Cofinal diagrams

Definition

Two diagrams $D,D\prime$ in a category $C$ (e.g.: directed systems, sub-posets, … ) are cofinal if they have equivalent colimits.

In most cases where the word “cofinal” is used, it seems to be the case $D\prime$ is a subdiagram of $D$ in whatever sense of subdiagram appears suitable. In that case the cofinality is equivalent to the inclusion functor being a cofinal functor.

As defined above, no relation is posited between $D$ and $D\prime$, and so it seems not too evil to define cofinalness¹ as “a single object is a colimit for both diagrams”.

It is not said that they are final if they have equivalent limits — the “co”s are not freely mutable, although the dual situation is doubtless just as interesting.

Examples

• Let $D\prime =DF$, and let $L:D\times (0\to 1)\to C$ be an initial cocone over $D$; it seems natural to ask that $L\circ (F\times (0\to 1))$ be an initial cocone over $D\prime$.

• Nonempty subsets of a finite total order are cofinal iff they have the same maximum.

• Every infinite subset of $\omega$ is cofinal with $\omega$, as diagrams in $\omega +1$.

Abuses

It is often said that two diagrams are cofinal even when neither has a colimit, if they acquire a common colimit on passing to a suitable completion of $C$. This can probably be phrased internally to $C$, at the cost of intuition.

Related concepts

• cofinal functor, cofinal diagram

• homotopy final functor

• final (∞,1)-functor