A Model 2-Category of Enriched Combinatorial Premodel Categories
Barton, Reid William
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CitationBarton, Reid William. 2019. A Model 2-Category of Enriched Combinatorial Premodel Categories. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences.
AbstractQuillen equivalences induce equivalences of homotopy theories and therefore form a natural choice for the "weak equivalences" between model categories.
In , Hovey asked whether the 2-category Mod of model categories has a "model 2-category structure" with these weak equivalences.
We give an example showing that Mod does not have pullbacks, so cannot be a model 2-category.
We can try to repair this lack of limits by generalizing the notion of model category.
The lack of limits in Mod is due to the two-out-of-three axiom, so we define a premodel category to be a complete and cocomplete category equipped with two nested weak factorization systems.
Combinatorial premodel categories form a 2-category CPM with excellent algebraic properties: CPM has all limits and colimits and is equipped with a tensor product (representing Quillen bifunctors) which is adjoint to an internal Hom.
The homotopy theory of a model category depends in an essential way on the weak equivalences, so it does not extend directly to premodel categories.
We build a substitute homotopy theory under an additional axiom on the premodel category, which holds automatically for a premodel category enriched in a monoidal model category V.
The 2-category of combinatorial V-premodel categories VCPM is simply the 2-category of modules over the monoid object V, so VCPM inherits the algebraic structure of CPM.
We construct a model 2-category structure on VCPM for V a tractable symmetric monoidal model category, by adapting Szumiło's construction of a fibration category of cofibration categories .
For set-theoretic reasons, constructing factorizations for this model 2-category structure requires a technical variant of the small object argument which relies on an analysis of the rank of combinatoriality of a premodel category.
Citable link to this pagehttp://nrs.harvard.edu/urn-3:HUL.InstRepos:42013127
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