2020/04/27 by Reid William Barton, Barton, Reid William · 7 citations
Mathematics · Medicine · #18N40 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #Computer science #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications #Mathematics #math.AT #math.CT #msc:18N40
paper · pdf · doi:10.48550/arxiv.2004.12937
published in arXiv (Cornell University) (Cornell University) · 170 pages; author's PhD thesis with formatting changes
arxiv created 2020/04/27 · openalex publication_date 2020/04/27 · arxiv updated 2020/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
In his book on model categories, Hovey asked whether the 2-category Mod of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that Mod does not have pullbacks and so cannot form a model 2-category. This lack of pullbacks can be traced to the two-out-of-three axiom on the weak equivalences of a model category. Accordingly, we define a premodel category to be a complete and cocomplete category equipped with two nested weak factorization systems. Combinatorial premodel categories form a complete and cocomplete closed symmetric monoidal 2-category CPM whose tensor product represents Quillen bifunctors. For a monoidal combinatorial premodel category V, the 2-category VCPM of V-enriched combinatorial premodel categories is simply the category of modules over V (viewed as a monoid object of CPM), and therefore inherits the algebraic structure of CPM. The homotopy theory of a model category depends in an essential way on the weak equivalences, so it does not extend directly to a general premodel category. We develop a substitute homotopy theory for premodel categories satisfying an additional property which holds automatically for model categories and also for premodel categories enriched in a monoidal model category. In particular, for a monoidal model category V, we obtain a notion of Quillen equivalence of V-premodel categories which extends the one for V-model categories. When V is a tractable symmetric monoidal model category, we construct a model 2-category structure on VCPM with these Quillen equivalences as the weak equivalences, by adapting Szumiło's construction of a fibration category of cofibration categories.