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The tone of the parent comment is a bit aggressive, but the content is relevant. Just in case it's not clear, the union of sets "merges" elements that are iden

The tone of the parent comment is a bit aggressive, but the content is relevant. Just in case it's not clear, the union of sets "merges" elements that are identical whereas in the (direct) sum each element gets tagged with where it comes from. Formally, we have a set that indexes over the sets we want to sum and then we just Cartesian product each set with it's index before unioning. In abstract nonsense we usually write A ⊕ B = (A × {0}) ∪ (B × {1}) where ⊕ is the direct sum operator. So there direct sum of a set with itself is quite different from a plain union. For more than two sets we just index over things in the obvious way and write ⨁ᵢ Aᵢ = ∪ᵢ (Aᵢ × {i}) What parent means by universal property is that any that collection of functions {fᵢ : Aᵢ → C} can be decomposed into functions {gᵢ : Aᵢ → ⨁ᵢ Aᵢ} and a single, uniquie g: ⨁ᵢ Aᵢ → C where fᵢ = g ∘ gᵢ. With a little thought, the universal property above seems obvious, and with a little more work, you can see that it is actually equivalent to the definition of direct sum above. Thus we can actually take this universal property as the definition of direct sum and derive the original. The cool thing is that the universal property doesn't refer to set-specific operations---just functions/maps---so we actually have a definition for the direct sum of things more general than sets. That is, we can form the direct sum of things like monoids, groups, certain classes of differential functions, etc. I could go on, but this road leads to madness^W category theory.

deleted 9y ago
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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 ago
  Whoops. 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 ago
  A 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 ago
      It 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.
sevensor 9y ago
> In abstract nonsense we usually write > A ⊕ B = (A × {0}) ∪ (B × {1}) Hardly nonsense! This makes clear the connection to sets in a way that words weren't doing for me.

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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 ago
Whoops. 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 ago
A 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 ago
    It 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.

sevensor 9y ago

> In abstract nonsense we usually write > A ⊕ B = (A × {0}) ∪ (B × {1}) Hardly nonsense! This makes clear the connection to sets in a way that words weren't doing for me.