Algebraic model structures
By Emily Riehl. 2009
63 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6739a2b5c2827781383ae00a788cece9383a4002
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In Algebra and Number Theory
- Emily Riehl
- Richard Garner
- John Bourke
- T. Barthel
- Jon P. May
- Michael Shulman
- James Gillespie
- Kathleen Ponto
In Geometry and Topology
- Emily Riehl
- Richard Garner
- John Bourke
- T. Barthel
- Jon P. May
- Michael Shulman
- James Gillespie
- J. Sigurdsson
In Mathematical Physics
- Emily Riehl
- Richard Garner
- John Bourke
- Jon P. May
- T. Barthel
- Michael Shulman
- Kathleen Ponto
- James Gillespie
Citations in the corpus
- Six model structures for DG-modules over DGAs: model category theory in homological action (2013, background): “We refer the reader to [2, 35, 36] for the precise definitions and further discussion.”
- Algebraic weak factorisation systems I: Accessible AWFS☆ (2014, background): “Parts of this theory can be found developed across the papers [6, 22, 25, 43, 44]; our objective is to draw the most important of these results together, and to complete them with two new theorems that clarify and simplify both the theory and the practice of awfs.”