nLab filtered object

Definition

Definition

Given a category $\mathcal{C}$, then a filtered object is an object $X$ of $\mathcal{C}$ equipped with a filtration:

A descending filtration or decreasing filtrations of $X$ is a sequence of morphisms (often required to be monomorphisms) of the form

$$\cdots \longrightarrow X_{n+1} \longrightarrow X_n \longrightarrow X_{n-1} \longrightarrow \cdots \longrightarrow X$$

An ascending filtration or increasing filtration of $X$ is of the form

$$X \longrightarrow \cdots \longrightarrow X_{n-1} \longrightarrow X_n \longrightarrow X_{n+1} \longrightarrow \cdots$$

(In more generality, it is also possible to index using any ordered abelian group.)

Definition

A decreasing filtration $\{X_s\}_s$ of $X$ (def. ) is called

• exhaustive if $\underset{\longrightarrow}{\lim}_s X_s \simeq X$ ($X$ is the colimit of the filter stages)

• Hausdorff if $\underset{\longleftarrow}{\lim}_s X_s \simeq 0$ (the limit of the the filter stages is initial (zero in the case of an abelian category))

and for a filtration of abelian groups:

• complete if $\underset{\longleftarrow}{\lim}^1_s X_n = 0$ (also the first derived limit (lim^1) vanishes)

(Boardman 99, def. 2.1, see also Rognes 12, section 2.1)

Remark

Beware that for decreasing filtrations (def. ) often exhaustiveness (def. ) is understood by default, and often it is even assumed by default that $X_s = X$ for $s \leq 0$.

Properties

Proposition

If a decreasing filtration (def. ) of abelian subgroups

$$\cdots \hookrightarrow A_{n+1} \hookrightarrow A_n \hookrightarrow A_{n-1} \hookrightarrow \cdots \hookrightarrow A$$

is exhaustive and complete Hausdorff (def. ) then $A$ may be reobtained from the subquotients of the filtering as the limit/colimit

$$\begin{aligned} A & \simeq \underset{\longleftarrow}{\lim}_s (A/A_s) \\ & \simeq \underset{\longleftarrow}{\lim}_s (\underset{\longrightarrow}{\lim}_t ( A_t / A_s )) \end{aligned} \,.$$

(Boardman 99, prop. 2.5)

Related concepts

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

References

• Michael Boardman, section I.2 of Conditionally convergent spectral sequences, 1999 (pdf)

• John Rognes, The Adams spectral sequence (following Bruner 09), 2012 (pdf)

• Paolo Perrone, Starting Category Theory, World Scientific, 2024, Section 3.2.7. (website)