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

Contents

Contents

Idea

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

Definition

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

Definition

A filtering on an object X𝒞X \in \mathcal{C} a sequential diagram X :(,<)𝒞X_\bullet \colon (\mathbb{Z}, \lt) \to \mathcal{C}

X n1X nX n+1 \cdots X_{n-1} \to X_n \to X_{n+1} \to \cdots

such that

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

is the sequential homotopy colimit of the tower.

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

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

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)

tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Postnikov towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence

Last revised on February 12, 2016 at 04:55:21. See the history of this page for a list of all contributions to it.