filtered object in nLab

Context

Category theory

• Definition
• Properties
• Related concepts
• References

Definition

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

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

⋯⟶X n+1⟶X n⟶X n−1⟶⋯⟶X \cdots \longrightarrow X_{n+1} \longrightarrow X_n \longrightarrow X_{n-1} \longrightarrow \cdots \longrightarrow X

An ascending filtration or increasing filtration of XX is of the form

X⟶⋯⟶X n−1⟶X n⟶X n+1⟶⋯ X \longrightarrow \cdots \longrightarrow X_{n-1} \longrightarrow X_n \longrightarrow X_{n+1} \longrightarrow \cdots

Definition

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

• exhaustive if lim⟶ sX s≃X\underset{\longrightarrow}{\lim}_s X_s \simeq X (XX is the colimit of the filter stages)

• Hausdorff if lim⟵ sX s≃0\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 lim⟵ s 1X n=0\underset{\longleftarrow}{\lim}^1_s X_n = 0 (also the first derived limit (lim^1) vanishes)

Proposition

If a decreasing filtration (def. ) of abelian subgroups

⋯↪A n+1↪A n↪A n−1↪⋯↪A \cdots \hookrightarrow A_{n+1} \hookrightarrow A_n \hookrightarrow A_{n-1} \hookrightarrow \cdots \hookrightarrow A

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

A ≃lim⟵ s(A/A s) ≃lim⟵ s(lim⟶ t(A t/A s)). \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} \,.

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