nLab
crossed G-set

Contents

Definition

Definition

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

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

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

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

Definition

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

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

Braided monoidal structure on the category of crossed G-sets

Definition

Let X̲\underline{X} and Y̲\underline{Y} be crossed G-sets. The tensor product of X̲\underline{X} and Y̲\underline{Y} is the product of the underlying GG-sets, namely the product of sets X×YX \times Y equipped with the action g(x,y)=(gx,gy)g \cdot (x,y) = (g \cdot x, g \cdot y), equipped with the map X×YU(G)X \times Y \rightarrow U(G) given by (x,y)|x||y|(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 GG-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 GG-set structure.

Definition

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

The braiding of crossed GG-sets gives rise, together with the monoidal structure of Definition , to a braided monoidal structure on the category of crossed GG-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. See the history of this page for a list of all contributions to it.