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Note that these functors are not inverse equivalences. The upshot is that we can get an algebraic model structure from an ordinary cofibrantly generated model structure without changing the generating cofibrations. 13. 8 did not require that the awfs (C, Ft ) is cofibrantly generated, though in examples this is typically the case. 5, we will show that the components of the comparison map in a cofibrantly generated algebraic model structure satisfying additional, relatively mild, hypotheses are themselves C-coalgebras and hence cofibrations.

Suppose (L, R) is an awfs on M generated by a category J. , suppose cod f = dom g. 22. In the remaining sections, we will present new results relating awfs to model structures, taking frequent advantage of the machinery provided by Garner’s small object argument. 3 Algebraic model structures The reasons that most topologists care (or should care) about weak factorizations systems is because they figure prominently in model categories, which are equipped with an interacting pair of them. Using Garner’s small object argument, whenever these wfs are cofibrantly generated, they can be algebraicized to produce awfs.

Similar remarks apply to the fibrations. 24 M2 by sending an object X to the unique arrow from X to the terminal object. This inclusion is a section to the functor dom : M2 → M. Because the monad F is a monad over cod, it induces a monad R = (R, η, µ) on M which we call the fibrant replacement monad. The functor R is obtained from the previous functor R : M2 → M accompanying the functorial factorization of (Ct , F) by precomposing R by this inclusion. We regret that our notation is somewhat ambiguous.

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