nLab filtered category

Idea

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

The dual notion of filtered category is that of cofiltered category: a category whose opposite is filtered.

Definitions

Ordinary filteredness

Definition

A (finitely) filtered category is a category $C$ in which every finite diagram has a cocone.

More in details, this requirement is that: For any finite category $D$ and any functor $F \colon D\to C$, 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 $c$. If $D^+$ is the result of freely adjoining a terminal object to a category $D$, then the condition is the same as that any functor $F: D \to C$ with finite domain admits an extension $\tilde{F}: D^+ \to C$.

Equivalently, filtered categories can be characterized as those categories where, for every finite diagram $J$, the diagonal functor $\Delta \colon C \to C^J$ 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.

All this may be rephrased in more elementary terms by saying that:

• There exists an object of $C$ (the case when $D=\emptyset$)

• 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 $h f = h g$.

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.

In constructive mathematics, the elementary rephrasing above is equivalent to every Bishop-finite diagram admitting a cocone.

Higher filteredness

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

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

In ABLR, they use the term $\infty$-filtered for a category that is $\kappa$-filtered for every cardinal $\kappa$. Thus, an $\infty$-filtered category is equivalently one in which every small diagram admits a cocone.

Generalized filteredness

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

If $\mathcal{J}$ is the class consisting of the terminal category and the empty category — which is to say, the class of $\kappa$-small categories when $\kappa$ is the finite regular cardinal $2$ — then being $\mathcal{J}$-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 $\kappa=2$ (that would be simply inhabitation).

Examples

• A filtered preorder is the same as a directed one: a filtered (0,1)-category.

• Every category with a terminal object is filtered.

• Every category which has finite colimits is filtered.

• A product of filtered categories is filtered.

Related concepts

• sifted category, sifted (∞,1)-category

• directed set, filtered category, filtered (∞,1)-category

• direct category

• filtered colimit

References

• Jiří Adámek, Francis Borceux, Stephen Lack, and Jiří Rosický, A classification of accessible categories, Journal of Pure and Applied Algebra 175 1-3 (2002) 7-30 [doi:10.1016/S0022-4049(02)00126-3]