nLab
cofinal diagrams

Cofinal diagrams

Definition

Two diagrams D,DD,D' in a category CC (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 DD' is a subdiagram of DD 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 DD and DD', 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=DFD' = D F, and let L:D×(01)CL\colon D \times (0 \to 1) \to C be an initial cocone over DD; it seems natural to ask that L(F×(01))L \circ (F \times (0 \to 1)) be an initial cocone over DD'.

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

  • Every infinite subset of ω\omega is cofinal with ω\omega, as diagrams in ω+1\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 CC. This can probably be phrased internally to CC, at the cost of intuition.


  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.

Revised on December 31, 2011 20:55:20 by Urs Schreiber (82.113.121.71)