A subobject of an object $c$ in a category $C$ is an isomorphism class of monomorphisms
into $c$. (Two morphisms $i: a \to c$, $j: b \to c$ are isomorphic if there exists an isomorphism $k: a \to b$ such that $i = j k$.)
Monos into an object $c$ are preordered by a relation
defined by the condition that there exists $k: a \to b$ such that $i = j k$. (There is at most one such $k$ since $j$ is monic, and such $k$ is monic since $i$ is monic.)
A subobject of $c$ may equivalently be defined as an element of the posetal reflection $Sub(c)$ of this preorder.
Let $C_c$ be the full subcategory of the over category $C/c$ on monomorphisms. Then $C_c$ is the poset of subobjects of $c$ and the set of isomorphism classes of $C_c$ is the set of subobjects of $c$. However this “set” can be in fact a proper class in general, see well-powered category.
More generally, in some contexts we may take “subobject” to mean an isomorphism class of morphisms $i: a\to c$ 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.)
The partial order on the collection of subobjects internalizes into contexts more general than Set. For instance in every topos the subobject classifier $\Omega$ has the structure of an internal poset (see there).
For $C$ an accessible category, $c \in C$ any object, the poset $Sub(c)$ of subobjects of $c$ is a small category.
Suppose $E$ is a well-powered category. Denote by $Sub(X)$ the poset of subobjects of object $X$ in $E$. The correspondence $Sub:X\mapsto Sub(X)$ may be extended to a contravariant functor $E \to Pos$ (that is a functor $E^op \to Pos$), namely if $f: X\to Y$ is arbitrary and $m:S\hookrightarrow Y$ is an element of $Sub(Y)$, i.e. monic, then the pullback $f^*(m):f^*(S)\to X$ of $m$ along $f$ is automatically a monic; the correspondence $m\mapsto f^*(m)$ describes $Sub(f)$ at the level of representatives of subobjects.
Assume that the ambient category has all limits and colimits considered in the following.
For $X \in C$ an object, $Sub(X)$ the poset of subobjects and
$U_1, U_2 \hookrightarrow X$ two subobjects,
their product in $Sub(X)$ is denoted $U_1 \cap U_2$ or $U_1 \wedge U_2$ and called the intersection or meet of the two subobjects;
their coproduct in $Sub(X)$ is denoted $U_1 \cup U_2$ or $U_1 \vee U_2$ and called the union or join of the two subobjects.
Two subobjects $U_1, U_2 \in Sub_C(X)$ are called disjoint if $U_1 \cap U_2 = \emptyset$ is the initial object.
Let $C$ be a topos.
The intersection of two subobjects in $Sub_C(X)$ is their fiber product in $C$: the diagram
is a pullback diagram.
The union of two subobjects $U_1, U_2 \in Sub_C(X)$ is the pushout $U_1 \coprod_{U_1 \cap U_2} U_2$ in $C$: the diagram
is a pushout diagram.
This last pushout diagram is also a pullback diagram.
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 $C$ then implies its universal property as a product in $Sub_C(X)$.
For the second point: by the same kind of argument, it is sufficient to show that the canonical morphism $U_1 \coprod_{U_1 \cap U_2} U_2 \to X$ 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
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
To see this, again use universal colimits to get
and similarly
and
This proves the second point.
The third point is directly verified by checking the universal property.
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 $\mathbb{Z}_13$ – and even category theorists do so when they are available.
A subobject of a representation is a subrepresentation.
A subobject (a subfunctor) of a representable functor is a sieve.
A subobject of a bundle (hence in a slice category) is a sub-bundle.
Standard textbook references include section I.3 of
and