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I refer to "Sheaves in Geometry and Logic", by S. MacLane.

Let C be a category. Dealing with a subobject of an object $D \in \text{Ob}_{\mathbf C}$, one defines an equivalence relation between morphisms towards D:

Two monomorphisms $f:A\to D$, $g:B\to D$ with a common codomain D are called equivalent if there exists an isomorphism $h\colon A\to B$ such that gh= f. A sbobject of D is an equivalence class of monos towards D. The collection SubC(D) of subobject of D carries a natural partial order [...]. Then SubC(D) is the set of all subobjects of D in the category C.

I can't figure out why SubC(D) is a set, rather than a proper class! Indeed, we are considering something like an qeuivalence relation on

$\displaystyle \coprod_{A\in \text{Ob}} \text{Hom}_{\bf C}(A,D)$

which is not a set, as soon as C isn't small.

So, how can I avoid the problem?

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By adding the condition that it doesn't happen (people do that), by throwing in a universe or two or by defining the notion of presheaf so that it doesn't matter (I haven't checked details but I think that that is possible). Another, more common, problem (usually more easily solvable) is that the members of $\mathrm{Sub}_{\mathbb C}(D)$ are proper classes so that $\mathrm{Sub}_{\mathbb C}(D)$ doesn't even exist. – Torsten Ekedahl Jul 28 '10 at 10:24
> defining the notion of presheaf so that it doesn't matter How can I do that? "My" definition af a presheaf on C is "a contavariant functor between C and Sets". I obviously thought about throwing everything in a suitable universe, but MacLane never mention the Grothendieck's universes, so I believe there is another way. – Fosco Loregian Jul 28 '10 at 10:55
(P.S.: How can I indent the code as if it were a quotation? And how can I link stuff from wiki or google or something? The tips in editing help page don't work... :( ) – Fosco Loregian Jul 28 '10 at 10:56
"defining the notion of presheaf so that it doesn't matter" That was just a thought I had. The idea is to use the Grothendieck construction to turn a presheaf into a fibred category and then use that the fibred category works even if the purported presheaf doesn't make sense. Hence one should look at the category whose objects are pairs $(A,c)$ where $A$ is an object of the category and $c$ an element of what morally should be $F(A)$. – Torsten Ekedahl Jul 28 '10 at 11:26
Incidentally, a category whose every object has a small set of subobjects is called well-powered. (See e.g. – Finn Lawler Jul 28 '10 at 12:16
up vote 9 down vote accepted

For a general category the subobjects do indeed not have to form a set.

In the context of MacLane/Moerdijk you only look at toposes and there one has a natural isomorphism $Sub_{\mathbf{C}}(D) \cong Hom_{\mathbf{C}}(D,\Omega)$, where $\Omega$ is the subobject classifier.

So it follows from the axioms of a topos, (edit, thanks Mike:) if it is locally small, that $Sub_{\mathbf{C}}(D)$ is a set. When you prove that the basic examples, sheaves, finite sets, products of those, etc. are toposes you exhibit an object $\Omega$ and establish the above bijection. Before that point the left hand side could a priori be a proper class but the right hand side is a set, since you know that your category is locally small, and your bijection then shows that subobjects form a set.

Knowing this you can conclude that objects in full subcategories (edit, thanks again:) whose embedding preserves monos (e.g. if they are reflective) of toposes also have a set of subobjects, e.g. in all locally presentable categories...

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IIRC Mac Lane & Moerdijk (along with most other topos theorists) do not include local smallness in the definition of an elementary topos, so it is not necessarily true that they are locally small either (in fact an elementary topos is locally small iff it is well-powered, since the collection of morphisms A→B can be identified with a subcollection of subobjects of A×B). – Mike Shulman Sep 3 '10 at 7:17
Also, I don't see why this lets you conclude anything about full subcategories of toposes, since a morphism could in theory be monic in such a subcategory without being monic in the topos itself. – Mike Shulman Sep 3 '10 at 7:20
Whoa, I should have stopped to think before writing that answer - thanks! (I have hardly seen any non-Grothendieck toposes, hence my careless naivity :-) – Peter Arndt Sep 3 '10 at 10:35

A reasonable reformulation of the question is, if there exists a set of representatives for subobjects; i.e. if there is a set of monomorphisms into our object $D$, such that every other monomorphism into $D$ is isomorphic to one of them.

This is, of course, false. Take a preorder $P$, which is a proper class, and has a maximal element $\infty$ (for example, the ordinals plus a maximal element). Then $\infty$ has no set of representatives for its subobjects.

However, it happens very often that there is a set of representatives. The category of sets or topological spaces are examples. If $C$ is a category which has the property, then the same is true for every algebraic finitary category over $C$. Thus, for example, the category of (topological) groups has the property.

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