# nLab filtered (infinity,1)-category

Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# Contents

## Idea

This is the analog of a filtered category in the context of (∞,1)-categories.

The main purpose of considering filtered (∞,1)-categories is to define filtered (∞,1)-colimits, which are the colimits that commute with finite (∞,1)-limits.

## Definition

###### Definition

Let $\kappa$ be a regular cardinal, and let $C\in \sSet$ be an (∞,1)-category, incarnated as a quasicategory.

$C$ is called $\kappa$-filtered if for all $\kappa$-small $K\in\sSet$ and every morphism $f\colon K\to C$ there is a morphism $\hat p\colon \rcone(K)\to C$ extending $f$, where $\rcone(K)$ denotes the (right) cone of the simplicial set $K$. $C$ is called filtered if it is $\omega$-filtered.

## Properties

###### Proposition

An (∞,1)-category $K$ is filtered precisely if (∞,1)-colimits of shape $K$ in ∞Grpd commute with all finite (∞,1)-limits, hence if

${\lim_\to} : Func(K, \infty Grpd) \to \infty Grpd$

is a left exact (∞,1)-functor.

This is HTT, prop. 5.3.3.3.

###### Proposition

A filtered $(\infty,1)$-category is in particular a sifted (∞,1)-category.

This appears as (Lurie, prop. 5.3.1.20). Since sifted (∞,1)-colimits are precisely those that commute with finite products, this is a direct reflection of the fact that finite products are a special kind of finite (∞,1)-limits.

###### Corollary

For $C$ a filtered $(\infty,1)$-category, the diagonal (∞,1)-functor $\Delta : C \to C \times C$ if a cofinal (∞,1)-functor.

###### Proposition

A filtered $(\infty,1)$-category is weakly contractible, i.e. when incarnated as a quasicategory, it is weakly equivalent to a point in the Kan-Quillen model structure on simplicial sets.

###### Proof

This is (Lurie, Lemma 5.3.1.18).

## Reference

Section 5.3.1 of

Last revised on November 8, 2021 at 04:58:50. See the history of this page for a list of all contributions to it.