Re: monads on model categories

[Emily Riehl <eriehl@math.harvard.edu>]

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-the-kleisli-category

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.html

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.
>

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