Examples of Monoid Objects

The Unapologetic Mathematician

It’s all well and good to define monoid objects, but it’s better to see that they subsume a lot of useful concepts. The basic case is, of course, that a monoid object in $latex mathbf{Set}$ is a monoid.

Another example we’ve seen already is that a ring with unit is a monoid object in $latex mathbf{Ab}$ — the category of abelian groups with the tensor product of abelian groups as the monoidal structure. Similarly, given a commutative ring $latex K$, a monoid object in the category $latex Kmathbf{-mod}$ with tensor product of $latex K$-modules as its monoidal structure is a $latex K$-algebra with unit. For extra credit, how would we get rings and $latex K$-algebras without units?

Here’s one we haven’t seen (and which I’ll talk more about later): given any category $latex mathcal{C}$, the category of « endofunctors » $latex mathcal{C}^mathcal{C}$ has a monoidal structure given by composition…

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