nLab
subobject

Subobjects

Definition

As classes of monomorphisms

A subobject of an object cc in a category CC is an isomorphism class of monomorphisms

i:ac i: a \hookrightarrow c

into cc. (Two morphisms i:aci: a \to c, j:bcj: b \to c are isomorphic if there exists an isomorphism k:abk: a \to b such that i=jki = j k.)

Monos into an object cc are preordered by a relation

(i:ac)(j:bc) (i: a \to c) \leq (j: b \to c)

defined by the condition that there exists k:abk: a \to b such that i=jki = j k. (There is at most one such kk since jj is monic, and such kk is monic since ii is monic.)

A subobject of cc may equivalently be defined as an element of the posetal reflection Sub(c)Sub(c) of this preorder.

In terms of over categories

Let C cC_c be the full subcategory of the over category C/cC/c on monomorphisms. Then C cC_c is the poset of subobjects of cc and the set of isomorphism classes of C cC_c is the set of subobjects of cc. However this “set” can be in fact a proper class in general, see well-powered category.

Generalizations

  • More generally, in some contexts we may take “subobject” to mean an isomorphism class of morphisms i:aci: a\to c satisfying some suitable condition other than being a monomorphism (usually a stronger one). Common choices are strong monomorphisms, regular monomorphisms, or the right class of some orthogonal factorization system. (The latter choice has the advantage that then images will automatically exist.)

    • For example, in Top a monomorphism is just a continuous injective function, whereas the strong and regular monomorphisms coincide and are the subspace embeddings. In some contexts at least, one can argue that subspace embeddings are a more appropriate notion of “subobject” in TopTop (for example, if one wants to exhibit it as a locally bounded category). A similar thing happens in a quasitopos.
  • The partial order on the collection of subobjects internalizes into contexts more general than Set. For instance in every topos the subobject classifier Ω\Omega has the structure of an internal poset (see there).

Properties

The poset of subobjects.

  • For CC an accessible category, cCc \in C any object, the poset Sub(c)Sub(c) of subobjects of cc is a small category.

  • Suppose EE is a well-powered category. Denote by Sub(X)Sub(X) the poset of subobjects of object XX in EE. The correspondence Sub:XSub(X)Sub:X\mapsto Sub(X) may be extended to a contravariant functor EPosE \to Pos (that is a functor E opPosE^op \to Pos), namely if f:XYf: X\to Y is arbitrary and m:SYm:S\hookrightarrow Y is an element of Sub(Y)Sub(Y), i.e. monic, then the pullback f *(m):f *(S)Xf^*(m):f^*(S)\to X of mm along ff is automatically a monic; the correspondence mf *(m)m\mapsto f^*(m) describes Sub(f)Sub(f) at the level of representatives of subobjects.

Limits and colimits of subobjects

Assume that the ambient category has all limits and colimits considered in the following.

Definition

For XCX \in C an object, Sub(X)Sub(X) the poset of subobjects and
U 1,U 2XU_1, U_2 \hookrightarrow X two subobjects,

  • their product in Sub(X)Sub(X) is denoted U 1U 2U_1 \cap U_2 or U 1U 2U_1 \wedge U_2 and called the intersection or meet of the two subobjects;

  • their coproduct in Sub(X)Sub(X) is denoted U 1U 2U_1 \cup U_2 or U 1U 2U_1 \vee U_2 and called the union or join of the two subobjects.

Two subobjects U 1,U 2Sub C(X)U_1, U_2 \in Sub_C(X) are called disjoint if U 1U 2=U_1 \cap U_2 = \emptyset is the initial object.

Proposition

Let CC be a topos.

  1. The intersection of two subobjects in Sub C(X)Sub_C(X) is their fiber product in CC: the diagram

    U 1U 2 U 2 U 1 X \array{ U_1 \cap U_2 &\to& U_2 \\ \downarrow && \downarrow \\ U_1 &\to& X }

    is a pullback diagram.

