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topos theory

# Contents

## 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. 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.

When you say ‘a locale is not identical to its topos of sheaves’, you mean that the following square cannot be filled in with an equivalence in $2 Cat$?

$\array { (0,1) Top & \overset{topos\:of\:sheaves}\longrightarrow & (1,1) Top \\ \downarrow & & \downarrow \\ (0,1) Cat & \longrightarrow & 1 Cat }$

—Toby

flavors of higher toposes

Last revised on August 25, 2021 at 11:40:49. See the history of this page for a list of all contributions to it.