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By Emily Riehl

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The awfs in the projective model structure will not be the 34 pointwise awfs, though these awfs will make an appearance in the proof establishing this model structure. We first take a detour to describe Garner’s small object argument in more detail, as these details will be used in the proofs in this section and the next. 1 Garner’s small object argument Like Quillen’s, Garner’s small object argument produces a functorial factorization through a colimiting process that takes many steps, a key difference being that the resulting functorial factorization canonically underlies an awfs.

12) uu(ξ K )∗ uu u7 L-coalg s sss s s sy Similarly, the corresponding extensions to Kleisli and coKleisli categories commute. Somewhat surprisingly due to the numerous conditions required by their components, algebraic Quillen adjunctions exist in familiar situations. 13. Let T : M o ⊥ K : S be an adjunction. Suppose M has an algebraic model structure, generated by I and J, with comparison map ξM . Suppose K has the algebraic model structure, generated by T I and T J, with canonical comparison map ξK .

See [Gar09, §3]. As far as the author is aware, model category theorists have not written about the issue of comparing the two wfs provided by an ordinary model structure, a fact that first came to her attention through discussions with Martin Hyland. But the existence of such a comparison map is more reasonable than one might expect: Peter May notes [MP11] that the universal property of the colimits in Quillen’s small object argument gives such a natural transformation, provided we assume that the generating trivial cofibrations J are contained in the generating cofibrations I.

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Algebraic model structures by Emily Riehl

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