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Many years ago, Lawvere showed that the forgetful functor $U: \mathbf{Endo}\to \mathbf{Set}$ has a left adjoint $F$ if and only if $\mathbf{Set}$ has a natural numbers object, where $\mathbf{Endo}$ is the category of endos $f: A\to A$, where $A$ is a set. Then $T=UF$ is the functor part of a monad $(T,\mu,\eta)$. I learned some years ago too that $\mathbf{Endo}$ satisfies the Beck conditions in Beck's Theorem, so that the category of $\mathbf{Set}^T$ of $T$-algebras is equivalent to $\mathbf{Endo}$.

An exercise in Barr and Wells states that under certain conditions, the equivalence guaranteed by Beck's Theorem is in fact an isomorphism (see Exercise PPTT p. 116). My question is: whether, in the case of the monad $(T,\mu,\eta)$ mentioned above, $\mathbf{Endo}$ turns out to be isomorphic to $\mathbf{Set}^T$? --- either because the conditions mentioned in the exercise are satisfied or for some other reason.

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  • $\begingroup$ I edited slightly to help the question be more obvious. $\endgroup$
    – David Roberts ♦
    Commented Dec 4, 2016 at 18:36
  • $\begingroup$ I think if you just write down the definitions it's obviously isomorphic. In both $\mathbf{Endo}$ and $\mathbf{Set}^T$, every object has a uniquely-defined carrier object in $\mathbf{Set}$, and you can write down the bijection between $T$-algebra structures on $A$ and endomorphisms of $A$, which underlies the comparison functor (or rather -- since the comparison functor is well-defined only up to isomorphism -- the obvious choice of comparison functor). I don't see any problems unless maybe if you're trying to work in a weak metatheory or something... $\endgroup$
    – Tim Campion
    Commented Dec 5, 2016 at 7:33

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