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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?

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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. –  drhab Feb 6 at 11:30
    
see also my answer: math.stackexchange.com/questions/421215/… –  Oskar Feb 6 at 19:03

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up vote 7 down vote accepted

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. ;)

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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? –  dfasdfasfasfsd Feb 6 at 11:26
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There's nothing inside the object: the identity and composition show up in the morphisms of the object of the category associated to $M$. –  Kevin Carlson Feb 6 at 11:27
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"since the category will only have 1 object which is the monoid itself. " No. Please read again. –  Martin Brandenburg Feb 6 at 11:28

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