Suppose C is a category and T is a monad on C. One knows that one can factor T into a composite GF, where F,G are an adjoint pair of functors, and in fact one knows that there are two universal ways to do this, a Kleisli/initial construction and an Eilenberg-Moore/terminal construction. Now suppose C is a model category and T is a monad on C which preserves weak equivalences. One would like to know that T factors as GF, where F,G are a Quillen pair. Is this always possible and does one have Kleisli-like and Eilenberg-Moore-like constructions with appropriate universal properties? I am sure people have worked on these questions before; where can I read about this? Thanks, Andrew S. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
*The Eilenberg-Moore adjunction is the one studied in the model category literature, so F is the monad viewed as taking values in the category C[T] of T-algebras, and G is the forgetful functor. The category C[T] is complete, with limits created in C, but it must be proven that it is cocomplete. This holds if T preserves reflexive coequalizers (EKMMM II.7.4) or if C[T] has coequalizers (a result of Linton). Define the weak equivalences and fibrations in C[T] to be created by the forgetful functor G. Then G automatically preserves fibrations and acyclic fibrations, so the only question is whether or not C[T] is a model category. (One does not expect T to preserve all weak equivalences). When C is cofibrantly generated with sets I and J of generating cofibrations and acyclic cofibrations, one takes FI and FJ as proposed sets of generating cofibrations and acyclic cofibrations in C[T]. Then C[T] is a cofibrantly generated model category if two conditions hold. 1. FI and FJ are small. In practice, this is the easy point (or so it seems to me) and the literature expands on it ad nauseum. It obviously holds by adjunction if G preserves the colimits used in the small object argument. 2. Every relative FJ-cell complex X --> Y is a weak equivalence. This is the substantive point and concerns the preservation of weak equivalences under the colimits used in the small object argument. In many topological situations, the maps in J are inclusions of deformation retractions and the verification is simple. In others one uses the structure of the given monad. Since the proof differs technically in different contexts, I'm not sure that an axiomatization is all that helpful. *On 11/26/11 6:49 PM, Andrew Salch wrote:
Suppose C is a category and T is a monad on C. One knows that one can factor T into a composite GF, where F,G are an adjoint pair of functors, and in fact one knows that there are two universal ways to do this, a Kleisli/initial construction and an Eilenberg-Moore/terminal construction.
Now suppose C is a model category and T is a monad on C which preserves weak equivalences. One would like to know that T factors as GF, where F,G are a Quillen pair. Is this always possible and does one have Kleisli-like and Eilenberg-Moore-like constructions with appropriate universal properties? I am sure people have worked on these questions before; where can I read about this?
Thanks, Andrew S.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Andrew, I don't know that this question has been answered in the literature. (If I am wrong about this, would someone please reply to me as well. I would be very interested to hear otherwise.) Here are a few things I do know. When the original model structure is cofibrantly generated, one could always attempt to define a model structure on C^T or C_T by passing the generating (trivial) cofibrations along the left adjoint and defining weak equivalences to be those created by the right adjoint. Up to the caveat of the next paragraph, this procedure yields a model structure and a Quillen adjunction iff the so-called "acyclicity condition" (transfinite composites of pushouts of the generating trivial cofibrations are weak equivalences) is satisfied. I don't know of general results indicating when this might be true, but there has been some work done regarding the category of algebras for a particular sort of monad, which I'll describe below. One annoying difficulty with the Kleisli/Eilenberg-Moore constructions is that these categories might not be (co)complete. For the Eilenberg-Moore category, the only issue is with colimits. If the category of algebras has reflexive coequalizers then this is strong enough. I believe this result is due to Linton (cf "Coequalizers in categories of algebras"). In general, the Kleisli category will be neither complete nor cocomplete. Todd Trimble has some nice counterexamples in this mathoverflow answer: http://mathoverflow.net/questions/37965/completeness-and-cocompleteness-of-t... Now suppose the model structure on C is cofibrantly generated and C permits the small object argument. Then from any set of trivial cofibrations that detect fibrant objects, one can construct a fibrant replacement monad on C using Richard Garner's small object argument (cf "Understanding the small object argument"). Its algebras are so-called "algebraically fibrant objects" (eg, infinity-categories with chosen fillers for all inner horns). Any fibrant object admits at least one algebra structure and all objects admitting algebra structures are fibrant. Thomas Nikolaus has shown that if the trivial cofibrations are monomorphisms, then the category of algebras is cocomplete and admits a model structure constructed by the procedure described above in which all objects are fibrant. It is easy to see that in this case the monadic adjunction is a Quillen equivalence. So one upshot is that any cofibrantly generated model category for which the trivial cofibrations are monic is Quillen equivalent to one for which all objects are fibrant. See: http://golem.ph.utexas.edu/category/2010/03/_guest_post_by_thomas_nikolaus.h... Returning to the question for a generic monad T, a more general answer should be available when one doesn't require that the categories C^T and C_T admit full model structures but rather asks only for homotopical categories and left or right deformable functors, in the sense of Dwyer-Kan-Hirschhorn-Smith. I have been working on several aspects of this question with Andrew Blumberg. We hope to report an answer soon. I believe Justin Noel and Niles Johnson have forthcoming work on a related question. Best, Emily On Sat, 26 Nov 2011, Andrew Salch wrote:
Suppose C is a category and T is a monad on C. One knows that one can factor T into a composite GF, where F,G are an adjoint pair of functors, and in fact one knows that there are two universal ways to do this, a Kleisli/initial construction and an Eilenberg-Moore/terminal construction.
Now suppose C is a model category and T is a monad on C which preserves weak equivalences. One would like to know that T factors as GF, where F,G are a Quillen pair. Is this always possible and does one have Kleisli-like and Eilenberg-Moore-like constructions with appropriate universal properties? I am sure people have worked on these questions before; where can I read about this?
Thanks, Andrew S.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Andrew Salch -
Emily Riehl -
Peter May