# Filtered category

In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered category which will be recalled below.

## Filtered categories

A category ${\displaystyle J}$ is filtered when

• it is not empty,
• for every two objects ${\displaystyle j}$ and ${\displaystyle j'}$ in ${\displaystyle J}$ there exists an object ${\displaystyle k}$ and two arrows ${\displaystyle f:j\to k}$ and ${\displaystyle f':j'\to k}$ in ${\displaystyle J}$,
• for every two parallel arrows ${\displaystyle u,v:i\to j}$ in ${\displaystyle J}$, there exists an object ${\displaystyle k}$ and an arrow ${\displaystyle w:j\to k}$ such that ${\displaystyle wu=wv}$.

A small diagram is said to be of cardinality ${\displaystyle \kappa }$ if the morphism set of its domain is of cardinality ${\displaystyle \kappa }$. A category ${\displaystyle J}$ is filtered if and only if there is a cocone over any finite diagram ${\displaystyle d:D\to J}$; more generally, for a regular cardinal ${\displaystyle \kappa }$, a category ${\displaystyle J}$ is said to be ${\displaystyle \kappa }$-filtered if for every diagram ${\displaystyle d}$ in ${\displaystyle J}$ of cardinality smaller than ${\displaystyle \kappa }$ there is a cocone over ${\displaystyle d}$.

A filtered colimit is a colimit of a functor ${\displaystyle F:J\to C}$ where ${\displaystyle J}$ is a filtered category. This readily generalizes to ${\displaystyle \kappa }$-filtered limits.

Given a small category ${\displaystyle C}$, a presheaf of sets ${\displaystyle C^{op}\to Set}$ that is a small filtered colimit of representable presheaves, is called an ind-object of the category ${\displaystyle C}$. Ind-objects of a category ${\displaystyle C}$ form a full subcategory ${\displaystyle Ind(C)}$ in the category of functors (presheaves) ${\displaystyle C^{op}\to Set}$. The category ${\displaystyle Pro(C)=Ind(C^{op})^{op}}$ of pro-objects in ${\displaystyle C}$ is the opposite of the category of ind-objects in the opposite category ${\displaystyle C^{op}}$.

## Cofiltered categories

A category ${\displaystyle J}$ is cofiltered if the opposite category ${\displaystyle J^{\mathrm {op} }}$ is filtered. In detail, a category is cofiltered when

• it is not empty
• for every two objects ${\displaystyle j}$ and ${\displaystyle j'}$ in ${\displaystyle J}$ there exists an object ${\displaystyle k}$ and two arrows ${\displaystyle f:k\to j}$ and ${\displaystyle f':k\to j'}$ in ${\displaystyle J}$,
• for every two parallel arrows ${\displaystyle u,v:j\to i}$ in ${\displaystyle J}$, there exists an object ${\displaystyle k}$ and an arrow ${\displaystyle w:k\to j}$ such that ${\displaystyle uw=vw}$.

A cofiltered limit is a limit of a functor ${\displaystyle F:J\to C}$ where ${\displaystyle J}$ is a cofiltered category.