A pair of diagrams in a category (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 is a subdiagram of 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 and , 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.
Let , and let be an initial cocone over ; it seems natural to ask that be an initial cocone over .
Nonempty subsets of a finite total order are cofinal iff they have the same maximum.
Every infinite subset of is cofinal with , as diagrams in .
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 09:43:58. See the history of this page for a list of all contributions to it.