nLab
exact (infinity,1)-functor

Contents

Idea

The generalization of the notion of exact functor/flat functor from category theory to (∞,1)-category theory.

Definition

As for 1-categorical exact functors, there is a general definition of exact functors that restricts to the simple conditions that finite (co)limits are preserved in the case that these exist.

Definition

For κ\kappa a regular cardinal, an (∞,1)-functor F:CDF : C \to D is κ\kappa-right exact, if, when modeled as a morphism of quasicategories, for any right Kan fibration DDD' \to D with DD' a κ\kappa-filtered (∞,1)-category, the pullback C:=C× DDC' := C \times_D D' (in sSet) is also κ\kappa-filtered.

If κ=ω\kappa = \omega then we just say FF is right exact.

This is HTT, def. 5.3.2.1.

Proposition

If CC has κ\kappa-small colimits, then FF is κ\kappa-right exact precisely if it preserves these κ\kappa-small colimits.

So in particular if CC has all finite colimits, then FF is right exact precisely if it preserves these.

This is HTT, prop. 5.3.2.9.

Properties

Proposition
  1. κ\kappa-right exact (,1)(\infty,1)-functors are closed under composition.

  2. Every (∞,1)-equivalence is κ\kappa-right exact.

  3. An (,1)(\infty,1)-functor equivalent (in the (∞,1)-category of (∞,1)-functors) to a κ\kappa-right exact one is itsels κ\kappa-right exact.

This is HTT, prop. 5.3.2.4.

References

Section 5.3.2 of

Revised on January 1, 2014 12:20:25 by Anonymous Coward (70.114.150.222)