3 ms·
> Just in case it's not clear, the disjoint union of sets "merges" elements that are identical whereas in the (direct) sum each element gets tagged with where i
by dbaupp 9y ago
> Just in case it's not clear, the disjoint union of sets "merges" elements that are identical whereas in the (direct) sum each element gets tagged with where it comes from.
That's just a union. The disjoint union (https://en.wikipedia.org/wiki/Disjoint_union https://en.wikipedia.org/wiki/Disjoint_union) is what you're calling the direct sum. The direct sum (https://en.wikipedia.org/wiki/Direct_sum https://en.wikipedia.org/wiki/Direct_sum) is an operation on algebraic structures to create new algebraic structures (e.g., one can direct-sum two groups to get a new group, and similarly rings, vector spaces, etc.), not just sets.
- xelxebar 9y agoWhoops. That one slipped past. Thanks. In case it's not clear, I just specialize tensor sum to the category Set; that's all. At the end I do mention how the tensor sum applies on a wider class of objects. In fact, any Cartesian category has (finite) tensor products, even if the objects aren't traditionally thought of as "algebraic structures", e.g. certain classes of topological spaces, Frolicher manifolds, etc.
- catnaroek 9y agoA union type `A | B` in Flow is disjoint if and only if the types `A` and `B` are a prior known to be disjoint. There is no equipping `A` and `B` with fresh new tags to force them to be disjoint.
- dbaupp 9y ago... Which isn't relevant to the clarification of mathematical terms in my comment.
- catnaroek 9y agoIt is relevant, because it means that while `|` may produce a disjoint union if its arguments are already disjoint, it is not guaranteed to produce a disjoint union in the general case. Hence, my original comment had a point. As if that weren't enough, any talk about universal properties is utterly pointless as soon as object identities matter everywhere.