Question about an exercise concerning empty coproduct

Question

The question below is taken from Arrows, Structures and Functors by Arbib and Manes.

Can someone tell me if my solution is correct or if I am wrong, what am I not understanding or misinterpreting about the question. Also, I read online that the empty coproduct (coproduct indexed by the empty set) is the same as an initial object in $C$ (denotes category). I don't know what that means since I am still a beginner in category theory and the book I am using have not talked about initial or terminal objects.

The interesting thing about general definitions is that they are often more general than you realize. Consider what happens to $\prod_{i\in I}A_i$ and $\coprod_{i\in I}A_i$ when $I=\emptyset$,

$$\prod A_i=\{f|f:\emptyset \rightarrow \emptyset\}$$ and so have exactly one element.
What is the empty coproduct?

Attempt solution.

Is the question asking if $I=\emptyset$ and set $A_\emptyset=\emptyset$, then what is $A\sqcup A$?. In this case if we let $A'=\{(\emptyset,1)|\emptyset\in A\}$, then $A'\sqcup A'=\{(\emptyset,1)\}\cup\{(\emptyset,1)\}=\{(\emptyset,1)\}=\emptyset$

Thank you in advance

Answer 1

The universal property of the coproduct of indexed sets $\{A_i\}_{i\in I}$ is this:

"The" coproduct: $$\bigsqcup_{i\in I}A_i$$Is any set $\mathcal{A}$ with maps $\iota_i:A_i\hookrightarrow\mathcal{A}$ for all $i$, such that:

For any set $X$ and family of functions $A_i\overset{f_i}{\longrightarrow}X$, there is a unique function: $$\large\mathcal{A}=\bigsqcup_{i\in I}A_i\overset{\langle f_i\rangle}{\longrightarrow}X$$Such that: $$\langle f_i\rangle\circ\iota_j=f_j:A_j\to X,\quad\forall j\in I$$

So now let $I=\emptyset$. This is the meaning of 'empty coproduct'. What happens? We need to find a set $\mathcal{A}$ and... no functions at all, since there are no $A_i$... such that for every family of functions $f_i:A_i\to X$ - there is only one, the 'empty' family, existing for every set $X$ uniquely - we get a unique function $\mathcal{A}\to X$ for which... oh, there are no functions $f_i,\iota_i$ to test the composition on!

We see that the universal property massively reduces to the following:

For every set $X$, there is a unique function $\mathcal{A}\to X$

The only solution to this in $\mathsf{Set}$ is $\mathcal{A}:=\emptyset$.

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If you aren't satisfied with e.g. "the empty family of functions", consider that this is no more or less weird than the notion of 'empty function'. Note that we can also extend the notion of coproduct to $I=\emptyset$ in another natural way. For any set $I$, we could define the coproduct of the $A_i$ to be the colimit of the functor $\mathfrak{D}(I)\to\mathsf{Set}$ that assigns $i\mapsto A_i$, $\mathrm{id}\to\mathrm{id}$, where $\mathfrak{D}(I)$ is the discrete category on the elements of $I$.

This definition agrees with the usual one for nonempty $I$, and for $I=\emptyset$ it gives us the same answer $\mathcal{A}=\emptyset$, as $\mathfrak{D}(I)$ is empty and the colimit of any empty diagram is the initial object of that category, when it exists, by similar reasoning (the universal property reduces).

If you haven't learn about (co)limits of diagrams yet, you may ignore the above.