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 $\omega + 1$.

## References

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.

Last revised on May 6, 2022 at 05:43:58. See the history of this page for a list of all contributions to it.