Pseudo-functor

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 ${\displaystyle F(f\circ g)=F(f)\circ F(g)}$ and ${\displaystyle F(1)=1}$ do not hold as exact equalities but only up to coherent isomorphisms.

A typical example is an assignment to each pullback ${\displaystyle Ff=f^{*}}$, which is a contravariant pseudofunctor since, for example for a quasi-coherent sheaf ${\displaystyle {\mathcal {F}}}$, we only have: ${\displaystyle (g\circ f)^{*}{\mathcal {F}}\simeq f^{*}g^{*}{\mathcal {F}}.}$

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).