category theory

# Contents

## Idea

Generally, in a category $𝒞$, one of the more usual forms of a filtered object encountered is an object $X$ equipped with a filtration. A descending filtration a sequence of monomorphisms of the form

$\cdots ↪{X}_{\left(n\right)}↪\cdots ↪{X}_{\left(2\right)}↪{X}_{\left(1\right)}↪{X}_{\left(0\right)}=:X\phantom{\rule{thinmathspace}{0ex}}$\cdots \hookrightarrow X_{(n)}\hookrightarrow \cdots \hookrightarrow X_{(2)} \hookrightarrow X_{(1)} \hookrightarrow X_{(0)} =: X \,

and an ascending filtration of the form

${X}_{\left(0\right)}↪{X}_{\left(1\right)}↪{X}_{\left(2\right)}\cdots ↪{X}_{\left(3\right)}↪\subset X.$X_{(0)} \hookrightarrow X_{(1)}\hookrightarrow X_{(2)}\cdots \hookrightarrow X_{(3)} \hookrightarrow \subset X.

There are variants in which the sequence may be infinite ‘to the right’ or may be bounded or stationary in some way to the left. In situations where quotients make sense, extra conditions on the quotients ${X}_{\left(n\right)}/{X}_{\left(n+1\right)}$, are often imposed; see associated graded object.

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

## Examples

Revised on November 18, 2013 11:16:33 by Urs Schreiber (82.169.114.243)