Hi, Emily, Have you or your grad student noticed the example got by taking as your locally presentable category C the category of sets, and as your monad M the ultrafilter or Stone-Cech monad ß (with the Eilenberg-Moore category of ß-algebras being the category of compact Hausdorff spaces (and continuous maps))? Here, your "objects those X in C such that the unit X -> MX is an isomorphism" are just the finite sets, and, unless I misunderstand your limits question, I fear that only finite limits will work as you desire. Does that help in any way? Cheers, -- Fred --- ------ Original Message ------ Received: Fri, 10 May 2013 09:23:34 PM EDT From: Emily Riehl <eriehl@math.harvard.edu> To: categories@mta.ca Subject: categories: on a subcategory of algebras for a monad
Hi,
I received the following question from a grad student that I was unable to answer, but maybe you can (shared with permission). The subcategory Comp_M he introduces below can equally be defined to be the inverter of the counit of the monadic adjunction. But I don't see how this universal property helps understand limits in the subcategory. We suspect a left adjoint to the inclusion is unlikely.
Can you help? Or have you seen something like this before?
Best, Emily
*** ?? Hi folks, ?? I'm interested in closure properties of a particular subcategory of the category of algebras of a monad. To be more precise, let C be a locally presentable category and M be a monad on C. The category of algebras Alg_M has all limits, and they are computed in C. Denote by Comp_M the full subcategory of Alg_M of "M-complete objects" (does anyone have a better name?), with objects those X in C such that the unit X -> MX is an isomorphism, viewed in the natural way as M-algebras (using the inverse MX -> X). ?? My question: Is Comp_M closed under (actually: sequential) limits, computed as limits in Alg_M? ?? For some examples that come to mind immediately, the answer is clearly yes, because Comp_M is either trivial (e.g., if M is the free monoid monad on Sets) or all of Alg_M (i.e., if M is idempotent). A more interesting example is Bousfield-Kan R-completion, for which I don't know the answer. ?? In fact, I'm interested in left exact monads; in this case, the idempotent approximation is given by the equalizer of the two natural maps M -> M^2, but I'm not sure if this is relevant. What I'm hoping for is a sufficient criterion or a good counterexample in the abstract situation. ?? Many thanks!
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