nLab
sifted (infinity,1)-category

Definition

Definition

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

1. it is not empty, $K\ne \varnothing$

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

Proposition

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

This appears as (Lurie, prop. 5.3.1.20).

In categories of commutative monoids

Proposition

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

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

Related concepts

• sifted category, sifted $(\mathrm{\infty },1)$-category

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

References

Section 5.5.8 of

• Jacob Lurie, Higher Topos Theory