I hope this question is not too naive. Is it possible to think of $\text{Set}$ (or some significant full subcategory of it) as a monoidal category with set theoretic union as the tensor product ($A \otimes B = A \cup B$), and the unit object as the empty set ?
Does Set, with union, form a monoidal category
category-theorymonoidal-categoriesset-theory
Best Answer
Suppose $\mathbf {Set}$ was a monoidal category with tensor given by union. Generally, in any monoidal category, the tensor product is invariant under isomorphisms in the sense that $X\cong Y$ and $U\cong V$ implies $X\otimes U \cong Y\otimes V$. However, if you interpret that in $\mathbf {Set}$ as above, and take $X=U$, say a one-point set $\{a\}$, and $Y=\{b\}$, $V=\{c\}$ two other one-point sets, then they are all isomorphic but $X\otimes U=\{a\}$ while $Y\otimes V=\{b,c\}$. This shows it is rather hopeless to hope for the union to be a tensor product for any $\mathbf {Set}$-like category. To avoid the obstruction above you would have to either have severely non-isomorphic objects become isomorphic, or for highly isomorphic objects to become non-isomorphic.
What is certainly true is that with disjoint union the category $\mathbf {Set}$ is monoidal. This is true in general for any category with finite coproducts. Another aspect of interest here is that for any fixed set $S$, within the category $\mathcal P(S)$, the lattice of subsets of $S$, union is the categorical coproduct, and thus serves as a tensor product in $\mathcal P(S)$. However, the structure of $\mathcal P(S)$ as a boolean algebra is much more precise than the mere existence of this monoidal structure.