nLab 1-topos

Idea

A $1$-topos, or $(1,1)$-topos, is simply a topos in the usual sense of the word. The prefix $1$- may be added when also discussing higher categorical types of topoi in higher topos theory such as 2-topos, $(\infty,1)$-topos, or even $\infty$-topos. Compare that a (0,1)-topos is a Heyting algebra.

Similarly, a Grothendieck $1$-topos, or Grothendieck $(1,1)$-topos, is simply a Grothendieck topos. Compare that a Grothendieck (0,1)-topos is a frame (or locale).

Note that a 1-topos is not exactly a particular sort of 2-topos or $\infty$-topos, just as a Heyting algebra is not a particular sort of 1-topos. The (1,2)-category of locales (i.e. (0,1)-topoi) embeds fully in the 2-category of Grothendieck 1-topoi by taking sheaves, but a locale is not identical to its topos of sheaves (and in fact no nontrivial 1-topos can be a poset), in that the following diagram of functors can not be filled by a natural isomorphism:

Likewise, one expects every Grothendieck 1-topos to give rise to a 2-topos or $\infty$-topos of stacks, hopefully producing a full embedding of some sort.

Related concepts

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