Quoting from Categories for the Working Mathematician by Saunders Mac Lane:

All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of endofunctors and unit set by the identity endofunctor.

So we have the category $X$ with Endofunctor $T: X \rightarrow X $

And two natural transformations: $$\eta: 1_X \rightarrow T $$ $$ \mu: T^2 \rightarrow T $$

Question: Since $T$ is the object of this Monoid, what are its elements? I understand that the elements are:

$$ \forall x \in X: T(x), T^2(x), T^3(x),...,T^n(x),...$$

Is that correct?

This is a followup to Monoid in the category of endofunctors and Monoid as a category with one object

  • 3
    $\begingroup$ As I said in my answer to your first question : this monoid does not have elements. $\endgroup$ – Arnaud D. Nov 7 '18 at 17:11
  • 4
    $\begingroup$ If you skip ahead to chapter VII, you'll see the discussion of the kind of monoid Mac Lane is talking about. This monoid in the category of endofunctors is not "a set with a binary operation, etc", but one discussed in the sense of this chapter. Many categories have no sensible notion of "element", and functor categories are among such categories. $\endgroup$ – Malice Vidrine Nov 7 '18 at 18:30

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