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 an elementary topos), the subobjects of form a Heyting algebra, so we may speak of the algebra of subobjects.
In any geometric category (such as a Grothendieck topos), the subobjects of form a frame, so we may speak of the frame 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 the preimage or 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 .
The poset of subobjects is the posetal reflection of the preorder of monomorphisms into .
If one opts for the alternative1 definition that subobjects are monomorphisms into the object (not isomorphism classes thereof), then the reflection quotient map is an equivalence.
Every category with pullbacks of monomorphisms has a contravariant functor to the category of posets called the subobject poset functor, making it into a hyperdoctrine.
Every finitely complete category has a subobject poset functor to the category of semilattices.
Every regular category is a regular hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of infimum-semilattices .
Every coherent category is a coherent hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of distributive lattices .
Every geometric category is a geometric hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of frames .
Every Heyting category is a first-order hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of Heyting algebras .
Every Boolean category is a Boolean hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of Boolean algebras .
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 September 13, 2024 at 22:35:40. See the history of this page for a list of all contributions to it.