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.

Just as the archetypical example of an ordinary topos (i.e. a $(\mathrm{1,1})$-topos) is Set – the category of 0-categories – so the $\mathrm{\infty }$-category of n-categories or at least of $n$-groupoids should form the archetypical example of an $(n+\mathrm{1,1})$-topos.

General higher topos theory

• Michael Shulman, n-topos for large n .

$(\mathrm{\infty }\mathrm{,1})$-Topos theory

• (∞,1)-topos

Early frameworks for Grothendieck (as opposed to “elementary”) $(\mathrm{\infty }\mathrm{,1})$-topoi are due Charles Rezk and Toën–Vezzosi in two versions (preprints 2002), via simplically enriched categories and via Segal categories:

• Bertrand Toën, Gabrielle Vezzosi, Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193 (2005), no. 2, 257–372, arXiv:math.AT/0207028

• B. Toën , G. Vezzosi, Segal topoi and stacks over Segal categories, arXiv:math.AG/0212330

Jacob Lurie has given the abstract model-independent definition of (∞,1)-toposes of (∞,1)-sheaves in

• Jacob Lurie, Higher Topos Theory

and shown that Brown-Joyal-Jardine model structure on simplicial presheaves (for ordinary ∞-stacks) and more generally the Toën-Vezzosi model structure on simplicially enriched presheaves (for derived ∞-stacks) are indeed models for this theory.

In this context, indeed the (∞,1)-category ∞-Grpd or Top of ∞-groupoids or equivalently (see homotopy hypothesis) of (compactly generated weakly Hausdorff) topological spaces – is the archetypical example of an $(\mathrm{\infty }\mathrm{,1})$-topos.

2-Topos theory

See 2-topos.