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

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)

Related concepts

• filtered chain complex

• filtered spectrum

• tower of homotopy fibers

• persistent homotopy theory

tower diagram/filtering	spectral sequence of a filtered stable homotopy type
filtered chain complex	spectral sequence of a filtered complex
Postnikov tower	Atiyah-Hirzebruch spectral sequence
chromatic tower	chromatic spectral sequence
skeleta of simplicial object	spectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebra	Adams spectral sequence
filtration by support	…
slice filtration	slice spectral sequence

filtered objects

• filtered topological space

  • spectral sequence of a tower of fibrations

• filtered vector space

• filtered chain complex

  • spectral sequence of a filtered complex

  • Hodge filtration

• filtered ring

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

• filtered stable homotopy type

  • spectral sequence of a filtered stable homotopy type

• filtered probability space

associated graded objects

• associated graded vector space

• associated graded ring