Let $G$ be a group. A crossed G-set consists of the following data.

A G-set, that is to say, a set $X$ together with an action$\cdot : U(G) \times X \rightarrow X$ of $G$ upon it, where $U$ is the forgetful functor from Grp to Set.

A map of sets $\left| - \right|: X \rightarrow U(G)$.

We require that $\left| g \cdot x \right| = g \left| x \right| g^{-1}$ for all $g \in G$ and $x \in X$.

Definition

Let $\underline{X}$ and $\underline{Y}$ be crossed $G$-sets. A morphism from $\underline{X}$ to $\underline{Y}$ is a map of sets $f : X \rightarrow Y$ which is a morphism of $G$-sets, that is to say, $f(g \cdot x) = g \cdot f(x)$ for all $g \in G$ and $x \in X$, and which has the property that $\left| f(x) \right| = \left| x \right|$ for all $x \in X$.

Crossed $G$-sets and morphisms between them assemble into a category. The identity morphism on a crossed $G$-set $\underline{X}$ is defined by the identity map $X \rightarrow X$.

Braided monoidal structure on the category of crossed G-sets

Definition

Let $\underline{X}$ and $\underline{Y}$ be crossed G-sets. The tensor product of $\underline{X}$ and $\underline{Y}$ is the product of the underlying $G$-sets, namely the product of sets $X \times Y$ equipped with the action $g \cdot (x,y) = (g \cdot x, g \cdot y)$, equipped with the map $X \times Y \rightarrow U(G)$ given by $(x,y) \mapsto \left| x \right| \left| y \right|$.

Extending in the evident way to morphisms, the tensor product of crossed G-sets equips the category of crossed $G$-sets with the structure of a (not strict, and not symmetric) monoidal category. The unit is the set with one element equipped with its unique crossed $G$-set structure.

Definition

Let $\underline{X}$ and $\underline{Y}$ be crossed $G$-sets. The braiding of $\underline{X}$ and $\underline{Y}$ is the morphism of crossed $G$-sets $X \times Y \rightarrow Y \times X$ given by $(x,y) \mapsto (\left| x \right| \cdot y, x)$.

The braiding of crossed $G$-sets gives rise, together with the monoidal structure of Definition , to a braided monoidal structure on the category of crossed $G$-sets.

References

Peter Freyd and David Yetter, Braided compact closed categories with applications to low dimensional topology, Adv. Math. 77 (1989), 156–182. (Section 4.2)

André Joyal and Ross Street, Braided tensor categories , Adv. Math. 102 (1993), 20–78. (Example 5.1)

Last revised on April 21, 2018 at 17:22:03.
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