nLab
filtered category

Idea

A filtered category is a categorification of the concept of directed set. In addition to having an upper bound (but not necessarily a coproduct) for every pair of objects, there must also be an upper bound (but not necessarily a coequaliser) for every pair of parallel morphisms.

A diagram $F:D\to C$ where $D$ is a filtered category is called a filtered diagram. A colimit of a filtered diagram is called a filtered colimit.

A category whose opposite is filtered is called cofiltered.

Definitions

A (finitely) filtered category (sometimes called a filtrant category, as for instance in Kashiwara–Schapira's book Categories and Sheaves) is a category $C$ in which any finite diagram has a cocone. That is, for any finite category $D$ and any functor $F:D\to C$, there exists an object $c\in C$ and a natural transformation $F\to \Delta c$ where $\Delta c:D\to C$ is the constant diagram at $c$.

Equivalently, filtered categories can be characterized as those categories where, for every finite diagram $J$, the diagonal functor $\Delta :C\to C^J$ is final. This point of view can be generalized to other kinds of categories whose colimits are well-behaved with respect to a type of limit, such as sifted categories.

This can be rephrased in more elementary terms by saying that:

• There exists an object of $C$ (the case when $D=\varnothing$)
• For any two objects $c_1,c_2\in C$, there exists an object $c_3\in C$ and morphisms $c_1\to c_3$ and $c_2\to c_3$.
• For any two parallel morphisms $f,g:c_1\to c_2$ in $C$, there exists a morphism $h:c_2\to c_3$ such that $hf=hg$.

Just as all finite colimits can be constructed from initial objects, binary coproducts, and coequalizers, so a cocone on any finite diagram can be constructed from these three.

More generally, if $\kappa$ is a cardinal number, then a $\kappa$-filtered category is one such that any diagram $D\to C$ has a cocone where $D$ has fewer than $\kappa$ arrows. Note that a preorder is $\kappa$-filtered as a category just when it is $\kappa$-directed as a preorder.

Examples

• A filtered preorder is the same as a directed one.

• Every category with a terminal object is filtered.

• Every category which has finite colimits is filtered.

• A product of filtered categories is filtered.