nLab
filtered limit

Definitions

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

For $\kappa$ a regular cardinal a $\kappa$-filtered colimit is one over a $\kappa$-filtered category.

Similarly, a cofiltered limit is a limit of a functor $F:D\to C$ where $D$ is a cofiltered category, or equivalently of a contravariant functor $F:D\to C$ (that is a functor $F:D^{\mathrm{op}}\to C$) where $D$ is a filtered category. A cofiltered limit may also be called a filtered limit (although this can be unclear); the respective terms filtered direct limit and filtered inverse limit are also popular.

A functor that preserves all finitely filtered colimits is called finitary.

Properties

One of the reasons filtered colimits are important is that in Set, filtered colimits commute with finite limits (and in fact, the filtered colimits can be characterized as precisely those colimits that commute with all finite limits in $\mathrm{Set}$).

For more on this see also limits and colimits by example.

Filtered colimits are also important in the theory of locally presentable and accessible categories. See also pro-object and ind-object.

According to 1.5 and 1.21 in the book

• Adamek & Rosicky, Locally presentable and accessible categories

a category has $\kappa$-directed colimits precisely if it has $\kappa$-filtered ones, and a functor preserves $\kappa$-directed colimits iff it preserves $\kappa$-filtered ones.

The fact that directed colimits suffice to obtain all filtered ones may be regarded as a convenience similar to the fact that all colimits can be constructed from coproducts and coequalizers. Of course, a dual result holds for codirected limits.