An object of a category is small if it is -compact for some regular cardinal (and therefore also for all greater regular cardinals).
Here, is called -compact if the corepresentable functor preserves -directed colimits.
We unwrap the definition further. Let be a -filtered poset, i.e. one in which every sub-poset of cardinality has an upper bound in .
Let be a category and a diagram, called a -filtered diagram. Let be any object.
Then the condition that commutes with the colimit over means that the map of hom-sets
is an isomorphism, i.e. a bijection.
By the general properties of colimit (recalled at limits and colimits by example), the colimit
may be expressed as a coequalizer
hence as a quotient set of the the set of morphism in from into one of the objects . Being a quotient set, every element of it is represented by one of the original elements in .
This means that we have
Restatement
The map of hom-sets
is onto precisely if every morphism lifts to a morphism into one of the , schematically:
Let be a regular cardinal greater than . Then any -filtered category is also -filtered. For being -filtered means that any diagram in of size has a cocone; but any diagram of size is of course also . Thus, any -filtered colimit is also a -filtered colimit, so any functor which preserves -filtered colimits must in particular preserve -filtered colimits. It follows that any -compact object is also -compact.
By definition, in a locally presentable category every object is a colimit over small objects.
Smallness of objects plays a crucial role in the small object argument.
cosmall object?, which is just the dual concept, but is interesting in its own right.
Last revised on October 21, 2017 at 15:12:28. See the history of this page for a list of all contributions to it.