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
An ascending filtration or increasing filtration of $X$ is of the form
(In more generality, it is also possible to index using any ordered abelian group.)
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:
(Boardman 99, def. 2.1, see also Rognes 12, section 2.1)
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 $s \leq 0$.
If a decreasing filtration (def. ) of abelian subgroups
is exhaustive and complete Hausdorff (def. ) then $A$ may be reobtained from the subquotients of the filtering as the limit/colimit
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)
Last revised on July 21, 2024 at 18:05:19. See the history of this page for a list of all contributions to it.