Compatibility of braids as a simplicial set and as a braided monoidal category

Question

We can form a braided monoidal category by taking the groupoid coproduct of the Artin braid groups $B_n$. We can also make the braids into a simiplical set where the ith face operation is removing the i-th strand and the ith degeneracy operation doubling the ith strand. (These operations are not group homomorphisms, so we don't get a simplicial group.)

Are these two notions compatible? For example, can we use the tensor product of the braided monodial category to describe these face and degeneracy operations? or vice versa? is there a 2-category or something that can combine these two notions?

Answer 1

I think the best way to think about this is in terms of operads.

Braided monoidal categories are representations of an operad $\Pi$ in the category of (small) categories.

The category $\Pi(n)$ has objects parenthesised permutations of $\{1,\ldots,n\}$ like $(4(23))1$. The morphisms $(\sigma) \to (\tau)$ are braids $\beta \in B_n$ so that $\beta$ maps to $\tau \sigma^{-1}$ under $B_n\to S_n$.

The operad structure $$ \Pi(n) \times \Pi(k_1) \times \ldots \times \Pi(k_n) \to \Pi(k_1+\ldots+k_n) $$ is given by replacing the $i$-th sting by the braid on $k_i$-strings.

Now any representations $\Pi(n) \to \underline{Hom}(\mathcal{C}^n,\mathcal{C}) = End(\mathcal{C})(n)$ induces a braided monoïdal structure with

• $\otimes$ corresponding to the object $(12) \in \Pi(2)$
• the associativity morphism corresponding to the trivial braid $(12)3 \to 1(23)$ in $\Pi(3)$
• the braiding corresponding to the morphism $(12) \to (21)$ in $\Pi(2)$ induced by the generator of $B_2 = \mathbb{Z}$.

MacLane's coherence theorem tells you that this is an equivalence.

All the operations you're looking at come from this operad structure.

The $\Pi(n)$ have a nice geometric interpretation. They are the fundamental groupoids of the operad of little discs $C_2(n)$ (or equivalently of $F(\mathbb{C},n)$ the spaces of configurations of $n$ points in the plane) restricted to a suitable collection of basepoints. This generalizes the classical definition $P_n = \pi_1(F(\mathbb{C},n),p)$, $B_n = \pi_1(F(\mathbb{C},n)/S_n,\overline{p})$.

I think this makes the whole picture a lot clearer because we get all of these operations as part of the same structure and we get a universal characterisation of that structure and a geometrical interpretation for it.

This leads to other nice considerations. For example the Grothendieck-Teichmuller group $GT$ defined by Drinfeld is the automorphism group of the (prounipotent completion) operad $\Pi$. This explains why $GT$ is universal for quasi triangular quasi Hopf algebras as their representations form braided monoïdal categories and related to the Galois group of $\mathbb{Q}$ as $GT$ appears as an automorphism group of fundamental groupoïds of the algebraic varieties $F(\mathbb{A}^1_{\mathbb{Q}},n)$.