nLab
universal colimit

Context

(,1)(\infty,1)-Category theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

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 CC has universal colimits if for every morphism f:XYf : X \to Y in CC the induced pullback-(∞,1)-functor on over-(∞,1)-categories

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

preserves all colimits.

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

(lim kF k)× YXlim k(F k× YX). ({\lim_\to}_k F_k ) \times_Y X \simeq {\lim_\to}_k (F_k \times_Y X) \,.

Properties

Proposition

If CC 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)