In set theory, map $f:X\rightarrow Y$ is interpreted to be a subset of the product $X\times Y$ satisfying some properties. If $X=\varnothing$ then $f \subseteq \varnothing\times Y = \varnothing$ and all empty maps are the same regardless of whether they have different codomains $Y$.
However, it is said that it matters what the codomain of a map is. If $f:X\rightarrow Y$ and $f':X\rightarrow Y'$ and the two codomains are different, then $f\ne f'$.
So what gives? If $f:\varnothing\rightarrow Y$ and $f':\varnothing\rightarrow Y'$ are maps, are the two maps equal or not? Does the answer depend on the choice of foundations you use?
Edit: I think at this point, I'm just looking for a citation that defines functions in terms of set theory keeping domains and codomains in mind.
Best Answer
You could say the same thing about the inclusion $i:\Bbb Z\hookrightarrow\Bbb Q$ compared to the inclusion $i':\Bbb Z\hookrightarrow\Bbb R$ (where I'm assuming $\Bbb Z\subset\Bbb Q\subset\Bbb R$). As sets, they're both given by $\{(x,x) \mid x\in\Bbb Z\}$. Does this make them equal? If you take these to be equal, then you're correct: by your reasoning, all empty maps are the same.
However, if you somehow insist that the above maps should be different, then this should likely also mean that empty maps differ if their codomains differ. One way, I suppose, you could have this is by encoding a map $f:X\to Y$ as the pair $((X,Y),\{(x,f(x)) \mid x\in X\})$ or something.