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The awfs of this model structure are not the pointwise awfs on MA ; instead, the generating categories are discrete, at least when the original generators I and J are. The underlying model structure agrees with the usual projective model structure on a diagram category: weak equivalences are pointwise weak equivalences and fibrations are pointwise fibrations. The generating categories Iproj and Jproj for the projective model structure look familiar; in the case where I and J are discrete these are the usual generating sets in the classical theory.

6 Cofibrantly generated awfs There are a few naturally occurring examples of awfs where the familiar functorial factorizations for some wfs underlie a comonad and a monad. One toy example is the so-called “graph” factorization of an arrow through the product of its domain and codomain. There are more serious 18 examples, including the wfs from the Quillen model structure on ChR and the folk model structure on Cat. However, the examples topologists find in nature are less obviously “algebraic,” and consequently awfs have not generated a lot of interest among topologists.

Define J → M2 to be the composite J → M2 → M2 where C is the comonad generated by I. For each j ∈ J, the corresponding element of J is its left factor C j. We claim that J = {C j | j ∈ J} satisfies conditions (i) and (ii) above. For (ii), define ζ to be the map that assigns each C j its canonical free coalgebra structure (C j, δ j ). Cj G · Ðc s Ð For (i), note that for each j ∈ J the lifting problem j Ð Ft j has a solution s, which gives  Ð  · · us a retract diagram j · ·  G · · s · · Cj Ft j G · j .

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