Re: monads on model categories

[Peter May <may@math.uchicago.edu>]

*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.
>

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