A wide-subcategory colimit criterion | Higher Topos Theory notes

Proposition 4.60. ([Uemura 2025, Proposition 2.6])

Let $$C$$ be a category with small colimits, and let $D\subseteq C$ be a wide subcategory. Suppose that:

For every object $X\in C$ , the wide subcategory $D_{/X}\subseteq C_{/X}$ is closed under small colimits.
For every small diagram $X_{\bullet}\colon I\to D$ , if $X=\colim_i X_i$ is computed in $$C$$ , then all structure maps $X_i\to X$ lie in $$D$$ .

Then $$D$$ is closed under small colimits in $$C$$ .

Proof

Let

X_{\bullet}\colon I\to D

be a diagram and let

X=\colim_i X_i

\(C\)

. We need to show that

\(X\)

has the universal property of a colimit in

\(D\)

. For any object

\(Y\)

, a morphism

f\colon X\to Y

belongs to

\(D\)

if and only if all composites

X_i\to X\to Y

belong to

\(D\)

: the forward direction follows from (2), and the reverse direction follows from (1), since

\(f\)

is the colimit in

C_{/Y}

of the diagram of the composites

X_i\to Y

. This identifies the mapping anima from

\(X\)

\(Y\)

\(D\)

with the limit of the mapping animae from the

\(X_i\)

\(Y\)

\(D\)

, as desired.

References

Taichi Uemura. Colimits in the ∞-category of ∞-topoi and étale morphisms. 2025.