Under Construction
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.
A crossed category consists of two categories , , two functors , and a natural transformation .
With only a slight abuse of notation, we can identify the components of with their respective objects thus identifying objects of with morphisms of . This is justified since it amounts to writing a morphism as
where we think of the functor as the source of and as the target of .
Given a morphism in , it follows from naturality that
Given groups and , the crossed category is equivalent to a crossed module .
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…