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12
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.
Multivariable adjunctions and mates
, 2012
"... We present the notion of “cyclic double multicategory”, as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this situation to n + 1 functors of n variables. Furthermore, we ..."
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Cited by 1 (1 self)
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We present the notion of “cyclic double multicategory”, as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this situation to n + 1 functors of n variables. Furthermore, we generalise the mates correspondence, which enables us to pass between natural transformations involving left adjoints to those involving right adjoints. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of “double multicategories”. Moreover, we show that the analogous isomorphisms of double multicategories give a cyclic action on the multimaps, yielding the notion of “cyclic double multicategory”. The workis motivated byand appliedtoRiehl’s approach
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
A folk model structure on omegacat
, 2009
"... The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generat ..."
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The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while free objects are cofibrant. We further exhibit model structures of this type on ncategories for arbitrary n ∈ N, as specialisations of the ωcategorical one along right adjoints. In particular, known cases for n = 1 and n = 2 nicely fit into the scheme.
Contents
"... Abstract. The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoretic arguments, we present streamlined proofs of a number of useful technical results, which are well known to ..."
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Abstract. The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoretic arguments, we present streamlined proofs of a number of useful technical results, which are well known to folklore but difficult to find in the literature. While the results presented here are not new, our approach to their proofs is somewhat novel. Specifically, we reduce the much of the hard work involved to simpler computations involving weighted colimits and Leibniz (pushoutproduct) constructions. The general theory is developed in parallel with examples, which we use to prove that familiar formulae for homotopy limits and colimits indeed have the desired properties.
Preface Prerequisites Notational Conventions Acknowledgments
"... Categorical homotopy theory Emily RiehlTo my students, colleagues, friends who inspired this work.... what we are doing is finding ways for people to understand and think about mathematics. ..."
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Categorical homotopy theory Emily RiehlTo my students, colleagues, friends who inspired this work.... what we are doing is finding ways for people to understand and think about mathematics.
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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
Comparing
"... composites of left and right derived functors Michael Shulman Abstract. We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double ..."
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composites of left and right derived functors Michael Shulman Abstract. We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and right Quillen functors, respectively, and that passage to derived functors is functorial at the level of this double category. The theory of conjunctions and mates in double categories, which generalizes the theory of adjunctions and mates in 2categories, then gives us canonical ways to compare composites of left and right derived functors. We give a number of sample applications, most of which are improvements
Comparing
"... composites of left and right derived functors Michael Shulman Abstract. We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double ..."
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composites of left and right derived functors Michael Shulman Abstract. We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and right Quillen functors, respectively, and that passage to derived functors is functorial at the level of this double category. The theory of conjunctions and mates in double categories, which generalizes the theory of adjunctions and mates in 2categories, then gives us canonical ways to compare composites of left and right derived functors. We give a number of sample applications, most of which are improvements