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

A filtered colimit is a colimit of a functor $$F:J\to C$$ where $$J$$ is a filtered category.

Cofiltered categories
A category $$J$$ is cofiltered if the opposite category $$J^{\mathrm{op}}$$ is filtered. In detail, a category is cofiltered when
 * it is not empty,
 * for every two objects $$j$$ and $$j'$$ in $$J$$ there exists an object $$k$$ and two arrows $$f:k\to j$$ and $$f':k \to j'$$ in $$J$$,
 * for every two parallel arrows $$u,v:j\to i$$ in $$J$$, there exists an object $$k$$ and an arrow $$w:k\to j$$ such that $$uw=vw$$.

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

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

κ-filtered categories
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 a cocone over any diagram in $$J$$ of the form $$\{\ \ \}\rightarrow J$$, $$\{j\ \ \ j'\}\rightarrow J$$, or $$\{i\rightrightarrows j\}\rightarrow J$$. The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist for any finite diagram; in other words, a category $$J$$ is filtered (according to the above definition) if and only if there is a cocone over any finite diagram $$d: D\to J$$.

Extending this, given a regular cardinal κ, a category $$J$$ is defined to be κ-filtered if there is a cocone over every diagram $$d$$ in $$J$$ of cardinality smaller than κ. (A small diagram is of cardinality κ if the morphism set of its domain is of cardinality κ.)

A κ-filtered colimit is a colimit of a functor $$F:J\to C$$ where $$J$$ is a κ-filtered category.