Pseudo-functor

 ( Functors ) 

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 $F(f\; \backslash circ\; g)\; =\; F(f)\; \backslash circ\; F(g)$ 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 $\backslash mathcal\{F\}$, we only have: $(g\; \backslash circ\; f)^*\; \backslash mathcal\{F\}\; \backslash simeq\; f^*\; g^*\; \backslash mathcal\{F\}.$

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).

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

such that

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

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

:and similarly for $Ff\; \backslash circ\; F\; (\backslash operatorname\{id\}\_x)$.

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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

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

Each pseudofunctor $C\; \backslash to\; \backslash textbf\{Cat\}$ 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).

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See also

• Lax functor

• C. Sorger, Lectures on moduli of principal G-bundles over algebraic curves

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External links

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

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