Re: orthogonal factorization systems
Richard Garner <richard.garner <at> mq.edu.au>
2012-04-14 22:27:57 GMT
You are in luck. Your result is an instance of the following:
If C is a category bearing the orthogonal factorisation system (E,M),
and T is a monad on C whose underlying functor preserves E-maps, then
C^T bears the orthogonal factorisation system (U^-1(E), U^-1(M)).
A full proof of which is given as Proposition 20.28 in:
Abstract and concrete categories: The joy of cats (Wiley, 1990)
Jiri Adamek, Horst Herrlich and George Strecker
Maybe there is an older reference than this but I am not aware of such.
On 14 April 2012 08:09, Emily Riehl <eriehl <at> math.harvard.edu> wrote:
> I've placed a bet with a colleague that the following result appears in
> the literature. Please help me win.
> Claim: Suppose (E,M) is an orthogonal factorization system (unique lifts)
> on a symmetric monoidal category and X is a fixed monoid. If tensoring
> with X preserves maps in the class E, then (E,M) lifts to an orthogonal
> factorizaiton system on the category of X-modules.
> Emily Riehl