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Regarding the subobject classifier construction, why do we need the pullback?

Monos from $U$ to $X$ are called subobjects, but I see that there might be injections which just have elements of the X (viewed as a set like in set theory) permuted. This is therefore somewhat weak.

However, as far as I can see, $\text{Hom}(X,\Omega)$ are the characteristic functions (set theory terminology) and these are in bijective correspondence with what we understand as subsets. Why then do we need $U$, $j$, etc. to do set theory? The only purpose for the $U\rightarrow 1\rightarrow\Omega$ route I can come up with is to define $\chi_j$ in terms of function composition and therefore associating $\chi_j$'s with objects like $U$. Is it ment that the subobject classifier enables us to classify certain unique $U$'s and we can then associate objects (like $U$) as subobjects of objects (like X)? But why would that be necessary? Why not just consider $\text{Hom}(X,\Omega)$ as subobjects (vs, plural) of $X$? And in case we just don't want them as morphisms: I've seen hom-sets taken to a new category via a functor, why doesn't this suffice?

Secondly, it's always explicitly stated that we need a terminal object to do all of the above. But don't we also need "true" and "!" as well, must this also always be explicitly required, or are we sure some of them are implying by there being a terminal object?

Lastly, it is said that topos theory aviods the stacking of element-inclusion, i.e. $\in$ get replaced by axioms of function composition. But if Set contains any universe you would want to talk about, then surely these nested sets are to be find there too, and this just means that there are super long chaings involving subobject classifiert. Does this really reduce notational ballast, compared to any set theory with multiple types?

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"Why not consider $\hom(X,\Omega)$ as a subobject of $X$?" Are you mixing up of $\hom(X,\Omega)$ with what in $\mathbf{Set}$ are its elements? The elements of $\hom(X,\Omega)$, i.e. the subobjects, are picked out exactly by the diagram you show. –  Kevin Carlson Oct 27 '12 at 17:00
    
@KevinCarlson: I'm not sure if I read your question correctly, but what I mean is that every element of the set $hom(X,\Omega)$, which if $\Omega=\{0,1\}$ are characteristic functions, corresponds to a specification of a subset of $X$. Hence, the arrows from $X$ to $\Omega$ should be defined as (isomorphic to) the subsets. Seem to be unique too. But if this would suffice, the we'd not do all this pullback business - I wonder where the problem is. (cont.) –  NikolajK Oct 27 '12 at 17:16
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We already have a pre-existing notion of subobject, and not every category has such a wonderful object as a subobject classifier. (It is the most distinctive feature of a topos.) –  Zhen Lin Oct 27 '12 at 17:30
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@Nick: The commutative diagram is the equation that says $\chi_j$ is the characteristic function of the subobject (represented by) $j$. In the set-like internal language, it says that it is the subobject $$\{ x \in X \mid \chi_j(x) \} $$ –  Hurkyl Oct 27 '12 at 17:45
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@Nick: You might be trying to be too abstract. We know subsets and characteristic functions are both very useful in set theory. The notion of subset translates quite nicely to category theory in the form of subobjects (represented by monomorphisms). But since characteristic functions are also useful, we want to translate that notion as well. The commutative diagram is pretty much a direct translation of the usual set-theoretic definition of characteristic function. –  Hurkyl Oct 27 '12 at 18:32

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We do not need pullbacks to define subobject classifier. All we need is the notion of a logic over a category, or equally, the notion of subobjects. This may be abstractly characterized as giving a logical fibration over the category. The subobject classifier is then a universal object that allows us to recover all subobjects via substitutions.

Pullbacks are needed to define the logic of subobjects. With every category $\mathbb{C}$ with pullbacks one may associate the canonical logic fibration --- $\mathit{sub} \colon \mathit{Sub} \rightarrow \mathbb{C}$. It is showable that if $\mathit{sub}$ has the universal object $X \rightarrow \Omega$ in the above sense, than $X$ is necessary a terminal object in $\mathbb{C}$.

One may consider other logics over $\mathbb{C}$, and reach other notions of subobjects. For example, the crucial role in the quasitopos is played by a regular subobject classifier --- it is just the universal object induced by the logic of regular subobjects.

One may even consider non-logical fibrations over $\mathbb{C}$, and see how such universal classifiers look like. For example, the external family fibration $\mathit{fam}(\mathbb{C}) \colon \mathit{fam}(\mathbb{C}) \rightarrow \mathbf{Set}$ has a universal classifier precisely when $\mathbb{C}$ is small; it is given by $\mathbb{C}_0$ --- the set of all objects of $\mathbb{C}$.

You have also said:


Monos from U to X are called subobjects, but I see that there might be injections which just have elements of the X (viewed as a set like in set theory) permuted.


That is why we do not define subobjects as monos, but as abstraction classes of equivalent monos.

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