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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 17 (9 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 8 (4 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.
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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Cited by 2 (1 self)
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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.
ωOperads of Coendomorphisms for Higher Structures
 http://arxiv.org/pdf/1211.2310v1.pdf ,(2012
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MAPPING SPACES OF GrayCATEGORIES
"... Abstract. We define a mapping space for Grayenriched categories adapted to higher gauge theory. Our construction differs significantly from the canonical mapping space of enriched categories in that it is much less rigid. The two essential ingredients are a path space construction for Graycategori ..."
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Cited by 1 (0 self)
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Abstract. We define a mapping space for Grayenriched categories adapted to higher gauge theory. Our construction differs significantly from the canonical mapping space of enriched categories in that it is much less rigid. The two essential ingredients are a path space construction for Graycategories and a kind of comonadic resolution of the 1dimensional structure of a given Graycategory obtained by lifting the resolution of ordinary categories along the canonical fibration of GrayCat over Cat.
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
ITERATED ICONS
"... We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performi ..."
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We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of “2tuply monoidal categories ” missing from our earlier studies of the Periodic Table. We propose a generalisation to all ktuply monoidal ncategories.
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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
OPERADIC DEFINITIONS OF WEAK NCATEGORY: COHERENCE AND COMPARISONS
"... Abstract. This paper concerns the relationships between notions of weak ncategory defined as algebras for nglobular operads, as well as their coherence properties. We focus primarily on the definitions due to Batanin and Leinster. A correspondence between the contractions and systems of compositio ..."
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Abstract. This paper concerns the relationships between notions of weak ncategory defined as algebras for nglobular operads, as well as their coherence properties. We focus primarily on the definitions due to Batanin and Leinster. A correspondence between the contractions and systems of compositions used in Batanin’s definition, and the unbiased contractions used in Leinster’s definition, has long been suspected, and we prove a conjecture of Leinster that shows that the two notions are in some sense equivalent. We then prove several coherence theorems which apply to algebras for any operad with a contraction and system of compositions or with an unbiased contraction; these coherence theorems thus apply to weak ncategories in the senses of Batanin, Leinster, Penon and Trimble. We then take some steps towards a comparison between Batanin weak ncategories and Leinster weak ncategories. We describe a canonical adjunction between the categories of these, giving a construction of the left adjoint, which is applicable in more generality to a class of functors induced by monad morphisms. We conclude with some preliminary statements about a possible weak equivalence of some sort between these categories. 1.