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exact (infinity,1)-functor

Idea

The generalization of the notion of exact 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 $\kappa$ a regular cardinal, an (∞,1)-functor $F:C\to D$ is $\kappa$-right exact, if, when modeled as a morphism of quasicategories, for any right Kan fibration $D\prime \to D$ with $D\prime$ a $\kappa$-filtered (∞,1)-category, the pullback $C\prime :=C{\times }_{D}D\prime$ (in sSet) is also $\kappa$-filtered.

If $\kappa =\omega$ then we just say $F$ is right exact.

This is HTT, def. 5.3.2.1.

Proposition

If $C$ has $\kappa$-small colimits, then $F$ is $\kappa$-right exact precisely if it preserves these $\kappa$-small colimits.

So in particular if $C$ has all finite colimits, then $F$ is right exact precisely if it preserves these.

This is HTT, prop. 5.3.2.9.

Properties

Proposition

1. $\kappa$-right exact $(\mathrm{\infty },1)$-functors are closed under composition.

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

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

This is HTT, prop. 5.3.2.4.

References

Section 5.3.2 of

• Jacob Lurie, Higher Topos Theory