orthogonal factorization systems

orthogonal factorization systems

[Emily Riehl]

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.

Regards,
Emily Riehl





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Re: orthogonal factorization systems

[Richard Garner]

Dear Emily,

You are in luck. Your result is an instance of the following:

PROP
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.

Richard


On 14 April 2012 08:09, Emily Riehl <eriehl@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.
>
> Regards,
> Emily Riehl
>
>
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]