nLab
higher topos theory
Contents
Context
Topos Theory
topos theory

Background
Toposes
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Theorems
$(\infty,1)$ -Topos Theory
(∞,1)-topos theory

Background
Definitions
elementary (∞,1)-topos

(∞,1)-site

reflective sub-(∞,1)-category

(∞,1)-category of (∞,1)-sheaves

(∞,1)-topos

(n,1)-topos , n-topos

(∞,1)-quasitopos

(∞,2)-topos

(∞,n)-topos

Characterization
Morphisms
Extra stuff, structure and property
hypercomplete (∞,1)-topos

over-(∞,1)-topos

n-localic (∞,1)-topos

locally n-connected (n,1)-topos

structured (∞,1)-topos

locally ∞-connected (∞,1)-topos , ∞-connected (∞,1)-topos

local (∞,1)-topos

cohesive (∞,1)-topos

Models
Constructions
structures in a cohesive (∞,1)-topos

Contents
Idea
Higher topos theory is the generalisation to higher category theory of topos theory . It is partly motivated by Grothendieck ‘s program in Pursuing Stacks .

More generally, the concept $(n,r)$ -topos is to topos as (n,r)-category is to category .

Rather little is known about the very general notion of higher topos theory. A rich theory however exists in the context of (∞,1)-categories , see at (∞,1)-topos theory

Examples
Flavors of higher toposes
Archetypical higher toposes
Just as the archetypical example of an ordinary topos (i.e. a (1,1)-topos ) is Set – the category of 0-categories – so the $\infty$ -category of (n,r)-categories should form the archetypical example of an $(n+1,r+1)$ -topos:

References
Last revised on August 25, 2021 at 15:38:24.
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