nLab
filtered object

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 sequence of monomorphisms

$$\cdots \hookrightarrow X_{(n)}\hookrightarrow \cdots \hookrightarrow X_{(2)}\hookrightarrow X_{(1)}\hookrightarrow X_{(0)}=:X.$$\cdots \hookrightarrow X_{(n)}\hookrightarrow \cdots \hookrightarrow X_{(2)} \hookrightarrow X_{(1)} \hookrightarrow X_{(0)} =: 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 $X_{(n)}/X_{(n+1)}$, are often imposed; see associated graded object.

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

Examples

• filtered topological space
• filtered chain complex
• filtered ring

Related concepts

• associated graded object
• stratification