nLab
universal colimit

Idea

One says – at least in the context of Giraud's axioms for toposes and (∞,1)-toposes) – that a colimit is universal if it is stable under pullbacks. 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

Definition

A locally presentable (∞,1)-category $C$ has universal colimits if for every morphism $f:X\to Y$ in $C$ the induced pullback-(∞,1)-functor on over-(∞,1)-categories

$$f^{*}:C^{/Y}\to C^{/X}$$f^* : C^{/Y} \to C^{/X}

preserves all colimits.

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

$$({\underrightarrow{\lim }}_k{F}_k){\times }_{Y}X\simeq {\underrightarrow{\lim }}_k(F_k{\times }_{Y}X).$$({\lim_\to}_k F_k ) \times_Y X \simeq {\lim_\to}_k (F_k \times_Y X) \,.

Properties

Proposition

If $C$ is an (∞,1)-topos, then it has universal colimits.

This is HTT, theorem 6.1.0.6 (3) ii)

Related concepts

• van Kampen colimit

References

Section 6.1.1 of

• Jacob Lurie, Higher Topos Theory