Pseudo-functor

Short description: Category mapping

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

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

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).

Definition

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

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

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

  $F({\mathrm{id}}_x)\simeq {\mathrm{id}}_{F(x)}$ for each object x

such that

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

$F({\mathrm{id}}_x)\circ Ff\stackrel{\sim }{\to }F({\mathrm{id}}_x\circ f)=Ff$ is the same as $F({\mathrm{id}}_x)\circ Ff\simeq {\mathrm{id}}_{F(x)}\circ Ff=Ff$,

and similarly for $Ff\circ F({\mathrm{id}}_x)$.

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

$\text{<mrow data-mjx-texclass="ORD">Fct</mrow>}(C,\text{<mrow data-mjx-texclass="ORD">Cat</mrow>}).$

Each pseudofunctor $C\to \text{<mrow data-mjx-texclass="ORD">Cat</mrow>}$ 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

• C. Sorger, Lectures on moduli of principal G-bundles over algebraic curves
• Vistoli, Angelo (September 2, 2008). "Notes on Grothendieck topologies, fibered categories and descent theory". http://homepage.sns.it/vistoli/descent.pdf. 

External links

• http://ncatlab.org/nlab/show/pseudofunctor