Algebraic model structures

Abstract

We define a new notion of an algebraic model structure, 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 natural 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.

51 pages; final journal version; cosmetic changes only

References in corpus (3)

• Cofibrance and Completion
• Cofibrantly generated natural weak factorisation systems
• Algebraic models for higher categories

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• Lax orthogonal factorisation systems
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• t-structures are normal torsion theories
• Models of Martin-Löf type theory from algebraic weak factorisation systems
• The Coalgebraic Structure of Cell Complexes
• A homotopy-theoretic model of function extensionality in the effective topos
• Multivariable adjunctions and mates
• Lifting Problems in Grothendieck Fibrations
• Identity Types in Algebraic Model Structures and Cubical Sets
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• An orthogonal approach to algebraic weak factorisation systems
• On semiflexible, flexible and pie algebras
• Quillen-Segal objects and structures: an overview
• Stable factorization from a fibred algebraic weak factorization system
• t-structures on stable infinity-categories