(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
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.
For a colimit diagram, this says in particular that
If is an (∞,1)-topos, then it has universal colimits.
This is HTT, theorem 22.214.171.124 (3) ii)
Section 6.1.1 of
Revised on February 23, 2013 07:54:27
by Mike Shulman