The statement, "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?)" is incorrect. The Kleisli category is equivalent to the subcategory of free Omega algebras. Monads are discussed extensively in Barr and Wells, Toposes, Triples and Theories (Triples is an earlier name for monads) on the internet at http://www.cwru.edu/artsci/math/wells/pub/ttt.html Chapters 3 and 9 contain references to several examples of particular monads, some of them with proofs. I wish someone would write a survey of all the major examples These ideas are expressed alternatively as a type of sketch. There is a brief outline online at http://www.cwru.edu/artsci/math/wells/pub/pdf/sketch.pdf More extensive discussions of sketches are in Toposes, Triples and Theories (above) and in: Adamek and Rosicky, Locally Presentable and Accessible Categories, Cambridge University Press. Barr and Wells, Category Theory for Computing Science, available from http://crm.umontreal.ca/pub/Ventes/CatalogueEng.html (at the moment the ability to order directly on line seems to be broken -- we are investigating -- but you can print an order form to mail in). This book is not available from Amazon. Charles Wells
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
Charles Wells Oberlin, Ohio, USA blog: http://wabe.blogspot.com/ professional website: http://www.cwru.edu/artsci/math/wells/home.html personal website: http://www.oberlin.net/~cwells/index.html genealogical website: http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/ NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm 10-Feb-2005 15:21:36 -0400,1746;000000000000-00000000