By Emily Riehl

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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. This is one area where awfs behave better than ordinary wfs. 3 Cofibrantly generated case Given a cofibrantly generated awfs (L, R) on M, is the resulting pointwise awfs (LA , RA ) on MA cofibrantly generated? There are many reasons to suspect that this is not the case.

8. 3 holds more generally than stated. In particular, it is not necessary that the arrows in the positions of Ct0 f and C 0 g be coends over all possible squares. As long as these arrows are constructed as coends over some squares such that (iS , iD ) is an inclusion, the conclusion follows. In applications, we will often require this slightly more general result, for reasons that will become clear in a moment. 1. Our proof used a modified version of the small object argument, suggested by Richard Garner in private communication, that can be used whenever the elements of the left class of the underlying wfs are monomorphisms.

So Ft ⊂ F ∩ WK . 30 on two occasions. Suppose f ∈ F ∩ WK . 30, f has some algebra structure ( f, ψ) ∈ T J and by adjunction (S f, ψ ) ∈ J . 30, it follows that there is some algebra structure ζ such that (S f, ζ) ∈ I . By adjunction, ( f, ζ ) ∈ T I , where ζ denotes the adjunct of ζ, which says that f ∈ Ft , as desired. The above argument showed that S preserves fibrations and trivial fibrations. Hence, T S is a Quillen adjunction. 10, it is not surprising that quite a lot more can be said about the nature of the Quillen adjunction between them.

### Algebraic model structures by Emily Riehl

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