In mathematics, a pseudofunctor F is a mapping from a category to the category Cat of (small) categories that is just like a functor except that and do not hold as exact equalities but only up to coherent isomorphisms.
A typical example is an assignment to each pullback , which is a contravariant pseudofunctor since, for example for a quasi-coherent sheaf , we only have:
Since Cat is a 2-category, more generally, one can also consider a pseudofunctor between 2-categories, where coherent isomorphisms are given as invertible 2-morphisms.
The Grothendieck construction associates to a contravariant pseudofunctor a fibered category, and conversely, each fibered category is induced by some contravariant pseudofunctor. Because of this, a contravariant pseudofunctor, which is a category-valued presheaf, is often also called a prestack (a stack minus effective descent).
A pseudofunctor F from a category C to Cat consists of the following data
The notion of a pseudofunctor is more efficiently handled in the language of higher category theory. Namely, given an ordinary category C, we have the functor category as the âÂÂ-category
Each pseudofunctor belongs to the above, roughly because in an âÂÂ-category, a composition is only required to hold weakly, and conversely (since a 2-morphism is invertible).