Hello, Patrik, A correction to:
- The Eilenberg-Moore category over the (covariant?) double powerset monad is isomorphic to the category of complete atomic Boolean algebras. (What about the contravariant double?)
The category of complete atomic Boolean algebras is isomorphic to the Eilenberg-Moore category over the double CONTRAVARIANT powerset monad. (but I have no idea about any double covariant powerset monad). A general bunch of examples: each "equationally definable class of (possibly) infinitary algebras for which all free algebras exist," thought of as a category by using the relevant homomorphisms, is isomorphic to the Eilenberg-Moore category over the associated "free-algebra" monad. (Certainly the cases below of complete atomic Boolean algebras, and of compact Hausdorff spaces, can be thought of as arising in this way.) And the Kleisli category arising from that monad is equivalent to the full subcategory of FREE algebras from that class. Cheers, -- Fred --- Patrik Eklund wrote:
MONADS AND ALGEBRAS
We have, from various sources, found the following examples of algebras related to monads:
- The Kleisli category over the usual powerset monad is isomorphic to the category of sets and relations. - The Eilenberg-Moore category of the usual powerset monad is isomorphic to the category of complete lattices and join-preserving maps. - The Kleisli category of the term monad coincides with its Eilenberg-Moore category and is isomorphic to the category of $\Omega$-algebras. (Is this really true?) - The Eilenberg-Moore category over the (covariant?) double powerset monad is isomorphic to the category of complete atomic Boolean algebras. (What about the contravariant double?) - The Eilenberg-Moore category over the filter monad is isomorphic to the category of continuous lattices. - The Eilenberg-Moore category over the ultrafilter monad is isomorphic to compact Hausdorff spaces.
Are there further well- or worse-known examples? Are there survey papers on this, also including proofs?
Best regards,
Patrik Eklund Umea University, Sweden