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
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
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
On Mon, 7 Feb 2005, Patrik Eklund wrote:
- 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?)
No, this is not true: if $\Omega$ consists of two constant sybmols, the term monad is TX = X+1+1 and the Klesli category does not have a terminal object whereas the Eilenberg-Moore category does. The latter part of the statement is true, see e.g. MacLane. An additional example: Recently, the Eilenberg-Moore category of the infinite-term monad was described as the category of complete Elgot algebras. See J. Adamek, S. Milius and J. Velebil: On Iterative Algebras and Elgot Algebras (submitted) http://www.iti.cs.tu-bs.de/~milius Best, Jiri Adamek
participants (4)
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Charles Wells -
Fred E.J. Linton -
Jiri Adamek -
Patrik Eklund