Category theory; homological algebra Definition

Braided monoidal category

Builds on Monoidal category Natural transformation

Definition

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ be a monoidal category. A braiding on $\mathcal{C}$ is a natural isomorphism $\gamma$ with components $\gamma_{A,B} \colon A \otimes B \to B \otimes A$ , natural in $A$ and $B$ . It must satisfy the two hexagon identities for all objects $A, B, C$ :

$\alpha_{B,C,A} \circ \gamma_{A, B \otimes C} \circ \alpha_{A,B,C} = (\id_B \otimes \gamma_{A,C}) \circ \alpha_{B,A,C} \circ (\gamma_{A,B} \otimes \id_C)$ as morphisms $(A \otimes B) \otimes C \to B \otimes (C \otimes A)$ ;
$\alpha^{-1}_{C,A,B} \circ \gamma_{A \otimes B, C} \circ \alpha^{-1}_{A,B,C} = (\gamma_{A,C} \otimes \id_B) \circ \alpha^{-1}_{A,C,B} \circ (\id_A \otimes \gamma_{B,C})$ as morphisms $A \otimes (B \otimes C) \to (C \otimes A) \otimes B$ .

A braided monoidal category is a monoidal category with a braiding.

Remarks

Example: $\mathbf{Vect}_k$ with the usual tensor product and $\gamma_{V,W}(v \otimes w) = w \otimes v$ . On $(u \otimes v) \otimes w$ both sides of the first hexagon give $v \otimes (w \otimes u)$ , and both sides of the second hexagon send $u \otimes (v \otimes w)$ to $(w \otimes u) \otimes v$ . Non-example: on $\mathbf{Vect}_k$ with $\operatorname{char} k \ne 2$ , the natural isomorphism $\gamma_{V,W}(v \otimes w) = -\, w \otimes v$ is not a braiding. In each hexagon the left side contains one copy of $\gamma$ and the right side two, so for nonzero spaces the two sides differ by the sign $-1$ . The first hexagon says that moving $A$ past $B \otimes C$ in one step equals moving it past $B$ and then past $C$ . The second says the same for moving $C$ past $A \otimes B$ .

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`7MZG4JQ04Y`
URI	`m2://7MZG4JQ04Y`
Kind	definition
Writing	W0 stub
MSC	18M15
Defines	Braided monoidal category
Uses	Monoidal category, Natural transformation

Braided monoidal category

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here