nLab poset of subobjects



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In higher category theory




Given any object XX in any category CC, the subobjects of XX 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 XX, or the subobject poset of XX.

Sometimes it is the poset of regular subobjects that really matters (although these are the same in any (pre)topos).


If CC 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 XX 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 f:XYf : X \to Y is a morphism that has pullbacks along monomorphisms, then pullback along ff induces a poset morphism f *:Sub(Y)Sub(X)f^* : Sub(Y) \to Sub(X), called the preimage or inverse image. This is functorial in the sense that if g:YZg : Y \to Z also has this property, then f *g *=(gf) *f^* \circ g^* = (g \circ f)^*.

If CC has pullbacks of monomorphisms, SubSub is often used to denote the contravariant functor C opPosetC^{op} \to Poset whose action on morphisms is Sub(f)=f *Sub(f) = f^*.

Relation to the preorder of monomorphisms

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

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)Sub(X)Mono(X) \to Sub(X) is an equivalence.

Subobject poset functor and relation to hyperdoctrines


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

Last revised on November 14, 2022 at 17:03:44. See the history of this page for a list of all contributions to it.