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We also have a faithful functor J → J J lying classes of arrows; this one, however, takes a bit more effort to define. Define J → M2 to Ct J be the composite J → M2 → M2 . We have a functor γ : J → Ct -coalg over M2 that assigns each arrow its free coalgebra structure. Mirroring the argument above, elements of J are retracts of elements of J , so we have J →J over M2 . 27). 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.

31 Similarly, the comparison map guarantees that Ft ⊂ F. 30. By adjunction (S g, φ ) ∈ I , where the arrows of φ are the adjuncts of the corresponding arrows of φ. So S g is a trivial fibration for the model structure on M. In particular, S g ∈ WM , which says that g ∈ WK . 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.

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. 2) are the same as before. At this point, Quillen’s small object argument has us freely attach “cells” to fill “spheres” in the object R1 f by repeating steps zero and one for the map F 1 f .

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