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27). 32. 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.

In retrospect, the solution to the above concern is obvious: the generating category JA for the pointwise awfs will not be discrete (unless A is)! This is the first example known to the author where the extra generality allowed in Garner’s small object argument is useful. 3. Let J : J → M2 be a small category over M2 , where M permits the small object argument, and let (L, R) be the awfs generated by J. Let JA be the category Aop × J equipped with the functor y×J −·− Aop × J −→ SetA × M2 −→ (MA )2 , 37 where y denotes the Yoneda embedding and − · − denotes the copower8 (tensor) of an arrow in M by a Set-valued functor from A.

Let M have an algebraic model structure ξM : (Ct , F) → (C, Ft ) and let K G have an algebraic model structure ξK : (Lt , R) → (L, Rt ). An adjunction T : M o ⊥ K : S is an algebraic Quillen adjunction if there exist adjunctions of awfs ξM  (C, Ft ) (T,S ) G (Lt , R) €€€ (T,S )€€ ξK €€@  G (L, Rt ) (T,S ) (Ct , F) € such that both triangles commute. Note the left adjoint of an adjunction of awfs preserves coalgebras and hence (trivial) cofibrations and dually the right adjoint preserves algebras and hence (trivial) fibrations.

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