Fibered Category

A fibered category over a topological space consists of

1. a category for each open subset ,

2. a functor for each inclusion , and

3. a natural isomorphism

for each pair of inclusions , .

The restriction data are normalized: for every open set , and is the identity natural transformation whenever or is an identity inclusion.

In addition, for any three composable inclusions , , and , the diagram shown above is a commutative diagram.

Sometimes, the pair is used to denote a fibered category with more precision while the shorthand is sometimes used for , , .