nLab higher topos theory

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

flavors of higher toposes

• 1-topos

  • (0,1)-topos

  • (1,1)-topos = topos

  • (2,1)-topos

  • $(n,1)$-topos

  • (∞,1)-topos ($n$-localic)

    $\;\;\;\;$ model topos

• 2-topos

  • (2,2)-topos ($n$-localic)

  • (∞,2)-topos

• $n$-topos

  • $(\infty,n)$-topos

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:

Example

(examples of archetypical higher toposes)

• Grpd– the (2,1)-category of groupoids, hence of (1,0)-categories – should be the archetypical (2,1)-topos.

• Cat– the 2-category of categories, hence of (1,1)-categories – should be the archetypical 2-topos (i.e. (2,2)-topos).

• Pos– the category of posets and monotone maps, hence of (0,1)-categories – should be the archetypal (1,2)-topos.

• ∞Grpd– the (∞,1)-category of ∞-groupoids, hence of (∞,0)-categories – is the archetypical (∞,1)-topos;

• (∞,1)Cat– the (∞,2)-category of (∞,1)-categories – should be the archetypical (∞,2)-topos.

Related concepts

• higher structures

• topos theory

• 2-topos theory

• (∞,1)-topos theory

References

• Michael Shulman, n-topos for large n .