The braid category

The braid category $\mathcal{B}$ is the category defined as follows.

• Its objects are the natural numbers $0,1,2,\dots$.
• For $m\mathrm{\ne }n$, there are no morphisms $m\to n$.
• For each $n$, the endomorphism group ${\mathrm{End}}_{\mathcal{B}}(n)$ is the braid group $B_n$ from The braid group by Artin presentation.

Composition is the group multiplication in each $B_n$. The tensor product on objects is addition. On morphisms, juxtaposition is the homomorphism $B_m\times B_n\to B_{m+n}$ sending the first block generators ${\sigma }_i$ to ${\sigma }_i$ and the second block generators ${\sigma }_j$ to ${\sigma }_{m+j}$. The Artin relations show directly that this is well defined, strictly associative, and unital. Thus $\mathcal{B}$ is a strict monoidal category (Strict monoidal category).

The standard block crossing ${\beta }_{m,n}$ moves the first $m$ strands over the last $n$ strands. Isotopy of braid diagrams, equivalently the Artin braid relations, gives naturality and the two block hexagons, so these crossings form a braiding (Braided monoidal category). Under this convention ${\beta }_{1,1}={\sigma }_1\in B_2$.