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The Braid Category

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Braid Category

The braid category is a braided monoidal category whose objects are natural numbers (equivalently, collections of dots), and whose morphisms are automorphisms represented by elements of the braid groups . The monoidal product is given by addition on objects and juxtaposition on morphisms. The braiding is represented by the braid

Remark

One can see that indeed satisfies the braiding axioms by the braid relation. A graphical proof is provided in \cite[Theorem XIII.2.1]{kasselQuantumGroups1995}.

e.g. For example, the following is a morphism :

This means in , there are no morphisms other than automorphisms from the braid group. Thus this is a discrete category, and by Artin’s presentation \cite[Sec. 5.4]{prasolovKnotsLinksBraids1997} of the braid group, is therefore generated by the following morphisms

sphere

Theorem

The braid category is the free (strict) braided monoidal category generated by one object.

Proof For any strict braided monoidal category , let be an object. To define a strict braided monoidal functor , there is no choice but to set
Moreover, the elementary braid generator , which crosses the -th and -st strands, is obtained by vertically stacking straight strands, a single crossing, and additional straight strands. Therefore, it must be sent to
It remains to verify that this assignment respects the braid relations: The first relation holds because braidings acting on disjoint tensor factors commute. To illustrate this, consider the object and the braid generators . Then

Misplaced & F(\sigma_1\sigma_3) &= (\tau_{X,X} \otimes \id_X \otimes \id_X) \circ (\id_X \otimes \id_X \otimes \tau_{X,X}) \\ &= (\tau_{X,X} \circ (\id_X \otimes \id_X)) \otimes ((\id_X \otimes \id_X) \circ \tau_{X,X}) \\ &= (\tau_{X,X} \circ \id_{X\otimes X}) \otimes (\id_{X\otimes X} \circ \tau_{X,X}) \\ &= \tau_{X,X} \otimes \tau_{X,X}, \end{aligned} $$and similarly, $$ \begin{aligned} F(\sigma_3\sigma_1) &= (\id_X \otimes \id_X \otimes \tau_{X,X}) \circ (\tau_{X,X} \otimes \id_X \otimes \id_X) \\ &= ((\id_X \otimes \id_X) \circ \tau_{X,X}) \otimes (\tau_{X,X} \circ (\id_X \otimes \id_X)) \\ &= \tau_{X,X} \otimes \tau_{X,X}. \end{aligned} $$ For the second relation, note that by the [[Braided Monoidal Categories#^d7b677|corollary]], $\tau_{X,X}$ is a Yang--Baxter operator. Hence $F(\sigma_j)$ satisfies the second braid relation by the [[Yang-Baxter Equation#^7c4536|proposition]].

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