nLab crossed G-set

Definition

Definition

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

1. 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.

2. 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)