poset of subobjects
Could not include topos theory - contents
Given any object in any category , the subobjects of form a poset, called (naturally enough) the poset of subobjects of , or the subobject poset of .
Sometimes it is the poset of regular subobjects that really matters (although these are the same in any (pre)topos).
If is finitely complete, then the subobjects form a meet-semilattice, so we may speak of the semilattice of subobjects.
In any coherent category (such as a pretopos), the subobjects form a distributive lattice, so we may speak of the lattice of subobjects.
In any Heyting category (such as a topos), the subobjects of form a Heyting algebra, so we may speak of the algebra of subobjects.
The reader can probably think of other variations on this theme.
Revised on March 15, 2012 15:47:35
by Urs Schreiber