nLab sifted (infinity,1)-category

Definition

Definition

An (∞,1)-category is sifted if a quasi-category $K \in sSet$ modelling it has the property that

1. it is not empty, $K \neq \emptyset$

2. The diagonal $K \to K \times K$ (in sSet) models a cofinal (∞,1)-functor.

Properties

Proposition

Let $C$ be an (∞,1)-category such that products preserve sifted (∞,1)-colimits (for instance an (∞,1)-topos, see universal colimits).

Then sifted (∞,1)-colimits preserve finite products.

This is (Lurie, lemma 5.5.8.11).

Examples

General

Proposition

The opposite category $\Delta^{op}$ of the simplex category is a sifted $(\infty,1)$-category.

(Lurie HTT, prop. 5.5.8.4).

Proposition

Every filtered (∞,1)-category is sifted.

(Lurie HTT, prop. 5.3.1.20).

In categories of commutative monoids

Proposition

In a category of commutative monoids in a symmetric monoidal $(\infty,1)$-category, sifted colimits are computed as sifted colimits in the underlying $(\infty,1)$-category.

See commutative monoid in a symmetric monoidal (∞,1)-category for details.

Related concepts

• sifted category, sifted $(\infty,1)$-category

• filtered category, filtered (∞,1)-category

References

• Jacob Lurie, Section 5.5.8 of: Higher Topos Theory