poset of subobjects



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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 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^*.

  • If one opts for the alternative1 definition that subobjects are monomorphisms into the object (not isomorphism classes thereof), then one gets a preorder of subobjects instead. In any case, the poset of subobjects Sub(X)Sub(X) in our sense is the posetal reflection of the preorder Mono(X)Mono(X) of subobjects in the alternative sense, and of course the reflection quotient map Mono(X)Sub(X)Mono(X) \to Sub(X) is an equivalence.


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

Last revised on June 12, 2021 at 07:24:43. See the history of this page for a list of all contributions to it.