(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An (∞,1)-topos is $n$-localic if
regarded as a little topos it behaves like a generalized n-groupoid;
it behaves like the (∞,1)-category of (∞,1)-sheaves over an (∞,1)-site that is a (n,1)-category.
More precisely: if (∞,1)-geometric morphisms into it are fixed by their restriction to the underlying (n,1)-toposes of (n-1)-truncated objects.
To the tower of (n,1)-toposes of (n-1)-truncated objects
of a given (∞,1)-topos $\mathcal{X}$ corresponds a tower of $n$-localic toposes $\mathcal{X}_n$ such that $\tau_{\leq n -1} \mathcal{X} \simeq \tau_{\leq n-1} \mathcal{X}_n$. We may think of the $n$-localic $\mathcal{X}_n$ as being $n$th stage in the Postnikov tower decomposition of $\mathcal{X}$.
A 0-localic $(1,1)$-topos is a localic topos from ordinary topos theory.
We write (∞,1)Topos for the (∞,1)-category of (∞,1)-toposes and (∞,1)-geometric morphisms between them.
For $\mathcal{X}$ an (∞,1)-topos we denote by
the (n,1)-topos of $(n-1)$-truncated objects of $\mathcal{X}$.
We write $(n,1)Topos$ for the (n+1,1)-category of (n,1)-toposes and $(n,1)$-geometric morphisms between them.
($n$-localic $(\infty,1)$-topos)
An (∞,1)-topos $\mathcal{X}$ is $n$-localic if for any other $(\infty,1)$-topos $\mathcal{Y}$ the canonical morphism
is an equivalence of (∞,1)-categories (of ∞-groupoids).
More generally,
a (k,1)-topos $\mathcal{X}$ is $n$-localic for $0 \leq n \leq k \leq \infty$ if for any other $(k,1)$-topos $\mathcal{Y}$ the canonical morphism
is an equivalence of (∞,1)-categories (of ∞-groupoids).
This is (HTT, def. 6.4.5.8).
This implies that an $n$-localic $(\infty,1)$-topos is also $(n+1)$-localic and generally $k$-localic for all $k \geq n$.
The (∞,1)-category of (∞,1)-sheaves over an (∞,1)-site $C$ with finite limits which is an (n,1)-category is $n$-localic.
This is (HTT, lemma 6.4.5.6).
For $n = 0$ this implies the familiar statement from ordinary topos theory: a category of sheaves over a posite=(0,1)-site is a localic topos (= 0-localic $(1,1)$-topos).
This is (LurieStructured, lemma 2.3.16).
For $n \in \mathbb{N}$ and $\mathcal{X}$ an $n$-localic $(\infty,1)$-topos, the over-(∞,1)-topos $\mathcal{X}/U$ is $n$-localic precisely if the object $U$ is $n$-truncated.
This is (StrSp, lemma 2.3.14).
For $\mathcal{X}$ an $n$-localic $(\infty,1)$-topos let $U \in \mathcal{X}$ be an object. Then the following are equivalent
the restriction of the inverse image $U^* : \mathcal{X} \to \mathcal{X}/U$ (of the etale geometric morphism from the over-(∞,1)-topos) to $(n-1)$-truncated objects is an equivalence of (∞,1)-categories;
the object $U$ is $n$-connected.
This is (StrSp, lemma 2.3.14).
Every (n,1)-topos $\mathcal{Y}$ is the (n,1)-category of $(n-1)$-truncated objects in an $n$-localic $(\infty,1)$-topos $\mathcal{X}_n$
This is (HTT, prop. 6.4.5.7).
Let $\mathcal{G}$ be a geometry (for structured (∞,1)-toposes).
If $\mathcal{G}$ is an (∞,n)-category then a $n$-localic $\mathcal{G}$-structured (∞,1)-topos is an $n$-truncated object in the (∞,1)-category $Topos(\mathcal{G})$.
This is StrSp, lemma 2.6.17
The general noion is the topic of section 6.4.5 of
Remarks on the application of $n$-localic $(\infty,1)$-toposes in higher geometry are in
Last revised on June 9, 2019 at 22:19:43. See the history of this page for a list of all contributions to it.