Let $F$ denote a monad on $\mathbf{Set}$. Write $\mathbf{Set}_F$ for the corresponding Kleisli category and $\mathbf{Set}^F$ for the Eilenberg-Moore category. On the train home today, it occured to me that functors $\mathbf{Set}_F^{\mathrm{op}} \rightarrow \mathbf{Set}$ that preserve all small products ought to be the same as objects of $\mathbf{Set}^F$, for the simple reason that arrows of $\mathbf{Set}_F^{\mathrm{op}}$ are kind of like the operations of a Lawvere theory, hence we expect functors $\mathbf{Set}_F^{\mathrm{op}} \rightarrow \mathbf{Set}$ that preserve small products to be kind of like algebras. A bit of searching around yielded this and this, but I couldn't find a completely clear answer.

Question. Is the category of small-product preserving functors $\mathbf{Set}_F^{\mathrm{op}} \rightarrow \mathbf{Set}$ equivalent to the Eilenberg-Moore category $\mathbf{Set}^F$?

  • I guess you mean to ask about the category of functors that preserve small products. The answer is yes, the proof is in the comments at the second linked page. – Zhen Lin Aug 6 '15 at 15:01
up vote 4 down vote accepted

Let $\mathcal{K}$ be the Kleisli category and let $\mathcal{A}$ be the Eilenberg–Moore category. We regard $\mathcal{K}$ as a full subcategory of $\mathcal{A}$ in the obvious way, and this induces a Yoneda representation $$\mathcal{A} \to [\mathcal{K}^\mathrm{op}, \mathbf{Set}]$$ which is fully faithful and essentially surjective onto the full subcategory of functors $\mathcal{K}^\mathrm{op} \to \mathbf{Set}$ that preserve small products.

Indeed, it is clear that, for any $(A, \alpha)$ in $\mathcal{A}$, the representable presheaf $\mathrm{Hom} (-, (A, \alpha))$ gives a functor $\mathcal{K}^\mathrm{op} \to \mathbf{Set}$ that preserves small products. Moreover, the counit $\epsilon_{(A, \alpha)} : (F A, \mu_A) \to (A, \alpha)$ is a (regular) epimorphism, so the Yoneda representation $\mathcal{A} \to [\mathcal{K}^\mathrm{op}, \mathbf{Set}]$ is faithful. A similar idea can be used to show that the Yoneda representation is full.

The essential surjectivity claim remains to be verified. Let $\mathcal{P}$ be the full subcategory of functors $\mathcal{K}^\mathrm{op} \to \mathbf{Set}$. Evaluation at $(F 1, \mu_1)$ gives a forgetful functor $U : \mathcal{P} \to \mathbf{Set}$. Let $L : \mathbf{Set} \to \mathcal{P}$ be given by $L X = \mathrm{Hom} (-, (F X, \mu_X))$. By the Yoneda lemma, $$\mathcal{P} (L X, K) \cong K (F X, \mu_X) \cong \mathbf{Set} (X, U K) $$ so $L \dashv U$, and the induced monad is isomorphic to $(F, \eta, \mu)$. To complete the proof, it is enough to show that $U : \mathcal{P} \to \mathbf{Set}$ is monadic; but $U : \mathcal{P} \to \mathbf{Set}$ is faithful (easy) and creates $U$-split coequalisers (harder), so this is a consequence of Beck's monadicity theorem.

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