New York Journal of Mathematics
Volume 17 (2011) 173-231


Emily Riehl

Algebraic model structures

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Published: March 10, 2011
Keywords: Model categories; factorization systems
Subject: 55U35, 18A32

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.


Supported by a NSF Graduate Research Fellowship

Author information

Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637