Categorial definition of subsets

Solution 1:

I'll try to get the idea across. Instead of thinking of a subset $\{1\}\subseteq\{1,2,3\}$ as just the set $\{1\}$, try to emphasize the inclusion $\{1\}\hookrightarrow\{1,2,3\}$ in your head.

Why? From the categorical point of view, there is no real distinction between isomorphic sets (and in the category of sets, "isomorphic" means "of the same cardinality"). Indeed, $\{1\}$, $\{2\}$ and $\{\text{elephant}\}$ are all isomorphic as sets. This means that if you just define a subobject of $A$ to be an object that has a monomorphism to $A$, $\{\text{elephant}\}$ will be a subobject of $\{1,2,3\}$, as will $\{1\}$ and $\{2\}$.

Now you want to somehow distinguish $\{1\}$ and $\{2\}$ as being different subsets (and you're maybe tempted to try to come up with a definition in which $\{\text{elephant}\}$ is not a subset of $\{1,2,3\}$). You would want to say "$B$ is a subobject of $A$ if there is a monomorphism from $B$ to $A$ and the elements of $B$ really are in $A$". This is impossible in categorical terms, because "elements of" does not make sense in a category.

And let's face it, we don't really care about the names of the elements in all of our sets. If we want to give a new name (say $\text{elephant}$) to the element 1 of our sets, we can do so and nothing will explode. In this spirit, we don't really care whether the element in $\{1\}$ is actually called 1 and the element in $\{2\}$ is really called 2. What we care about is how the sets $\{1\}$ and $\{2\}$ have injections (monomorphisms) to $\{1,2,3\}$ (in this case, what the images of the morphisms are). This is what distinguishes $\{1\}\hookrightarrow\{1,2,3\}$ from $\{2\}\hookrightarrow\{1,2,3\}$: although $\{1\}$ and $\{2\}$ are isomorphic as sets (in this case even in a unique way), there is no isomorphism between them that will make the inclusion $\{1\}\hookrightarrow\{1,2,3\}$ correspond to the inclusion $\{2\}\hookrightarrow\{1,2,3\}$.

Now let's look at the injection $\{\text{elephant}\}\hookrightarrow\{1,2,3\}$ mapping $\text{elephant}$ to $2$. In the spirit of category theory, there is no difference between including $\text{elephant}$ into $\{1,2,3\}$ this way or the element $2$ in the usual way. And indeed, the inclusions $\{2\}\hookrightarrow\{1,2,3\}$ and $\{\text{elephant}\}\hookrightarrow\{1,2,3\}$ are equivalent for the equivalence relation explained on the Wikipedia article (there is an isomorphism between $\{2\}$ and $\{\text{elephant}\}$ making the triangle consisting of this isomorphism and the two inclusions commute -- in other words, if we suddenly rename $\text{elephant}$ to 2, then the inclusion $\{\text{elephant}\}\hookrightarrow\{1,2,3\}$ suddenly becomes the same as the inclusion $\{2\}\hookrightarrow\{1,2,3\}$).

Let me try to sum up the situation in a single sentence: we consider subsets of $A$ as (equivalence classes of) inclusions of any set into $A$, where we consider two inclusions to be the same if they have the same image (indeed, this is what the equivalence relation amounts to).

I find it hard to explain this, but I hope this is close enough to the heart of the matter to be useful. Let me just give you one confusing remark: one could inject $\{1\}$ into $\{1,2,3\}$ by mapping 1 to 2. In that case, we should regard this inclusion as the same subobject as the usual inclusion of $\{2\}$ into $\{1,2,3\}$ (meaning the two monomorphisms are equivalent).

Solution 2:

When one starts to learn category theory one often learns how think about concepts, which one already knows (or claims to know) in a new, almost always more conceptual way. And one also learns that some notions, which are used all the time outside of category theory, don't really capture what is really going on.

For example, this is the case for the inclusion of sets. In any of the common axiomatizations of set theory it makes sense to ask if two sets $X,Y$ are contained in each other. This is just a property. But in some cases this is quite awkward: Is $42$ contained in $\pi_4(S^2)$? Probably no, but this depends on the precise definitions of these sets, and in any case the answer won't give us any insight about these two objects. With the von Neumann definition of natural numbers, we have $2 \subseteq 3$ (namely, $2=\{0,1\}$ and $3=\{0,1,2\}$). But again, is this a useful property? What is the reason for this inclusion? And is this reason unique at all? A set with three elements has three subsets with two elements. In other words, there are three injective maps $2 \to 3$, and no one can be preferred from the other. Thus, instead of saying that $2 \subseteq 3$ holds or holds not, it is more meaningful if some map $2 \to 3$ witnesses $2 \subseteq 3$. And this is exactly what happens in category theory. Similarily, it doesn't really make sense to ask if the set theoretic foundations guarantee that $\mathbb{Z} \subseteq \mathbb{Q}$. It is more meaningful to ask if a given map $\mathbb{Z} \to \mathbb{Q}$ provides a reason for us to write $\mathbb{Z} \subseteq \mathbb{Q}$. And of course we should also specify which structure this inclusion is refering to, for example it could refer to $\mathbb{Z}$ and $\mathbb{Q}$ as commutative rings. Then the map is required to be a homomorphism of commutative rings. Perhaps I over-emphasize these trivialities here, but students always ask themselves these questions which are well-defined from a set-theoretical point of view, but in reality have no meaning at all. And I wonder why this still happens.

Anyway, after this motivation it is easy to understand the definition of a subobject in an arbitrary category $C$. If $X,Y$ are objects of $C$, it doesn't make sense to ask if $X$ is a subobject of $Y$. Instead, one may ask which monomorphism $X \to Y$ provides a reason for this, and we remember it, i.e. it belongs to the data of the subobject! Moreover, two reasons $X \to Y$ and $X' \to Y$ are equivalent if there is an isomorphism $X \cong X'$ such that the obvious diagram commutes. An equivalence class with respect to this relation is called a subobject of $Y$.

If $C$ is the category of sets, then there is a bijection between the subsets of $Y$ and the subobjects of $Y$. Similar statements hold for categories of algebraic structures. This does not mean that we have found a more complicated definition of substructures; instead, we have found the correct one. Open an arbitrary book on galois theory and field extensions and replace everywhere "field extension" by "morphism of fields", it will make the subject a lot more transparent and understandable (if anyone is interested I can expand this).