A -topos, or -topos, is simply a topos in the usual sense of the word. The prefix - may be added when also discussing higher categorical types of topoi in higher topos theory such as 2-topos, -topos, or even -topos. Compare that a (0,1)-topos is a Heyting algebra.
Similarly, a Grothendieck -topos, or Grothendieck -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 -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 -topos of stacks, hopefully producing a full embedding of some sort.
Last revised on September 18, 2022 at 08:44:04. See the history of this page for a list of all contributions to it.