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 <math>F(f \circ g) = F(f) \circ F(g)</math> and <math>F(1) = 1</math> do not hold as exact equalities but only up to ''coherent isomorphisms''.

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

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).

== Definition == A pseudofunctor ''F'' from a category ''C'' to '''Cat''' consists of the following data *a category <math>F(x)</math> for each object ''x'' in ''C'', *a functor <math>Ff</math> for each morphism ''f'' in ''C'', *a set of coherent isomorphisms for the identities and the compositions; namely, the invertible natural transformations *:<math>F(f \circ g) \simeq F f \circ Fg</math>, *:<math>F(\operatorname{id}_x) \simeq \operatorname{id}_{F(x)}</math> for each object ''x'' :such that ::<math>F(fgh) \overset{\sim}\to F(fg) Fh \overset{\sim}\to Ff Fg Fh </math> is the same as <math>F(fgh) \overset{\sim}\to Ff F(gh) \overset{\sim}\to Ff Fg Fh </math>, ::<math>F (\operatorname{id}_x) \circ Ff \overset{\sim}\to F(\operatorname{id}_x \circ f) = Ff</math> is the same as <math>F (\operatorname{id}_x) \circ Ff \simeq \operatorname{id}_{F(x)} \circ Ff = Ff</math>, ::and similarly for <math>Ff \circ F (\operatorname{id}_x)</math>.<ref>{{harvnb|Vistoli|2008|loc=Definition 3.10.}}</ref>

== Higher category interpretation == 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 :<math>\textbf{Fct}(C, \textbf{Cat}).</math> Each pseudofunctor <math>C \to \textbf{Cat}</math> 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).

== See also == *Lax functor

== References == {{reflist}} {{refbegin}} *C. Sorger, [http://users.ictp.it/~pub_off/lectures/lns001/Sorger/Sorger.pdf Lectures on moduli of principal G-bundles over algebraic curves] *{{cite web |first=Angelo |last=Vistoli |author-link=Angelo Vistoli |url=http://homepage.sns.it/vistoli/descent.pdf |title=Notes on Grothendieck topologies, fibered categories and descent theory |date=September 2, 2008}} {{refend}}

== External links == *http://ncatlab.org/nlab/show/pseudofunctor

Category:Functors