Eric Forgy Crossed Category

Under Construction

Contents

Idea

A crossed category is to a strict 2-category as a crossed module is to a strict 2-group. In fact, since a strict 2-group is a special case of a strict 2-category where all morphisms are invertible, we will show that a crossed module is a special case of a crossed category.

Definition

A crossed category $(C,D,s,t,\eta)$ consists of two categories $C$, $D$, two functors $s,t:D\to C$, and a natural transformation $\eta:s\Rightarrow t$.

With only a slight abuse of notation, we can identify the components of $\eta$ with their respective objects thus identifying objects of $D$ with morphisms of $C$. This is justified since it amounts to writing a morphism $f\in C$ as

where we think of the functor $s$ as the source of $f$ and $t$ as the target of $f$.

Given a morphism $f\to g$ in $D$, it follows from naturality that

Relation to Crossed Modules

Given groups $G$ and $H$, the crossed category $(\mathbf{B}G,\mathbf{B}H,s,t,\eta)$ is equivalent to a crossed module $(G,H,t,\alpha)$.

First, note that homomorphisms between groups are equivalent to functors between 1-object groupoids. I’m not sure if this is coincidence or not, but the homomorphism

corresponds to the functor

To be continued…