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In "The Geometry of Iterated Loop Spaces", May shows that any monad that is coming from an operad may be "whiskered", so that the unit map becomes a closed cofibration. The ability to do this is vital to constructing simplicial resolutions. In particular, it guarantees that the bar resolution $B_\bullet(T, T, X)$ is a proper (or good, in Segal's terminology) simplicial space.

I am wondering in what generality this can be done. Given a topological monad $T$, when can I find a monad $T'$ whose unit is a closed cofibration, along with a morphism of monads $T' \rightarrow T$ inducing homotopy equivalences $T'(X) \simeq T(X)$ for all spaces $X$?

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  • $\begingroup$ The homotopy theory of monads! $\endgroup$
    – Fernando Muro
    Commented Nov 25, 2014 at 17:12
  • $\begingroup$ @Fernando Muro Is there a model category structure on topological monads where weak equivalences are objectwise homotopy equivalences and cofibrant objects are monads whose units are closed cofibrations? This seemed like a lot to ask for, but I would be pleasantly surprised if this is the case. $\endgroup$
    – Matthew Sartwell
    Commented Nov 25, 2014 at 17:28
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    $\begingroup$ Me too, I find the question very interesting. General monads are not very 'algebraic' so I'd say no. Maybe special kinds of monads (but more general than operads) like polyonomial monads, etc. These monads have a nice homotopy theory of algebras, as Batanin and Berger recently showed (under usual assumptions). $\endgroup$
    – Fernando Muro
    Commented Nov 25, 2014 at 17:42

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