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).

A pseudofunctor F from a category C to Cat consists of the following data

• a category ${\displaystyle F(x)}$ for each object x in C,
• a functor ${\displaystyle Ff}$ for each morphism f in C,
• a set of coherent isomorphisms for the identities and the compositions; namely, the invertible natural transformations

  ${\displaystyle F(f\circ g)\simeq Ff\circ Fg}$,

  ${\displaystyle F(\operatorname {id} _{x})\simeq \operatorname {id} _{F(x)}}$ for each object x

such that

${\displaystyle F(fgh){\overset {\sim }{\to }}F(fg)Fh{\overset {\sim }{\to }}FfFgFh}$ is the same as ${\displaystyle F(fgh){\overset {\sim }{\to }}FfF(gh){\overset {\sim }{\to }}FfFgFh}$,

${\displaystyle F(\operatorname {id} _{x})\circ Ff{\overset {\sim }{\to }}F(\operatorname {id} _{x}\circ f)=Ff}$ is the same as ${\displaystyle F(\operatorname {id} _{x})\circ Ff\simeq \operatorname {id} _{F(x)}\circ Ff=Ff}$,

and similarly for ${\displaystyle Ff\circ F(\operatorname {id} _{x})}$.¹

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 ∞-category

${\displaystyle {\textbf {Fct}}(C,{\textbf {Cat}}).}$

Each pseudofunctor ${\displaystyle C\to {\textbf {Cat}}}$ belongs to the above, roughly because in an ∞-category, a composition is only required to hold weakly, and conversely (since a 2-morphism is invertible).