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.

At the same time a Monoid is a category with one object. Given a Monoid in the category of endofunctors of X as above, how do we get a category with one object from there? Please specify exactly what the object and the morphisms of this category are.

Followup question: Elements of the Monoid in the category of endofunctors


1 Answer 1


The quote you mention does not say that monads are monoids in the usual sense; it says that they are monoids in the monoidal category of endofunctors of $X$. A monoid in a monoidal category (or monoid object, as suggested by Derek Elkins in a comment) is not a monoid; that's a red herring. In general, a monoid object is an oject of a category which is not necessarily concrete, so it need not be any kind of set; in particular, it does not need to have any kind of elements. But seeing a (usual) monoid as a one-object category means precisely that you identify its elements with the arrows of a category; so you can't do that for monoid objects.

You can however, identify such monoids with one-object enriched categories, exactly in the same way that classical monoids can be identified with classical one-object categories (in fact the classical case is just enrichment over sets with the cartesian product).

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    $\begingroup$ I tend to use the term "monoid object" to mean a model of the theory of monoids in a category potentially different from $\mathbf{Set}$. $\endgroup$ Nov 8, 2018 at 6:45

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