nLab (0,1)-topos

1. Idea

The notion of $(0,1)$-topos is that of topos in the context of (0,1)-category theory.

The notion of $(0,1)$-topos is essentially equivalent to that of Heyting algebra; similarly, a Grothendieck $(0,1)$-topos is a locale.

Notice that every $(1,1)$-Grothendieck topos comes from a localic groupoid, i.e. a groupoid internal to locales, hence a groupoid internal to $(0,1)$-toposes. See classifying topos of a localic groupoid for more.

2. 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

3. References

• J. C. Baez, M. Shulman, Lectures on n-categories and cohomology , pp.1-68 in J. C. Baez, P. May (eds.), Towards Higher Categories, Springer Heidelberg 2010. (preprint; section 5.3)

• Jacob Lurie, Higher Topos Theory , Princeton UP 2009. (section 6.4.2)

.