  2. The union of two subobjects U 1,U 2Sub C(X)U_1, U_2 \in Sub_C(X) is the pushout U 1 U 1U 2U 2U_1 \coprod_{U_1 \cap U_2} U_2 in CC: the diagram

    U 1U 2 U 2 U 1 U 1U 2 \array{ U_1 \cap U_2 &\to& U_2 \\ \downarrow && \downarrow \\ U_1 &\to& U_1 \cup U_2 }

    is a pushout diagram.

  3. This last pushout diagram is also a pullback diagram.

Proof

For the first point: Since monomorphism are (as discussed there) stable under pullback and composition, the fiber product is a subobject. Its universal property as a limit in CC then implies its universal property as a product in Sub C(X)Sub_C(X).

For the second point: by the same kind of argument, it is sufficient to show that the canonical morphism U 1 U 1U 2U 2XU_1 \coprod_{U_1 \cap U_2} U_2 \to X exhibits the coproduct as a subobject.

Since monomorphisms (as discussed there) are characterized by the fact that the pullback along themselves is their domain, it is sufficient to show that

U 1 U 1U 2U 2 Id U 1 U 1U 2U 2 id U 1 U 1U 2U 2 i X \array{ U_1 \coprod_{U_1 \cap U_2} U_2 &\stackrel{Id}{\to}& U_1 \coprod_{U_1 \cap U_2} U_2 \\ {}^{\mathllap{id}}\downarrow && \downarrow \\ U_1 \coprod_{U_1 \cap U_2} U_2 &\stackrel{i}{\to}& X }

is a pullback diagram. For showing this we use that in a topos we have universal colimits, so that equivalently it is sufficient to show that

U 1 U 1U 2U 2(i *U 1) i *(U 1U 2)(i *U 2). U_1 \coprod_{U_1 \cap U_2} U_2 \simeq (i^* U_1) \coprod_{i^* (U_1 \cap U_2)} (i^* U_2) \,.

To see this, again use universal colimits to get

i *U 1 U 1× X(U 1 U 1U 2U 2) (U 1× XU 1) U 1× X(U 1U 2)(U 1× XU 2) U 1 U 1U 2(U 1× XU 2) U 1 U 1U 2(U 1U 2) U 1 \begin{aligned} i^* U_1 & \simeq U_1 \times_X (U_1 \coprod_{U_1 \cap U_2} U_2) \\ & \simeq (U_1 \times_X U_1) \coprod_{U_1 \times_X (U_1 \cap U_2)} (U_1 \times_X U_2) \\ & \simeq U_1 \coprod_{U_1 \cap U_2} (U_1 \times_X U_2) \\ & \simeq U_1 \coprod_{U_1 \cap U_2} (U_1 \cap U_2) \\ & \simeq U_1 \end{aligned}

and similarly

i *U 2U 2 i^* U_2 \simeq U_2

and

i *(U 1U 2)(U 1U 2). i^* (U_1 \cap U_2) \simeq (U_1 \cap U_2) \,.

This proves the second point.

The third point is directly verified by checking the universal property.

Comparison with the notion of “subset”

The notion of subobject figures prominently in topos theory and in other approaches to set theory based on categories. It is not an exact translation of the usual notion of “subset” in traditional set theory; in ordinary set theory, the notion of subset is defined in terms of a global elementhood relation between sets, which one doesn’t have in categorical set theory (and which one wouldn’t necessarily want: it’s “evil” in the sense of not being invariant with respect to isomorphism).

Category-theoretically, the traditional notion of subset gives a way of picking out a canonical representative or “normal form” among all the monos in an isomorphism class. As we intimated, there is no intrinsic way of defining such representatives in the theory of toposes: such would have to be considered an extra structure on a topos. Mathematically, there is no particular gain in having such structure around; at best it enables a traditional mode of discourse in which subsets are concrete maps, and to this end it can function as a linguistic or psychological convenience.

On the other hand, there is no particular harm either in having such structure around, as long as one remembers that it is not an isomorphism invariant. People will instinctively turn to canonical representatives whenever they can – think of what we would tell a student who asks for help understanding how to multiply elements in 13\mathbb{Z}_13 – and even category theorists do so when they are available.

Examples

References

Standard textbook references include section I.3 of

and

Revised on January 20, 2014 13:00:32 by Urs Schreiber (82.113.121.232)