The book I'm working with (Mitchell's Theory of Categories (1965)) defines a $A'$ to be a subobject of an object $A$ (in some category $\mathbf{C}$ †) iff there exists some monomorphism $\alpha\!:\!A' \rightarrowtail A$. (I quote Mitchell's exact definition along with one important caveat in another question on math.SE). In addition, Mitchell calls such $\alpha$ the inclusion of $A'$ in $A$, and defines the notation $A' \subset A$ to mean $A'$ is a subobject of $A$ ‡. Furthermore, he adds that in this case "we shall say that $A'$ is contained in $A$, or that $A$ contains $A'$".
Mitchell's choices of notation and nomenclature strongly suggest that he intends the concept of subobject to be a generalizations of the concept of subset from standard set theory. If this is indeed the case, I find the whole idea really puzzling, because the analogy with subsets seems to me more likely to be confusing than illuminating.
What is the benefit of pushing this analogy between subsets and subobjects? Is there a way to modify Mitchell's definitions that retain this benefit without the flaws described below?
I find that, when working with subobjects as Mitchell defines them, my intuition frequently trips on one or the other of two important differences between the is-subset-of and is-subobject-of relations, as described below; they look like significant flaws to me.
First, in set theory $(A \subset B) \wedge (B \subset A)$ is equivalent to $A = B$, but this equivalence does not hold with Mitchell's $\subset$. Certainly, $A = B$ implies that $(A \subset B) \wedge (B \subset A)$ (in Mitchell's sense), but the reverse implication does not hold. Indeed Mitchell's $(A \subset B) \wedge (B \subset A)$ ensures only the existence of monomorphisms $\alpha\!:\!A \rightarrowtail B$ and $\beta\!:\!B \rightarrowtail A$. (In fact, as far as I can tell, one cannot even say that such $A$ and $B$ are isomorphic!)
Second, suppose that $A$ is a subobject of $B$ and $B$ is a subobject of $C$, with inclusions* $\alpha\!:\!A\rightarrowtail B$ and $\beta\!:\!B\rightarrowtail C$. In this context, the statement $A$ is a subobject of $C$ admits two inequivalent interpretations:
- it simply asserts a special case of the transitivity of the is-subobject-of relation (which follows from the fact that the composition of monomorphisms is also a monomorphism);
- it asserts that there exists a monomorphism $\gamma\!:\!A\rightarrowtail C$ (irrespective of the existence of $\beta\;\alpha$).
Clearly, the first interpretation implies the second one, but the converse is not true.
Contrast this situation with the one in which $A$ is a subset of $B$ and $B$ is a subset of $C$. In this case, the statement $A$ is a subset of $C$ can be interpreted only as a special case of the transitivity of the is-subset-of relation. There is no alternative way in which $A$ can be a subset of $C$ (given that $A$ is a subset of $B$ and $B$ is a subset of $C$).
†Actually, Mitchell's book deals only with categories having the property that for every pair $(A, B)$ of category objects, the class of all morphisms $A\to B$ is a set. The Wikipedia page for category theory calls such categories locally small, but I'm refraining from using this term here because Mitchell defines locally small categories differently. The definition chasing never ends...
‡In my experience, in the context of standard set theory, $A \subset B$ usually implies that $A \neq B$. This is not the case with Mitchell's $\subset$, since, for any object $A$, the identity $1_A$ is a monomorphism, so $A \subset A$.
*For the sake of my own sanity if nothing else, in this post I'll stick with Mitchell's notation and nomenclature as given above, even though I realize they may not be standard. Hence, inclusion, $\subset$, etc.
Llast poi$n$t: your answer (over your various comments) got closest to what I was after; please repost it as an "official" answer so I can mark it as the "accepted answer". Also, sorry for the "extended discussion in comments"; I see the "Would you like to automatically move this discussion to chat?" link, but clicking it doesn't do anything resembling what it proposes (it doesn't do anything at all, in fact, afaict). – 2011-09-25