nLab filtered object in an (∞,1)-category

Contents

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

Contents

Idea

In (∞,1)-category theory the typical notion of filtered object in category theory simplifies: instead of a sequential diagram of monomorphisms, it is just any tower diagram.

Definition

Let $\mathcal{C}$ be an (∞,1)-category.

Definition

A filtering on an object $X \in \mathcal{C}$ is a sequential diagram $X_\bullet \colon (\mathbb{Z}, \lt) \to \mathcal{C}$

$\cdots X_{n-1} \to X_n \to X_{n+1} \to \cdots$

such that

$X \simeq \underset{\rightarrow}{\lim} X_\bullet$

is the sequential homotopy colimit of the tower.

Dually a cofiltering of $X$ is a tower $X_\bullet$ such that

$X \simeq \underset{\leftarrow}{\lim} X_\bullet$

is the homotopy limit.

This appears as (Higher Algebra, def. 1.2.2.9).

Applications

Spectral sequence

If $\mathcal{C}$ is a stable (∞,1)-category with sequential limits/sequential colimits and with a t-structure, then every filtering/cofiltering on $X$ induces a spectral sequence of a filtered stable homotopy type which converges to the homotopy groups of $X$.

The spectral sequence of a filtered stable homotopy type associated with the chromatic tower (regarded as a filtered object in an (infinity,1)-category) is the chromatic spectral sequence (Wilson 13, section 2.1.2)

Last revised on May 23, 2020 at 06:03:55. See the history of this page for a list of all contributions to it.