Incategory theory,filtered categories generalize the notion ofdirected 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 ofcofiltered category, which will be recalled below.

Acategory${\displaystyle J}$ isfiltered when

• it is not empty,
• for every two objects${\displaystyle j}$ and${\displaystyle j^{\prime }}$ in${\displaystyle J}$ there exists an object${\displaystyle k}$ and two arrows${\displaystyle f:j\to k}$ and${\displaystyle f^{\prime }:j^{\prime }\to k}$ 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${\displaystyle wu=wv}$.

Afiltered colimit is acolimit of afunctor${\displaystyle F:J\to C}$ where${\displaystyle J}$ is a filtered category.

A category${\displaystyle J}$ is cofiltered if theopposite category${\displaystyle J^{\mathrm{o}\mathrm{p}}}$ is filtered. In detail, a category is cofiltered when

• it is not empty,
• for every two objects${\displaystyle j}$ and${\displaystyle j^{\prime }}$ in${\displaystyle J}$ there exists an object${\displaystyle k}$ and two arrows${\displaystyle f:k\to j}$ and${\displaystyle f^{\prime }:k\to j^{\prime }}$ 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${\displaystyle uw=vw}$.

Acofiltered limit is alimit of afunctor${\displaystyle F:J\to C}$ where${\displaystyle J}$ is a cofiltered category.

Given asmall category${\displaystyle C}$, apresheaf of sets${\displaystyle C^{op}\to Set}$ that is a small filtered colimit of representable presheaves, is called anind-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${\displaystyle C^{op}}$.

There is a variant of "filtered category" known as a "κ-filtered category", defined as follows. This begins with the following observation: the three conditions in the definition of filtered category above say respectively that there exists acocone over any diagram in${\displaystyle J}$ of the form${\displaystyle \{\}\to J}$,${\displaystyle \{j{j}^{\prime }\}\to J}$, or${\displaystyle \{i\rightrightarrows j\}\to J}$. The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist forany finite diagram; in other words, a category${\displaystyle J}$ is filtered (according to the above definition)if and only if there is a cocone over anyfinite diagram${\displaystyle d:D\to J}$.

Extending this, given aregular cardinal κ, a category${\displaystyle J}$ is defined to be κ-filtered if there is a cocone over every diagram${\displaystyle d}$ in${\displaystyle J}$ ofcardinality smaller than κ. (A smalldiagram is of cardinality κ if themorphism set of its domain is of cardinality κ.)

A κ-filtered colimit is a colimit of afunctor${\displaystyle F:J\to C}$ where${\displaystyle J}$ is a κ-filtered category.

• Category theory