Background: I'm self-studying Aluffi's Algebra: Chapter 0; currently on Section I.5.5 on coproducts. I've worked through proving that the disjoint union in $\mathsf{Set}$ is indeed a coproduct and I understand that any two coproducts in $\mathsf{Set}$ are isomorphic. I am asking whether the converse is true; that is, whether all coproducts in $\mathsf{Set}$ are disjoint unions.
I am aware that there are categories out there whose coproducts do not coincide with disjoint unions; see the answer here. My question concerns coproducts in $\mathsf{Set}$, specifically.
I am asking this question because, before learning category theory at all, the definitions of products and disjoint unions that I had in mind were:
- A product of sets $X$ and $Y$ is a triple $(U,\pi_X,\pi_Y)$ where $U$ is a set and $\pi_X:U\to X$ and $\pi_Y:U\to Y$ are functions (called canonical projections) such that for any other product $(V,f_X,f_Y)$, there exist a unique function $\sigma:V\to U$ such that $f_X=\pi_X\circ\sigma$ and $f_U=\pi_Y\circ\sigma$.
- A disjoint union of sets $X$ and $Y$ is a triple $(W,\iota_X,\iota_Y)$ where $W=X'\cup Y'$ for any sets $X'$ and $Y'$ such that $X\cong X'$, $Y\cong Y'$, and $X'\cap Y'=\varnothing$, and $\iota_X:X\to W$ and $\iota_Y:Y\to W$ are functions such that for any other disjoint union $(Z,g_X,g_Y)$, there exists a unique function $\rho:W\to Z$ such that $g_X=\rho\circ\iota_X$ and $g_Y=\rho\circ\iota_Y$.
The definition for products aligned perfectly with final objects in the slice category $\mathsf{Set}_{X,Y}$. However, for disjoint unions, there seems to be additional restrictions on $W$ (namely $W=X'\cup Y'$ …) that are absent in the categorical definition that coproducts are initial objects in the coslice category $\mathsf{Set}^{X,Y}$. Naturally (I hope…), I am led to wonder if there are coproducts in $\mathsf{Set}$ that are not disjoint unions (but are, of course, isomorphic to disjoint unions).
Please correct me if my definitions are wrong. Thank you in advance.
Best Answer
Any universal property determines an objects only up to an isomorphism. Therefore, the coproduct of $X$ and $Y$ in the category of sets, is not uniquely defined. It can be defined in infinitely many ways, and any two such choices will be isomorphic. In particular, if $C=X\coprod Y$ is one such choice, with the canonical injections $X\to C$ and $Y\to C$, then any other set $D$ isomorphic to $C$ also yields a coproduct of $X$ and $Y$. $D$ is isomorphic to $C$ if, and only if, $C$ and $D$ have the same cardinality; a pretty weak property.
Now, weather you interpret this to mean that the coproduct in $\mathbf {Set}$ much be the disjoint union (by which you mean that any choice of coproduct is canonically isomorphic to the disjoint union) or if you take that to mean that the coproduct does not have to be the disjoint union (since the disjoint union is but one of many choices for the coproduct) is a linguistic matter. I would say that in $\mathbf{Set}$ disjoint unions are not canonically definable. If the sets are disjoint, there is a nice choice for the disjoint union. In any case, disjoint unions are nice models for coproducts.