Applications

Filtered categories

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 diagramF:D→CF:D\to C where DD 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

Ordinary filteredness

Definition

That is, for any finite category DD and any functorF:D→CF:D\to C, there exists an object c∈Cc\in C and a natural transformationF→ΔcF\to \Delta c where Δc:D→C\Delta c:D\to C is the constant diagram at cc. If D+D^+ is the result of freely adjoining a terminal object to a category DD, then the condition is the same as that any functor F:D→CF: D \to C with finite domain admits an extension F˜:D+→C\tilde{F}: D^+ \to C.

Equivalently, filtered categories can be characterized as those categories where, for every finite diagram JJ, the diagonal functor Δ:C→CJ\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 CC (the case when D=∅D=\emptyset)

For any two objects c1,c2∈Cc_1,c_2\in C, there exists an object c3∈Cc_3\in C and morphisms c1→c3c_1\to c_3 and c2→c3c_2\to c_3.

For any two parallel morphismsf,g:c1→c2f,g:c_1\to c_2 in CC, there exists a morphism h:c2→c3h:c_2\to c_3 such that hf=hgh f = h g.

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.

Higher filteredness

More generally, if κ\kappa is an infinite regular cardinal (or an arity class), then a κ\kappa-filtered category is one such that any diagram D→CD\to C has a cocone where DD has <κ\lt \kappa arrows, or equivalently that any functor F:D→CF: D \to C whose domain has fewer than κ\kappa morphisms admits an extension F˜:D+→C\tilde{F}: D^+ \to C. The usual filtered categories are then the case κ=ω\kappa = \omega, i.e., where the DD have fewer than ω\omega morphisms (in other words are finite). (We could also say in this case “ℵ0\aleph_0-filtered”, but ω\omega-filtered is more usual in the literature.)

Note that a preorder is κ\kappa-filtered as a category just when it is κ\kappa-directed as a preorder.

Generalized filteredness

Even more generally, if 𝒥\mathcal{J} is a class of small categories, a category CC is called 𝒥\mathcal{J}-filtered if CC-colimits commute with 𝒥\mathcal{J}-limits in Set. When 𝒥\mathcal{J} is the class of all κ\kappa-small categories for an infinite regular cardinal κ\kappa, then 𝒥\mathcal{J}-filteredness is the same as κ\kappa-filteredness as defined above. See ABLR.

If 𝒥\mathcal{J} is the class consisting of the terminal category and the empty category — which is to say, the class of κ\kappa-small categories when κ\kappa is the finite regular cardinal 22 — then being 𝒥\mathcal{J}-filtered in this sense is equivalent to being connected. Note that this is not what the explicit definition given above for infinite regular cardinals would specialize to by simply setting κ=2\kappa=2 (that would be simply inhabitation).