equivalences in/of $(\infty,1)$-categories
The generalization of the notion of exact functor/flat functor from category theory to (∞,1)-category theory.
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.
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 $D' \to D$ with $D'$ a $\kappa$-filtered (∞,1)-category, the pullback $C' := C \times_D D'$ (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.
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.
$\kappa$-right exact $(\infty,1)$-functors are closed under composition.
Every (∞,1)-equivalence is $\kappa$-right exact.
An $(\infty,1)$-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.
Section 5.3.2 of