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A slick proof of a pesky lemma about monoidal categories

Posted by Akhil Mathew under

category theory | Tags:

monoidal categories,

unital objects |

[4] Comments
Let be a monoidal category—I don’t actually want to define what that means, so I refer you to John Armstrong’s post–with as the monoidal operation. Suppose is a unital object.

The following is well-known:

**Theorem 1** * is a commutative monoid. *

Oftentimes, one has an additive structure on as well, and one actually wants to be a field. The result is interesting, because it strikes a parallel with the following:

**Proposition 2** *The endomorphisms of the identity functor in a category form a commutative monoid. *

The proof is different though. In some places, it’s not even properly mentioned; in others, it’s always seemed extremely non-intuitive.

I learned of a neat proof of the first theorem in the first chapter of a book by Saavedra on Tannakian categories. It is as follows. By definition, , so it is enough to prove commutative. Let . Since the functors and are equivalences of categories, and in particular fully faithful, we can write for appropriate . But then

which proves commutativity.

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February 14, 2010 at 3:53 pm

Aren’t all of these results morally provable by the Eckmann-Hilton argument? (I would be very interested if the answer was “no.”)

February 14, 2010 at 6:10 pm

What would you choose as the two operations?

February 20, 2010 at 8:45 pm

Composition and the tensor product? (I’m not sure about this, though.)

February 20, 2010 at 8:55 pm

Whoa, I think this does work.