Given any object $X$ in any category $C$, the subobjects of $X$ 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 $X$, or the subobject poset of $X$.
Sometimes it is the poset of regular subobjects that really matters (although these are the same in any (pre)topos).
If $C$ 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 $X$ 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 $X$ form a frame, so we may speak of the frame of subobjects.
The reader can probably think of other variations on this theme.
If $f : X \to Y$ is a morphism that has pullbacks along monomorphisms, then pullback along $f$ induces a poset morphism $f^* : Sub(Y) \to Sub(X)$, called the preimage or inverse image. This is functorial in the sense that if $g : Y \to Z$ also has this property, then $f^* \circ g^* = (g \circ f)^*$.
If $C$ has pullbacks of monomorphisms, $Sub$ is often used to denote the contravariant functor $C^{op} \to Poset$ whose action on morphisms is $Sub(f) = f^*$.
The poset of subobjects $Sub(X)$ is the posetal reflection of the preorder $Mono(X)$ of monomorphisms into $X$.
If one opts for the alternative^{1} definition that subobjects are monomorphisms into the object (not isomorphism classes thereof), then the reflection quotient map $Mono(X) \to Sub(X)$ is an equivalence.
Every category with pullbacks of monomorphisms has a contravariant functor $\mathrm{Sub}:C^\op \to Pos$ to the category of posets called the subobject poset functor, making it into a hyperdoctrine.
Every finitely complete category has a subobject poset functor $\mathrm{Sub}:C^\op \to SemiLat$ to the category of semilattices.
Every regular category $C$ is a regular hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of infimum-semilattices $\mathrm{Sub}:C^\op \to InfSemiLat$.
Every coherent category $C$ is a coherent hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of distributive lattices $\mathrm{Sub}:C^\op \to DistLat$.
Every geometric category $C$ is a geometric hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of frames $\mathrm{Sub}:C^\op \to Frm$.
Every Heyting category $C$ is a first-order hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of Heyting algebras $\mathrm{Sub}:C^\op \to HeytAlg$.
Every Boolean category $C$ is a Boolean hyperdoctrine induced by the subobject poset functor, where the subobject poset functor has codomain the subcategory of Boolean algebras $\mathrm{Sub}:C^\op \to BoolAlg$.
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.