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, nonalgebraic, theory.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 arXiv:
 arXiv:0910.2733
 Bibcode:
 2009arXiv0910.2733R
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Algebraic Topology;
 55U35;
 18A32
 EPrint:
 51 pages