Cofinal diagrams

category theory

# Cofinal diagrams

## Definition

A pair of diagrams $D,D'$ 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 $D'$ 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 $D'$, and so it seems not too much in violation of the principle of equivalence 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 $D' = D F$, and let $L\colon D \times (0 \to 1) \to C$ be an initial cocone over $D$; it seems natural to ask that $L \circ (F \times (0 \to 1))$ be an initial cocone over $D'$.

• 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 $\omega + 1$.

1. Cofinality is an ordinal invariant of ordinals, which doesn’t make sense in the present generality, so we need another name for the adjective.