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At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/direct sum of modules seems to satisfy all of the prerequisites for the monoidal product operation (because I think the identity would be the trivial module, which is always a terminal object in $Set$, and therefore we should have that $V \times \{0\} \simeq V$, I think).

Moreover, modules are sets, and the Cartesian product works well as a monoidal product for $Set$, so seemingly if the product operation from $Set$ were compatible with the structure of the subcategory of modules, which it does appear to be (hence my mention of direct sums earlier), wouldn't it make the most sense to just use the monoidal product from the "mother category" (i.e. the Cartesian product/direct sum) in the subcategory (of modules)?

Why is the introduction of the notion of tensor product not an unnecessary complication (in the context of monoidal categories only)?

The reason why this bothers me so much is that rejecting the Cartesian product in favor of the tensor product to use as the monoidal product for the monoidal category of modules seems like the only reason for introducing the notion of Hopf algebra.

The Cartesian product is completely compatible with the simpler and more obvious notion of group object, so if we are going to spend so much time and effort trying to work around the absence of the nice features of the Cartesian imposed by using the tensor product instead, there ought to be a good reason, but I can't figure out what it is.

I.e. if we were using Cartesian products as the monoidal product for the category of modules, instead of the tensor product, it seems like we could define group objects on them with little to no effort, rather than having to invent a whole new concept of Hopf algebra.

(I.e. so we can define something similar to a group object on vector spaces while using the tensor product as the monoidal product instead of the Cartesian product.)

Why is it so important to use the tensor product as the monoidal product when considering modules as a monoidal category?

(See, e.g. https://www.youtube.com/watch?v=zZn9ZETVkF8)

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    $\begingroup$ Basically because rings are defined as semigroups wrt the tensor product of abelian groups, and modules are defined as semigroup action wrt tensor product again. In any case it is a basic and very important piece of structure on the category of modules, so why abandon it? The cartesian product corresponds to the product in $Mod_R$, which is simple and boring. $\endgroup$ – Anton Fetisov Jul 25 '16 at 20:25
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You ask:

wouldn't it make the most sense to just use the monoidal product from the "mother category"?

but this question does not make sense, really. To judge if something makes more sense or less sense than something else, you have to spell out what you are trying to achive.

Sometimes, the direct sum is the correct operation to turn modules into a monoidal category —this is used in the context of $K$-theory all the time, for example. Sometimes, the tensor product is the correct operation: for example, in the context of the Eilenberg-Watts theorem, modules appear in the form of a description of functors, and then the tensor product of modules corresponds to the composition of functors.

Almost nothing makes sense «in the abstract»: when we pick a structure or another, we have an objective in mind, and it is with respect to that objective that we evaluate our choices.

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Every object in an additive category is a group and a cogroup, canonically, with respect to the monoidal structure of direct sum. This reflects the fact that vector spaces, and more generally, modules, are actually groups. So there's nothing there.

Hopf algebras weren't invented to cause pain, much less because of the desire to decorate the monoidal structure given by the tensor product, but to describe the structure that arises, for instance, in certain cohomology rings. I can't help but wonder where you're reading on Hopf algebras that leaves the actual motivation so obscure. Generally, if you can't figure out the motivation for some concept on your own, Google around! The Wikipedia article would already seem to be an improvement over your current motivation.

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The direct sum of $k$-vector spaces makes $\mathbf{Vec}_k$ into a cocartesian monoidal category, not a cartesian one.

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  • $\begingroup$ Um, really? Even though direct sum is both a product and coproduct? $\endgroup$ – Kevin Carlson Jul 25 '16 at 19:42

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