How do one object Categories Induce Monoids?

Question

From pg. 11 of Categories for the Working Mathematician:

Monoids. A monoid is a category with one object. Each monoid is thus determined by the set of all of its arrows, by the identity arrow, and by the rule for the composition of arrows. Since any two arrows have a composite, a monoid may then be described as a set $M$ with a binary operation $M \times M \rightarrow M$ which is asociative and has an identity (= unit).

Question: I do not see how we go from arrows $\rightarrow$ on a category with one object to a monoid $M$, which makes use of the notion of $\times$ (as described in the quote). So how exactly do categories with one object induce monoids (in the algebraic sense of the word)? Or, how do we move from $\rightarrow$ to $\times$ as expressed in the quote?

Answer 1

It might be more helpful to first see how any given monoid (in the usual sense) can be turned into a category.

Suppose $\mathcal{M}=(M, *, e)$ is a monoid. Then we consider the following category $\hat{\mathcal{M}}$:

• $\hat{\mathcal{M}}$ has one object, $\alpha$.

• $\hat{\mathcal{M}}$ has one arrow $f_m$ for every $m\in M$.

• The composition law in $\hat{\mathcal{M}}$ is given by $$f_m\circ f_n=f_{m*n}.$$ (Note that in particular this makes $f_e=id_\alpha$.)

You can think of $\alpha$ as representing the set $M$, and $f_m$ as representing the map $x\mapsto mx$; but you can also just think of the abstract data above as standing on its own.

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The construction described in your post is just the dual of this: from an appropriate category, we can construct a corresponding monoid.

Answer 2

Give a one-object category, define the corresponding monoid as follows: The underlying set of the monoid is the set of the arrows of the category. The product of two arrows $f$ and $g$ is the composite $gf$, which always exists because the category has only one object. Composition in a category is associative by definition, and one of the arrows in the set of arrows must be the unique identity arrow of the only object, so the resulting structure is a monoid.

Note that this works for small categories. For a large category you get a "large monoid".