# nLab universal colimit

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

## Models

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## 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

$\left({\underset{\to }{\mathrm{lim}}}_{k}{F}_{k}\right){×}_{Y}X\simeq {\underset{\to }{\mathrm{lim}}}_{k}\left({F}_{k}{×}_{Y}X\right)\phantom{\rule{thinmathspace}{0ex}}.$({\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)

## References

Section 6.1.1 of

Revised on February 23, 2013 07:54:27 by Mike Shulman (192.16.204.218)