Given any object in any category , the subobjects of form a poset (though in general it may be a large one, i.e. a partially ordered proper class; cf. well-powered category). This is 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.
If is a morphism that has pullbacks along monomorphisms, then pullback along induces a poset morphism , called inverse image. This is functorial in the sense that if also has this property, then .
If has pullbacks of monomorphisms, is often used to denote the contravariant functor whose action on morphisms is .
Martin Brandenburg, Concise definition of subobjects (mathoverflow:184196)
Discussions of this can be found in A.1.3 of Johnstone’s Elephant, and also this MO discussion. ↩
Last revised on May 25, 2022 at 13:39:53. See the history of this page for a list of all contributions to it.