Sorry, I am new to category theory (actually all fields of math...). When I was learning the concept of monoid in college, the identity element is roughly defined as "an element e of set which satisfy $e * a = a = a * e$, for every a in that set". But when I am learning this concept again for understanding monad. The identity element of monoid is defined as a morphism (see here). Could anyone please explain why these two definitions of monoid (in particular the identity of monoid) are equivalent. Thanks!


Some other definition that represents the identity of monoid as morphism:

A definition from a math website

Another definition in stackoverflow (in the highest vote answer)

  • $\begingroup$ Can you please post the exact definition of monoid and identity element you're referring to? I can probably answer this question given more details, but I can't extrapolate what definition you're trying to clarify from what you've written. $\endgroup$ – user231101 Sep 23 '16 at 16:15
  • $\begingroup$ The identity element "e" is such $e*a=a=a*e$. Compare your definition of the identity element. $\endgroup$ – amWhy Sep 23 '16 at 16:16
  • $\begingroup$ Hi @MikeHaskel, sorry I am not familiar with latex notation(it's my first time to post in this site). I have attach a link to wikipedia page where the identity of monoid is defined as a natural transformation. Thank you for you help. $\endgroup$ – Lifu Huang Sep 23 '16 at 16:27
  • $\begingroup$ In the linked definition, the unit is a morphism, not a natural transformation. Did you misread it, did you have a different definition in mind, or do you not see the difference between the two? $\endgroup$ – user231101 Sep 23 '16 at 16:43
  • $\begingroup$ @MikeHaskel the definition in the link is what I want to know. I have edited my question. I just realize that the reason why I thought identity as a natural transformation is that I read a definition somewhere else talking about monad. So could you please explain why this definition (identity as morphism) is equivalent to the one I learned in college? Thanks! $\endgroup$ – Lifu Huang Sep 23 '16 at 16:57

The definition you linked to is doing more than just defining a monoid. It's actually defining a monoid object in a (monoidal) category. The usual notion of a monoid is a monoid object in $\operatorname{Set}$. Let's see how the definition plays out in that context.

According the the linked definition, the identity is a morphism from $1$—the unit object of the monoidal category—into the object of the monoid, $M$. In $\operatorname{Set}$, the unit object is any one-element set. The only data associated with a function from a one-element set to $M$ is knowledge of where the function sends that one element. That is, a function from a one-element set to $M$ is essentially just an element of $M$, so the definition aligns with the usual one.

The reason the general definition is phrased in terms of morphisms is because, in a general monoidal category, there might not be a notion of an "element." A morphism from the unit object to $M$ will still make sense, however.


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