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As a consequence, the new notion of “cofibrantly generated” agrees with the usual one, in the case where they ought to overlap. 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.

For Garner’s small object argument, this process is modified to include additional quotienting. At step two and all subsequent steps, the beginning is the same. We pushout L0 R1 f along the canonical arrow to obtain L1 R1 f . But then L2 f is defined to be L1 f composed with the coequalizer of two arrows from L1 f to L1 R1 f . As in previous steps, this is a quotient of Quillen’s definition. In the language of cell complexes, the arrow L1 R1 f freely attaches new “cells” to the “spheres” in the domain of R1 f , while L1 f includes those “cells” attached to “spheres” in the domain of f into their image in the domain of R1 f .

Dually, we can include M into M2 by slicing under the initial object. Using this inclusion, the comonad C induces a comonad Q = (Q, , δ) on M which we call the cofibrant replacement comonad. Once again, the functor Q : M → M is obtained from the previous functor Q : M2 → M by precomposing Q by this inclusion. Algebras for R are called algebraically fibrant objects and coalgebras for Q are called algebraically cofibrant objects. 3) is in comparing fibrant-cofibrant replacements of an object. Let M be a category with an algebraic model structure and let X ∈ M.

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