The category $\mathbf{Set}$ of sets can be viewed as the category of models for a Lawvere theory, and hence it is equivalent to the Eilenberg-Moore category of algebras of an (associated to the Lawvere theory) monad $\mathbb{T}:\mathbf{Set}\to\mathbf{Set}$.

What is the Kleisli category of $\mathbb{T}$?

The Kleisli category of $\mathbb T$ is the subcategory of the Eilenberg-Moore category of all free algebras. Now, every set is the free set over itself (if I'm not mistaken).

  • 2
    Yes, just recall that the actual monad in this case is the identity functor. – Kevin Carlson Jul 25 '16 at 19:51

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