Eric Forgy
Crossed Category

Under Construction

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

$${\eta }_f=f:s(f)\to t(f),$$\eta_f = f:s(f)\to t(f),

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

$$t(f\to g)\circ f=g\circ s(f\to g).$$t(f\to g)\circ f = g\circ s(f\to g).

Relation to Crossed Modules

Given groups $G$ and $H$, the crossed category $(BG,BH,s,t,\eta )$ 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

$$t:H\to G$$t:H\to G

corresponds to the functor

$$t:BH\to BG.$$t:\mathbf{B}H\to\mathbf{B}G.

To be continued…