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I have read two definitions of Monoid:

  1. wiki link In category theory, a monoid (or monoid object) $(M, μ, η)$ in a monoidal category $(C, ⊗, I)$ is an object $M$ together with two morphisms
    $μ: M ⊗ M → M$ called multiplication,
    $η: I → M$ called unit

  2. wiki link Suppose that $S$ is a set and $•$ is some binary operation $S × S → S$, then $S$ with $•$ is a monoid if it satisfies the following two axioms:
    Associativity:
    For all $a, b, c \in S$, the equation $(a • b) • c = a • (b • c)$ holds.
    Identity element:
    There exists an element $e \in S$ such that for every element $a \in S$, the equations $e • a = a • e = a$ hold.

I know the 1st definition is from category theory's perspective, the 2nd one is from abstract algebra's. In my opinion, these two definitions defines the same concept from different way, but how could a monoid $(S, •)$ be consider as a category $(M, μ, η)$?

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    Aside: while a monoid *can* be recovered from $(S, \bullet)$, it is more appropriate to describe it with $(S, \bullet, e)$, since the identity is part of the structure. (e.g. it must be preserved by homomorphism) – 2018-01-29

1 Answers 1

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A monoid in the second sense is a monoid object in the category of sets with its cartesian monoidal structure.

Specifically, take the category $\operatorname{Set}$ of sets, with the monoidal structure $A\otimes B := A \times B$ the cartesian product, and unit is a set with one element, $I:=\{ *\}$.

Then a monoid object in this category is a set $M$ with a multiplication $\mu: M\times M \to M:(a,b)\mapsto a\cdot b$ and a unit $\eta:\{*\} \to M: *\mapsto e$. The axioms of the monoid object give you the associativity of $\mu$ and the unitality of $e$.

Remember in general the triple $(M,\mu,\eta)$ refers to an object $M$ in the monoidal category, not a category itself. So for example, we could take the monoidal category $\operatorname{AbGp}$ of abelian groups, with the tensor product as its monoidal structure. Then a monoid object in $\operatorname{AbGp}$ is precisely the same thing as a ring (with unit). Similarly algebras are monoid objects in the category $\operatorname{Vect}$ of vector spaces.