Results 1  10
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44
Algebras and Modules in Monoidal Model Categories
 Proc. London Math. Soc
, 1998
"... In recent years the theory of structured ring spectra (formerly known as A #  and E # ring spectra) has been signicantly simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be dened as a monoid with respect t ..."
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Cited by 145 (27 self)
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In recent years the theory of structured ring spectra (formerly known as A #  and E # ring spectra) has been signicantly simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be dened as a monoid with respect to the smash product in one of these new categories of spectra. In order to make use of all of the standard tools from homotopy theory, it is important to have a Quillen model category structure [##] available here. In this paper we provide a general method for lifting model structures to categories of rings, algebras, and modules. This includes, but is not limited to, each of the new theories of ring spectra. One model for structured ring spectra is given by the Salgebras of [##]. This example has the special feature that every object is brant, which makes it easier to fo...
StateSum Invariants of 4Manifolds
 J. Knot Theory Ram
, 1997
"... Abstract: We provide, with proofs, a complete description of the authors ’ construction of statesum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery ..."
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Cited by 30 (6 self)
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Abstract: We provide, with proofs, a complete description of the authors ’ construction of statesum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery invariants [Br1,Br2] using techniques developed in the case of the semisimple subquotient of Rep(Uq(sl2)) (q a principal 4r th root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4manifolds equipped with 2dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. 1 1
On ZeroCurvature Representations of Partial Differential Equations
, 1993
"... . This paper concerns the problem of finding all possible zerocurvature representations of a given system of partial differential equations. A new computational procedure is proposed, designed to give complete answers. The procedure is applicable to nonoverdetermined systems when the number m of i ..."
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Cited by 19 (8 self)
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. This paper concerns the problem of finding all possible zerocurvature representations of a given system of partial differential equations. A new computational procedure is proposed, designed to give complete answers. The procedure is applicable to nonoverdetermined systems when the number m of independent variables is two, while for m ? 3 we prove a theorem of nonexistence. The approach, inspired by A.M. Vinogradov's Cspectral sequence, is based on finding representatives of gauge equivalence classes. Keywords. Nonlinear partial differential equation, zerocurvature representation, linear pseudopotential, linear covering, AKNS formulation, diffiety, Cspectral sequence, gauge complex, gauge equivalence, jet comonad. MS classification numbers. 35A30, 58G05; 35G20, 18C15. Introduction Consider a system of partial differential equations f l (x i ; u k ; u k I ) = 0; (0.1) where x i , i = 1; : : : ; m are independent variables, u k 's are unknown functions, and u k I...
Calculus III: Taylor Series
, 2003
"... We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal nexcisive approximation, which may be thought of as its nexcisive part. Homogeneous functors, meaning nexcisive functors with trivial (n − 1)excisive part, ..."
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Cited by 18 (0 self)
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We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal nexcisive approximation, which may be thought of as its nexcisive part. Homogeneous functors, meaning nexcisive functors with trivial (n − 1)excisive part, can be classified: they correspond to symmetric functors of n variables that are reduced and 1excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen’s algebraic Ktheory.
New Model Categories From Old
 J. Pure Appl. Algebra
, 1995
"... . We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categor ..."
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Cited by 13 (5 self)
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. We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas  most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in nonabelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such nonabelian derived functors is the E 2 term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E 2 term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra (see x7.4). The original purpose of this note w...
Normal Forms and CutFree Proofs as Natural Transformations
 in : Logic From Computer Science, Mathematical Science Research Institute Publications 21
, 1992
"... What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax g ..."
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Cited by 12 (4 self)
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What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cutelimination and asymmetrical interpretations of cutfree proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity ([27], [2]), and to the KellyLambekMac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interconnections of proof theory, typed lambda calculus (as a functional programming paradigm) and category theory. Some of these connections can be surprisingly subtle. Here we a...
Grassmann variables on quantum spaces
, 2005
"... In this article we consider quantum spaces which could be of particular importance in physics, i.e. the 2dimensional quantum plane, the qdeformed Euclidean space with 3 or 4 dimensions as well as the qdeformed Minkowski space. For each of these spaces we present some standard techniques for deali ..."
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Cited by 10 (10 self)
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In this article we consider quantum spaces which could be of particular importance in physics, i.e. the 2dimensional quantum plane, the qdeformed Euclidean space with 3 or 4 dimensions as well as the qdeformed Minkowski space. For each of these spaces we present some standard techniques for dealing with qdeformed Grassmann variables. Especially, we give formulae for multiplying two supernumbers and show how symmetry generators and fermionic derivatives act on antisymmetrized quantum spaces. Furthermore, we review for all types of quantum spaces their Hopf structures. From the corresponding formulae for the coproduct we are then able to read off a realization of the Lmatrices in terms of the symmetry generators. This means that the commutation relations between all types of quantum spaces are calculable as soon as the actions of the symmetry generators are known.
Notes on A∞algebras, A∞categories and noncommutative geometry, Homological mirror symmetry
 Lecture Notes in Phys
, 2009
"... 1.1 A∞algebras as spaces........................ 2 ..."
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Cited by 9 (0 self)
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1.1 A∞algebras as spaces........................ 2
CW simplicial resolutions of spaces, with an application
"... Abstract. We show how a certain type of CW simplicial resolutions of spaces by wedges of spheres may be constructed, and how such resolutions yield an obstruction theory for a given space to be a loop space. 1. ..."
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Cited by 7 (5 self)
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Abstract. We show how a certain type of CW simplicial resolutions of spaces by wedges of spheres may be constructed, and how such resolutions yield an obstruction theory for a given space to be a loop space. 1.