nLab
(infinity,1)-pretopos

Contents

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

The concept of (,1)(\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 some finite (∞,1)-limits and for finite (∞,1)-colimits.

Definition

Definition

Let 𝒞\mathcal{C} be an (∞,1)-category. This is called an (,1)(\infty,1)-pretopos if

  1. 𝒞\mathcal{C} has a terminal object and homotopy fiber products;

  2. 𝒞\mathcal{C} has finite (∞,1)-colimits;

  3. finite coproducts in 𝒞\mathcal{C} are universal and disjoint;

  4. groupoid objects in 𝒞\mathcal{C} are effective:

  5. realization of groupoid objects is universal.

If these conditions hold except possibly for the existence of a terminal object, then 𝒞\mathcal{C} is a local (,1)(\infty,1)-pretopos.

Lurie, def. A:6.1.1

Examples

Example

Every Grothendieck (∞,1)-topos is an (,1)(\infty,1)-pretopos (def. ).

(Lurie, example A.6.1.5)

Example

Let H\mathbf{H} be a Grothendieck (∞,1)-topos then the full sub-(∞,1)-category

H cohH \mathbf{H}_{coh} \hookrightarrow \mathbf{H}

on the coherent objects is a local (,1)(\infty,1)-pretopos (def. ).

If moreover H\mathbf{H} is an coherent (∞,1)-topos, then H coh\mathbf{H}_{coh} is an (,1)(\infty,1)-pretopos.

(Lurie, prop. A.6.1.6, cor. 6.1.7)

References

Last revised on January 29, 2018 at 09:50:25. See the history of this page for a list of all contributions to it.