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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 e ..."
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Cited by 7 (5 self)
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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 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. Contents
On the construction of functorial factorizations for model categories
, 2012
"... Abstract. We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use “algebraic ” characterizations of fibrations to produce factorizations that have the desired lifting properties in a com ..."
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Cited by 3 (2 self)
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Abstract. We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use “algebraic ” characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored, and cotensored in spaces, proving the existence of Hurewicztype model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) Gspaces, and diagram spectra among others. 1.
THE COALGEBRAIC STRUCTURE OF CELL COMPLEXES
"... Abstract. The relative cell complexes with respect to a generating set of cofibrations are an important class of morphisms in any model structure. In the particular case of the standard (algebraic) model structure on Top, we give a new expression of these morphisms by defining a category of relative ..."
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Abstract. The relative cell complexes with respect to a generating set of cofibrations are an important class of morphisms in any model structure. In the particular case of the standard (algebraic) model structure on Top, we give a new expression of these morphisms by defining a category of relative cell complexes, which has a forgetful functor to the arrow category. This allows us to prove a conjecture of Richard Garner: considering the algebraic weak factorisation system given in that algebraic model structure between
Multitensor lifting and strictly unital higher category theory
"... Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories ..."
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Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories and the Crans tensor product of Gray categories as part of this framework. We define weak ncategories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak ncategories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)categories with strict units. 1.
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"... All Rights ReservedTo L, who let me go.Table of Contents Acknowledgments....................................... Abstract............................................ Preface............................................. vi ..."
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All Rights ReservedTo L, who let me go.Table of Contents Acknowledgments....................................... Abstract............................................ Preface............................................. vi