By Emily Riehl

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3). The solutions to these lifting problems are the components of a natural transformation RQ ⇒ QR comparing the two fibrant-cofibrant replacements. This natural map is particularly well-behaved; hence the following theorem. 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.

28 → J over M2 , necessarily faithful, that is bijective on the underlying (i) There is a functor J classes of arrows. It follows that J and J generate the same underlying wfs. (ii) There is a functor ζ : J → C-coalg over M2 . The set J generates an awfs (Ct , F ). 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.

L A α G M2 ' M2 All of the other natural transformations are defined similarly. It is easy to see that LA = (LA , A , δA ) and RA = (RA , ηA , µA ) define an awfs because all the definitions are given by simply post-composing a natural transformation with the old comonad and monad. 2. Note however that the underlying wfs of the pointwise awfs (LA , RA ) is not itself given pointwise by the underlying wfs of (L, R). This is because, unlike the case for the left and right factors, generic pointwise maps will not have natural lifts.

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