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By Emily Riehl

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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.

It follows from the universal property of the functor J → Ct -coalg that I and J generate an algebraic model structure on M2 with the same underlying model structure (C, F, W). J J C Proof. Define J → M2 to be the composite J → M2 → M2 where C is the comonad generated by I. For each j ∈ J, the corresponding element of J is its left factor C j. We claim that J = {C j | j ∈ J} satisfies conditions (i) and (ii) above. For (ii), define ζ to be the map that assigns each C j its canonical free coalgebra structure (C j, δ j ).

5. The functor Q lifts to a cofibrant replacement comonad on the category R-alg of algebraically fibrant objects. Dually, the functor R lifts to a fibrant replacement monad on the category Q-coalg of algebraically cofibrant objects. ” Proof. 1). 4 satisfies the desired property: The defining lifting problem shows that χ is compatible with the unit and counit for R and Q. , that RQRp R2 QQ ` Rχ xxxx x x xx QQ µQ QQQ % RQ pp χ pp R pp p4 χ 26 G QR2    Qµ Õ  QR commutes, and a dual condition for the comultiplication.