6
$\begingroup$

Looking at the definition of Monoids, it looks like they are an object inside a category with one object. I have also noticed that they have operations like composition and identity which must be associative. Is a monoid a kind of category hidden in an object of another category?

$\endgroup$
  • 2
    $\begingroup$ A monoid can be thought of as a category with exactly one object. This way they can be considered to be full subcategories of a given category. Completely determined by the choice of the unique object. $\endgroup$ – drhab Feb 6 '14 at 11:30
  • $\begingroup$ see also my answer: math.stackexchange.com/questions/421215/… $\endgroup$ – Oskar Feb 6 '14 at 19:03
10
$\begingroup$

The connection between monoids and categories is as follows:

  • To every monoid $M$ we can associate a category $BM$ with exactly one object $\star$ and $\mathrm{End}_{BM}(\star)=M$. The identity and composition comes from $M$.
  • In fact, a category with one object is the same as a monoid, and a functor between such categories is the same as a monoid homomorphism.
  • Given a category $C$ and an object $x \in C$, then $\mathrm{End}_C(x)$ is a monoid.

Inside joke: Therefore categories are just monoidoids. ;)

$\endgroup$
  • $\begingroup$ Thank you for your answer. When you say "the identity and composition comes from M" what do you mean since the category will only have 1 object which is the monoid itself. I picture this as a usual object in a category without caring what is inside and having an identity arrow. What am I missing? $\endgroup$ – dfasdfasfasfsd Feb 6 '14 at 11:26
  • 1
    $\begingroup$ There's nothing inside the object: the identity and composition show up in the morphisms of the object of the category associated to $M$. $\endgroup$ – Kevin Carlson Feb 6 '14 at 11:27
  • 1
    $\begingroup$ "since the category will only have 1 object which is the monoid itself. " No. Please read again. $\endgroup$ – Martin Brandenburg Feb 6 '14 at 11:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.