Joyal's version of Quillen's Theorem A

Theorem 21.5.1 (Joyal’s version of Quillen’s Theorem A, [Lurie (2009), Theorem 4.1.3.1]). For a functor \(\alpha \colon I \to J\), the following two conditions are equivalent:

(1)

A functor \(F\colon J \to C\) has a colimit if and only if the functor \(F \circ \alpha \colon I \to C\) has a colimit, and in this case the canonical map \[ \colim _I (F \circ \alpha ) \to \colim _J F \] is an isomorphism in \(C\);

(2)

For every object \(j\) of \(J\), the relative slice category \(I_{j/} = I \times _J J_{j/}\) is weakly contractible, meaning that its geometric realization \(\geom {I_{j/}}\) is contractible.

The dual result is also true, where in (1) we use limits and in (2) we use the over categories \(I_{/j}\).