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So more or less what I ask in the title: is it possible to identify (uniquely up to bijection) the two-element set in the category of sets as an object that has a particular (categorical) property?

EDIT: Following Hanno's answer, I would like to mention that I know $2$ is a subobject classifier. What I am instead searching for here is some other property that $2$ has (if there is one) and which is not intrinsic to toposes.

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    $\begingroup$ Note: the two answers you have been given so far are basically descriptions of $\mathrm{Hom}(2,-)$ and $\mathrm{Hom}(-,2)$. It doesn't get much more categorical than that. $\endgroup$ – Mees de Vries Oct 10 '16 at 11:38
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As the coproduct of the (unique up to canonical isomorphism) one element set with itself perhaps. If we call this two element set $X$, then we have $$X=*\sqcup *,$$ where $*$ is the one-point set.

We can characterise this using the universal property of the coproduct by saying that given maps $f,g:*\rightarrow Y$ for some set $Y$, then there is a unique map, sometimes called $f+g:X\rightarrow Y$, such that for the two inclusion maps $i_1, i_2:*\rightarrow X$, we have $f=(f+g)\circ i_1$, and $g=(f+g)\circ i_2$.

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The two-element set, call it $\Omega$, represents the subset-functor: For any set $X$, there is a natural bijection $$(\ddagger):\qquad\text{Hom}_{\textsf{Set}}(X,\Omega)\ \ \cong\ \ \text{Subset}(X).$$ Because of this, it is called a Subobject Classifier. By the Yoneda-Lemma, an object is determined up to unique isomorphism by the datum of a natural isomorphism $(\ddagger)$.

The two elements $0,1:\ast\to\Omega$ of $\Omega$ can be recovered from $(\ddagger)$ as those corresponding to $\emptyset,\ast\in\text{Subset}(\ast)$.

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  • $\begingroup$ Is there a name for $\operatorname{Hom}_{\mathsf{Top}}(X,\Omega)$? I know connectedness of $X$ can be defined in terms of such maps. $\endgroup$ – Kyle Miller Oct 10 '16 at 7:11
  • $\begingroup$ @KyleMiller: If you want, continuous maps $X\to\Omega$ (where $\Omega=\{0,1\}$ endowed with the discrete topology) naturally identifie with the 'clopen' - closed and open - subsets of $X$. $\endgroup$ – Hanno Oct 10 '16 at 7:13
  • $\begingroup$ Thanks for your reply. However I was afraid of this answer. I know it is the subobject classifier, this is how I arrived to this question: I wondered whether there is some other property that $2$ has that is not intrinsic to topoi. Perhaps I should mention this in the question... $\endgroup$ – SeanKolm Oct 10 '16 at 7:14
  • $\begingroup$ @SeanKolm: Hm, I can't think of more than explicitly describing both the covariant and the contravariant functor represented by $2$ currently. $\endgroup$ – Hanno Oct 10 '16 at 7:16
  • $\begingroup$ @Hanno Thanks, that's about what I was looking for. So for $\mathsf{Top}$ the subobject classifier is in natural bijection with $\operatorname{Open}(X)\cap \operatorname{Closed}(X)$. $\endgroup$ – Kyle Miller Oct 10 '16 at 7:19

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