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15
Quotients of the multiplihedron as categorified associahedra
 Homotopy, Homology and Appl
, 2008
"... Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associah ..."
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Cited by 7 (2 self)
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Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We also present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of
Fixed point theory and trace for bicategories
, 2007
"... The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point inde ..."
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Cited by 4 (1 self)
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The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is the identification of these different invariants. We define a generalization of the trace in symmetric monoidal categories to a trace in bicategories with shadows. We show the invariants used in the converse of the Lefschetz fixed point theorem are examples of this trace and that the functoriality of the trace provides some of the necessary identifications. The methods used here do not use simplicial techniques and so generalize readily to other contexts. iii Contents
Parametrized spaces model locally constant homotopy sheaves
 Topology Appl
, 2008
"... Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification ..."
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Cited by 4 (0 self)
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Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopytheoretic version of the correspondence between covering spaces and π1sets. We then use these two equivalences to study base change functors for parametrized spaces. Contents
Twisted differential nonabelian cohomology Twisted (n−1)brane nbundles and their ChernSimons (n+1)bundles with characteristic (n + 2)classes
, 2008
"... We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shif ..."
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Cited by 3 (3 self)
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We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian ngroup B n−1 U(1). Notable examples are String 2bundles [9] and Fivebrane 6bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spinstructures to Stringstructures [13] and further to Fivebranestructures [133, 52], are abelian ChernSimons 3 and 7bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞Lieintegrating the L∞algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2 and twisted Fivebrane 6bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted Ktheory. We explain the GreenSchwarz mechanism in heterotic string theory in terms of twisted String 2bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6bundles. We close by transgressing differential cocycles to mapping
DIAGRAM SPACES, DIAGRAM SPECTRA, AND SPECTRA OF UNITS
, 908
"... Abstract. We compare the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM Smodules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω ∞. We prove that these models for sp ..."
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Cited by 2 (0 self)
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Abstract. We compare the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM Smodules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω ∞. We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors Ω ∞ agree. This comparison is then used to show that two different constructions of the spectrum of units gl1R of a structured ring spectrum R agree. Contents
PARAMETRIZED SPACES ARE LOCALLY CONSTANT HOMOTOPY SHEAVES
, 706
"... Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification ..."
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Cited by 1 (1 self)
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Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopytheoretic version of the correspondence between covering spaces and π1sets. We then use these two equivalences to study base change functors for parametrized spaces. Contents
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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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
Elmendorf’s Theorem for Cofibrantly Generated Model Categories
, 2010
"... Elmendorf’s Theorem in equivariant homotopy theory states that ..."
Nonabelian homotopical cohomology,
"... higher fiber bundles with connection, and their σmodel QFTs ..."