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My apologies in advance if this question is to vague, but here goes.... In the category of vector spaces, products are given by direct sums. In general category theory, the existence of products is a property of the category, there is no choice going on. On the other hand the tensor products of vector spaces is an examples of a monoidal structure. In general category theory, monoidal structures are chosen, there can exist more than one for any category. How is one to understand the distinction. Taking direct sums seems as "natural" as taking tensor products, so why is their general categorical interpretation so different?

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    $\begingroup$ Look up cartesian monoidal structures, for instance here: ncatlab.org/nlab/show/cartesian+monoidal+category That said, this question is probably a better fit for Math.SE. $\endgroup$ Commented Dec 20, 2020 at 23:58
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    $\begingroup$ Just a small remark: in the category of vector spaces the categorical product is the Cartesian product, while the direct sum corresponds to the coproduct. Both coincide for finitely many objects, but are distinct for infinitely many. I wouldn't put direct sum and tensor product on the same footing, though, since the latter is a quotient. $\endgroup$
    – M.G.
    Commented Dec 20, 2020 at 23:59

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One way to think about what the monoidal structure on vector spaces is doing is that it is telling us that vector spaces do not really form a category, or not "just" a category: they form a multicategory whose multimorphisms $V_1, \dots V_n \to W$ are given by multilinear maps $V_1 \times \dots \times V_n \to W$. We care a lot about multilinear maps and not just linear maps in practice so this is a very natural thing to do.

Multicategories are a strict generalization of monoidal categories; if the multihom functor $\text{Hom}(V_1, \dots V_n ; W)$ happens to be representable as a functor of $W$ for all $V_i$ then the representing object can be written $V_1 \otimes \dots \otimes V_n$ and this should define what is called an "unbiased" monoidal category (an axiomatization where we axiomatize all the $n$-fold tensor products at once rather than just binary ones).

In other words, the tensor product is describing an additional structure we care about in practice which is not captured by the category structure alone. Similar situations happen all the time, e.g. rings have additional structure given by multiplication which is not captured by addition alone (and it's not so bad to think of monoidal abelian categories, say, as categorified rings). What the category theory tells us is that the notion of direct sum can be defined using only the notion of linear map but the notion of tensor product requires the notion of multilinear map.

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    $\begingroup$ Instead of talking about multilinear maps directly we can also talk about the fact that the set $[V, W]$ of linear maps between two vector spaces is itself a vector space, which produces a closed category structure on $\text{Vect}$. Then we can ask for a left adjoint satisfying a tensor-hom adjunction $\text{Hom}(V_1 \otimes V_2, W) \cong \text{Hom}(V_1, [V_2, W])$. The connection to multilinear maps being that both of these functors describe bilinear maps $V_1 \times V_2 \to W$. $\endgroup$ Commented Dec 21, 2020 at 23:27
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    $\begingroup$ Thanks, this explanation actually makes more sense to me! $\endgroup$ Commented Dec 22, 2020 at 14:34

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