Contents

Idea

One says – at least in the context of Giraud's axioms for toposes and (∞,1)-toposes) – that colimits are universal in a context in which they are stable under pullback. This is described in more detail at commutativity of limits and colimits.

The statement “colimits are universal” is then one of Giraud's axioms that characterize Grothendieck toposes in the 1-categorical context and Grothendieck-Rezk-Lurie (∞,1)-toposes in the higher categorical context.

Definition

For $F:K\to C^{/Y}$ a colimit diagram, this says in particular that

Properties

This is HTT, theorem 6.1.0.6 (3) ii)

References

Section 6.1.1 of

• Jacob Lurie, Higher Topos Theory