objects such that commutes with certain colimits
Let be a poset such that every directed subset of has a join; that is, is a dcpo. A compact element, or finite element, of is a compact object in regarded as a thin category; that is, homs out of it commute with these directed joins.
In other words, is compact precisely if for every directed subset of we have
Of course, the part of this is automatic, so the real condition is the part. In more elementary terms:
- If for a directed subset, then for some .
Revised on March 28, 2012 11:18:52
by Toby Bartels