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I'm self-learning category theory from multiple sources. In Bartosz Milewski's lectures, he introduces the categorical product, via the universal construction. He then shows that for several useful categories, the product is defined for every pair of objects, and often resembles a Cartesian product.

On the other hand, in Coecke and Paquette's Categories for the practicing physicist, they introduce a symmetric monoidal category as a category that simply comes equipped with a bifunctor $\otimes$ and identity object $I$ that satisfy certain relations. This $\otimes$ operator also seems to be similar to (and sometimes the same as) a Cartesian product.

My question is, are these two concepts actually the same? By which I suppose I mean, is it the case that if a categorical product exists for every pair of objects, does it necessarily always give the structure of a (symmetric?) monoidal category, and conversely, is the $\otimes$ operator in a symmetric monoidal category always given by the categorical product?

More broadly, regardless of the answers to the above questions, how should I think about the relationship between these two ways of defining Cartesian product like operations in category theory?

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  • $\begingroup$ The product is always a symmetric monoidal structure, but most symmetric monoidal structured are not the product; a very important example is the tensor product on vector spaces. $\endgroup$ – Qiaochu Yuan Nov 20 '17 at 8:44
  • $\begingroup$ How about the example of carrot-potato mash? $\endgroup$ – Christoph Nov 20 '17 at 8:47
  • $\begingroup$ To view the categorical product as a tensor product in the monoidal structure, you also need a unit, that is, a terminal object in the category. $\endgroup$ – Phil. Z Nov 20 '17 at 8:47
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There can be many different symmetric monoidal structures $\otimes$ on a given category. For example, in any category admitting a product of two objects and a terminal object $F$, the operation $(X, Y) \mapsto X \times Y$ will give a symmetric monoidal structure, with unit $F$. In any category admitting coproducts of any two objects and an initial object $I$, the operation $(X, Y) \mapsto X + Y$ will give a symmetric monoidal structure, with unit $I$. There are further nontrivial examples specific to the category at hand: a commonly encountered one is the tensor product of vector spaces, which cannot be thought of as either a product or a coproduct in the category of vector spaces.

In contrast with a general symmetric monoidal structure, the categorical product is very special, and (should it exist) it is unique up to unique isomorphism. You should think of it by the universal property, which (imprecisely) says that giving a single arrow $f: Z \to X \times Y$ is the same data as a pair of arrows $f_X: Z \to X$ and $f_Y: Z \to Y$.

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  • $\begingroup$ Perhaps worth highlighting that the product is unique only up to canonical isomorphism. $\endgroup$ – Kevin Carlson Nov 20 '17 at 16:02
  • $\begingroup$ @Nex my mistake, I've fixed it. $\endgroup$ – Joppy Nov 20 '17 at 22:02
  • $\begingroup$ Thanks, this makes sense. So in short, the categorical product (if it exists) does define a symmetric monoidal category, but in general the monoidal structure doesn't have to come from the categorical product. This leaves me feeling that symmetric monoidal categories in which the monoidal structure does arise from a categorical product are "better"/"cleaner"/"more elegant" than ones where it doesn't. Would you say that's fair, or is it an error to think in that way? $\endgroup$ – Nathaniel Nov 21 '17 at 6:57
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    $\begingroup$ @Nathaniel: as someone in representation theory, I would have to disagree. The categorical product in vector spaces is the direct sum, which while important, is quite boring. The tensor product is much more interesting, and specific to the category: I don't know if there is a way to state its universal property in an arbitrary category. Also, I feel like saying the monoidal structure is misleading, you should really say a monoidal structure, since there could be many. Some are more interesting than others, just like some categories are more interesting than others. $\endgroup$ – Joppy Nov 21 '17 at 11:18

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