# nLab filtered derived category

## Idea

Given an abelian category, the filtered derived category is the category of chain complexes of filtered objects up to quasi-isomorphism.

## Definition

Given an abelian category $A$, one may consider the additive category $Fil(A)$ of filtered objects in $A$, whose objects are pairs $(a, F^*)$ with $a$ an object of $A$ and $F^*$ a filtration on $a$. Let $Comp(Fil(A))$ denote the category of chain complexes in $Fil(A)$. One defines a morphism in $Comp(Fil(A))$ to be a quasi-isomorphism if it induces quasi-isomorphisms on each degree of the associated graded objects. The filtered derived category of $A$,

$D_{fil}(A) = Comp(Fil(A))[qis^{-1}]$

is the localization of $Comp(Fil(A))$ at the class of quasi-isomorphisms.