Algebraic model structures

Emily Riehl*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

28 Citations (Scopus)


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 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, nonalgebraic, theory.

Original languageEnglish
Pages (from-to)173-231
Number of pages59
JournalNew York Journal of Mathematics
Publication statusPublished - 2011
Externally publishedYes


  • Factorization systems
  • Model categories


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