nLab poset of subobjects

Idea

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).

Properties

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 $Sub(f) = f^*$.

Relation to the preorder of monomorphisms

The poset of subobjects $Sub(X)$ is the posetal reflection of the preorder $Mono(X)$ of monomorphisms into $X$.

If one opts for the alternative¹ 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.

Subobject poset functor and relation to hyperdoctrines

• 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 $\mathrm{Sub}:C^\op \to BoolAlg$.

Related concepts

• lattice of subgroups

• well-powered category

• (n+1,1)-category of n-truncated objects

References

• Elephant

• Martin Brandenburg, Concise definition of subobjects (mathoverflow:184196)

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1. Discussions of this can be found in A.1.3 of Johnstone’s Elephant, and also this MO discussion. ↩