nLab
filtered (infinity,1)-category

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

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

Properties

Proposition

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

lim :Func(K,Grpd)Grpd {\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 (,1)(\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 CC a filtered (,1)(\infty,1)-category, the diagonal (∞,1)-functor Δ:CC×C\Delta : C \to C \times C if a cofinal (∞,1)-functor.

Reference

Section 5.3.1 of

Revised on May 20, 2014 01:52:25 by Urs Schreiber (185.37.147.12)