nLab (infinity,1)-pretopos

Idea

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.

Definition

Definition

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

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

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

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

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

5. realization of groupoid objects is universal.

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

Lurie, def. A:6.1.1

Examples

Example

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

(Lurie, example A.6.1.5)

Example

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

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

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)

Related concepts

• elementary (∞,1)-topos

References

• Jacob Lurie, appendix A of Spectral Algebraic Geometry (pdf)