(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The concept of $(\infty,1)$-pretopos (Lurie, appendix A) is a version of the concept of pretopos as one passes from toposes to (∞,1)-toposes. The definition is a variant of the characterization of Grothendieck (∞,1)-toposes, via the Giraud-Rezk-Lurie axioms, asking only for finite (∞,1)-limits and finite (∞,1)-colimits with some exactness properties relating them.
Let $\mathcal{C}$ be an (∞,1)-category. This is called an $(\infty,1)$-pretopos if
$\mathcal{C}$ has a terminal object and homotopy fiber products;
$\mathcal{C}$ has finite (∞,1)-colimits;
finite coproducts in $\mathcal{C}$ are universal and disjoint;
groupoid objects in $\mathcal{C}$ are effective:
realization of groupoid objects is universal.
If these conditions hold except possibly for the existence of a terminal object, then $\mathcal{C}$ is a local $(\infty,1)$-pretopos.
Every Grothendieck (∞,1)-topos is an $(\infty,1)$-pretopos (def. ).
Let $\mathbf{H}$ be a Grothendieck (∞,1)-topos then the full sub-(∞,1)-category
on the coherent objects is a local $(\infty,1)$-pretopos (def. ).
If moreover $\mathbf{H}$ is an coherent (∞,1)-topos, then $\mathbf{H}_{coh}$ is an $(\infty,1)$-pretopos.
(Lurie, prop. A.6.1.6, cor. 6.1.7)
Last revised on April 12, 2021 at 10:19:16. See the history of this page for a list of all contributions to it.