Compact Hausdorff spaces are algebras of the ultrafilter monad on Set.


Is the category of Tychonoff spaces also monadic over Set?



Is there a monad on Set whose algebras are Tychonoff spaces?
No. In fact any full subcategory of $\mathbf{Top}$ that contains all the discrete spaces cannot be monadic over $\mathbf{Set}$ unless it contains only discrete spaces. Indeed, for any such subcategory $\mathcal{C}$, the forgetful functor $\Gamma : \mathcal{C} \to \mathbf{Set}$ has a left adjoint $\Delta : \mathbf{Set} \to \mathcal{C}$ which sends a set to the corresponding discrete space. The monad induced by $\Delta \dashv \Gamma$ is the identity monad on $\mathbf{Set}$, so if $\Gamma$ were monadic, then $\mathcal{C}$ must be the full subcategory of discrete spaces.


