Colimit preserving if and only if anima colimits and weakly contractible colimits

Lemma 6.163.

A category $$C$$ admits small colimits if and only if it admits anima-indexed colimits and weakly contractible colimits. Similarly, if $$C$$ and $$D$$ are categories with small colimits, then a functor $F\colon C \to D$ preserves small colimits if and only if it preserves both anima-indexed colimits and weakly contractible colimits.

Proof

Assume that

\(C\)

admits anima-indexed colimits and weakly contractible colimits. We need to show that it admits a colimit for an arbitrary diagram

X\colon I \to C

, where

\(I\)

is a small category. To this end, consider the localization functor

p\colon I \to \abs{I}

inverting all morphisms in

\(I\)

. As discussed before, the finality of

\(p\)

implies that the relative slices of

\(p\)

are weakly contractible, so that the left Kan extension

p_!X \colon \abs{I} \to C

\(X\)

along

\(p\)

exists in

\(C\)

. Moreover, the colimit of

\(p_!X\)

exists in

\(C\)

since

\abs{I}

is an anima. Since Kan extensions compose, we conclude that the colimit of

\(X\)

exists in

\(C\)

.The argument for preservation of colimits is entirely analogous.