# nLab (infinity,1)-pretopos

Contents

### Context

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## 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. ).

###### 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.

## References

Last revised on April 12, 2021 at 06:19:16. See the history of this page for a list of all contributions to it.