Login
Login

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 \mathcal{F}, we only have:

(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).

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 F f \circ Fg,
  • :F(\operatorname{id}_x) \simeq \operatorname{id}_{F(x)} for each object x

:such that

::F(fgh) \overset{\sim}\to F(fg) Fh \overset{\sim}\to Ff Fg Fh is the same as F(fgh) \overset{\sim}\to Ff F(gh) \overset{\sim}\to Ff Fg Fh ,

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

::and similarly for Ff \circ F (\operatorname{id}_x).{{harvnb|Vistoli|2008|loc=Definition 3.10.}}

Higher category interpretation

The notion of a pseduofunctor 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

:\textbf{Fct}(C, \textbf{Cat}).

Each pseudofunctor 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).

See also

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 |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

{{categorytheory-stub}}