mar 24 2022 at 10:12):

We could call this structure a 'semibraiding': it's enough to swap things around, associatively, but it doesn't go all the way to a braiding.
So, first of all, is this notion sensible? Or am I missing some other 'obvious' requirement for $\beta$ that would make it a braiding?
Secondly, is this notion known?

Matteo Capucci (he/him) (Mar 24 2022 at 10:13):

Notice that one might as well drop the requirement that $B$ is identity-on-objects, and contemplate lax monoidal functors $\mathcal M \to \mathcal M^{rev}$

Antonin Delpeuch (Mar 26 2022 at 19:54):

I cannot answer any of the questions but I find it a very elegant and interesting way to define a braiding-like structure.

Matteo Capucci (he/him) (Feb 27 2023 at 10:49):

I just noticed that a 'semisymmetry', i.e. a semibraiding such that $\beta_{m,n} ; \beta_{n,m} = 1_{m \otimes n}$, is the same thing as a symmetry! In fact if you swap the two leftmost wires on both sides of the associativity law (the second one in the diagrams in the op) you get the hexagon axiom.

Matteo Capucci (he/him) (Feb 27 2023 at 10:51):

Hence:
image.png

Tim Campion (Feb 27 2023 at 14:42):

I think it's interesting to think about this from the perspective of operads.

(Having written the following, I think it's a bit off, because I don't think an $E_2$ algebra is necessarily an algebra for the operad $(E_1)_{C_2}$ I describe below. So in the following definition, "braided" does not imply "semibraided"! I think I will post this anyway, but take it with a grain of salt!)

Let $O_{C_2}$ be the orbit category for the group of order 2 $C_2$. So $$O_{C_2}$ has two objects: a strict terminal object denoted $C_2 / C_2$, and an object denoted $C_2$ which has an automorphism group $C_2$ and no other endomorphisms. (That's a complete description of the category!)

There is a functor $\underline E_1 : O_{C_2}^{op} \to Opd$ (where $Opd$ is the $\infty$-category of operads) which carries $C_2 // C_2$ to the $E_0$ operad (whose algebras are pointed objects) and carries $C_2$ to the $E_1$ operad (whose algebras are monoid objects). The $C_2$ action on $E_1$ is given by reversing the order of multiplication, and the map $E_0 \to E_1$ is the natural inclusion (on algebras, this induces the functor from monoid objects to pointed objects which forgets the multiplication and just remembers the unit).

The colimit (in $Opd$) of the functor $\underline E_1$ will be an operad which $(E_1)_{C_2}$. An algebra for $(E_1)_{C_2}$ is a monoid object $A$ equipped with an equivalence of monoid objects $A \cong A^{rev}$ which is the identity on the underlying object $|A|$, along with some higher coherence conditions on this equivalence, and conditions saying that the higher coherence conditions are all trivial on the unit of $A$.

I might claim (???) that the algebras for the operad $(E_1)_{hC_2}$ should be the "right" notion of "semibraided monoidal category". From what I've described, then, it's clear that a semibraided monoidal category should be a monoidal category $C$ equipped with a monoidal equivalence $\phi: C \simeq C^{rev}$ which restricts to the identity on the underlying category $C$, and such that the unit constraint of $\phi$ is the identity automorphism of the unit $I$ of the monoidal category $C$ -- plus some higher coherence.

Anyway, one of your questions was whether there were other compatibility conditions you "should" be imposing. I'm still not sure of the answer, but the above approach would give a way of formalizing your question.

Graham Manuell (Feb 27 2023 at 14:59):

For what it's worth, in our paper on distributive laws for lax functors we note that braidings can viewed as invertible distributive laws between the identity functors on a monoidal category (viewed as a 1-object bicategory).
From that perspective it would be natural to add some axioms that relate the semibraiding to the unitors when one of the objects is the unit of the monoidal category. I think it then follows from our results that semibraidings correspond to certain unitary lax monoidal structures on tensor product functor.