The University of Chicago Algebraic Model Structures a Dissertation Submitted to the Faculty of the Division of the Physical Sciences in Candidacy for the Degree of Doctor of Philosophy

In Part I of this thesis, we introduce algebraic model structures, a new context for homotopy theory in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads and prove “algebraic” analogs of classical results. Using a modified version of Quillen’s small object argument, we show that every cofibrantly generated model structure in the usual sense underlies a cofibrantly generated algebraic model structure. We show how to pass a cofibrantly generated algebraic model structure across an adjunction, and we characterize the algebraic Quillen adjunction that results. We prove that pointwise algebraic weak factorization systems on diagram categories are cofibrantly generated if the original ones are, and we give an algebraic generalization of the projective model structure. Finally, we prove that certain fundamental comparison maps present in any cofibrantly generated model category are cofibrations when the cofibrations are monomorphisms, a conclusion that does not seem to be provable in the classical, non-algebraic, theory. In Part II, we define monoidal algebraic model structures and discuss examples. The main structural component requires a new notion: an algebraic Quillen two-variable adjunction. The principal technical work is to develop the category theory necessary to define and characterize them. Our investigations reveal an important role played by “cellularity”—loosely, the property of a cofibration being a relative cell complex, not simply a retract of such—which we particularly emphasize. A main result is a simple criterion which shows that algebraic Quillen two-variable adjunctions correspond precisely to cellular structures on the pushout-products of generating (trivial) cofibrations, extending a similar result from Part I.