Most of the earliest instances of limits and colimits used in mathematics were for diagrams indexed by the partially ordered set of natural numbers, which we now call sequential (co)limits. Many of the nice features of these (co)limits apply more generally to codirected limits and directed colimits, where the indexing category is a (co)-directed set (much as sequences in topology generalise fruitfully to nets). A filtered (co)limit is a further generalisation of this, essentially removing the requirement that the indexing category be a poset while preserving the directedness aspect in a categorified way.
Another very important class of early limits and colimits involved situations that generalised intersections and unions. If one is looking at a family of subsets of some set, then one can close it up under finite intersections and/or unions (if they are not already included) and use it to index diagrams. For instance, the family of continuous functions defined on open neighbourhoods of some point in a topological space will have this property. It was noticed that these limits and colimits behaved very nicely and a closer look showed that it was the (co)filtering nature of the indexing category that was the key. This also leads us to filtered (co)limits.
So, a filtered colimit is a colimit over a diagram from a filtered category, and a filtered limit is a limit over a diagram from a cofiltered category. Taken in a suitable category such as Set, a colimit being filtered is equivalent to its commuting with finite limits.
More generally, for a regular cardinal, a -filtered colimit is one over a -filtered category (and dually), and when taken with values in Set these are precisely the colimits that commute with -small limits.
For a regular cardinal a -filtered colimit is one over a -filtered category.
The following is the crucial property of filtered colimits: that they commute with finite limits.
For and two diagram categories and
a diagram, there is a canonical morphism
is given by a cone, whose components
are in turn given by a cocone with components
This finally take to have as components
One checks that this indeed implies that all the components are natural and gives the existence of the original morphism.
Notice that in general is not an isomorphism.
A detailed components proof of the first part is in Borceux, theorem I2.13.4.
For more on this see also limits and colimits by example.
It is not true that filtered colimits and finite limits commute in any category which has them. A simple example is where is the poset of closed subspaces of the one-point compactification of the discrete space of natural numbers. If and ranges over finite subsets of , then , but .
According to 1.5 and 1.21 in LPAC, a category has -directed colimits precisely if it has -filtered ones, and a functor preserves -directed colimits iff it preserves -filtered ones. A proof of this result, following Adamek & Rosicky, may be found here?.
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.
is left exact. One may prove as a corollary that if is finitely complete, is flat if and only if it is left exact (preserves finite limits). Since this tensor product is automatically a left adjoint, we have the following basic result:
Section 2.13 in part I of