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Multiplication of natural numbers can be understood as iterated addition, and we can understand binary Cartesian products as set-indexed coproducts; for sets $X$ and $Y$,

$$X\times Y\cong\coprod_XY\cong\coprod_YX.$$

The internalization of indexed coproducts is dependent sums, so we need these to ask the question

What categories with dependent sums does this relationship fail in for objects $X$ and $Y$? Is there a name for categories where this property holds?

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  • $\begingroup$ In enriched category theory, this kind of phaenomena is linked to be tensored (or copowered). See ncatlab.org/nlab/show/powered+and+copowered+category. $\endgroup$
    – Ivan Di Liberti
    Commented Feb 24, 2021 at 8:35
  • $\begingroup$ @IvanDiLiberti That's interesting, but I'm not familiar enough with enriched category theory to suss out the connection; could you elaborate on it a bit more? $\endgroup$
    – Alec Rhea
    Commented Feb 24, 2021 at 8:40
  • $\begingroup$ Why doesn't the section "relation to the product" at ncatlab.org/nlab/show/dependent+sum answer your question? $\endgroup$
    – Achim Krause
    Commented Feb 24, 2021 at 9:16
  • $\begingroup$ @AchimKrause That definitely does answer it, I'm embarrassed. If you'd like to post that as an answer I'll accept it, or if you don't want to I can delete the question. $\endgroup$
    – Alec Rhea
    Commented Feb 24, 2021 at 9:19

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A discussion of the relation between dependent coproducts and cartesian products is found at

https://ncatlab.org/nlab/show/dependent+sum

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