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129
Functional Programming with Overloading and HigherOrder Polymorphism
, 1995
"... The Hindley/Milner type system has been widely adopted as a basis for statically typed functional languages. One of the main reasons for this is that it provides an elegant compromise between flexibility, allowing a single value to be used in different ways, and practicality, freeing the progr ..."
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Cited by 69 (3 self)
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The Hindley/Milner type system has been widely adopted as a basis for statically typed functional languages. One of the main reasons for this is that it provides an elegant compromise between flexibility, allowing a single value to be used in different ways, and practicality, freeing the programmer from the need to supply explicit type information. Focusing on practical applications rather than implementation or theoretical details, these notes examine a range of extensions that provide more flexible type systems while retaining many of the properties that have made the original Hindley/Milner system so popular. The topics discussed, some old, but most quite recent, include higherorder polymorphism and type and constructor class overloading. Particular emphasis is placed on the use of these features to promote modularity and reusability.
Higher rank graph C*algebras
, 2000
"... Building on recent work of Robertson and Steger, we associate a C ∗ –algebra to a combinatorial object which may be thought of as a higher rank graph. This C ∗ –algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the ..."
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Cited by 58 (11 self)
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Building on recent work of Robertson and Steger, we associate a C ∗ –algebra to a combinatorial object which may be thought of as a higher rank graph. This C ∗ –algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the higher rank graph for the associated C ∗ –algebra to be: simple, purely infinite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from “commuting” rank 1 graphs is given.
Spaces over a Category and Assembly Maps in Isomorphism Conjectures in Kand LTheory
"... : We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory of reduced group C algebras. The approach is through spectra over the orbit category of a discrete ..."
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Cited by 49 (12 self)
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: We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory of reduced group C algebras. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and describe such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and the study of the associated generalized homology and cohomology theories and homotopy limits. Key words: Algebraic K and Ltheory, BaumConnes Conjecture, assembly maps, spaces and spectra over a category AMSclassification number: 57 Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G=H, considered as left Gsets, and morphisms are Gmaps. This is a useful construct for o...
The Alexander duality functors and local duality with monomial support
 J. Algebra
"... Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nngraded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as co ..."
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Cited by 32 (13 self)
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Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nngraded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution of its Alexander dual, and contains all of the maps in the minimal free resolution of M over every Zngraded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcmlattices. Using injective resolutions, theorems of EagonReiner and Terai are generalized to all Nngraded modules: the projective dimension of M equals the supportregularity of its Alexander dual, and M is CohenMacaulay if and only if its Alexander dual has a supportlinear free resolution. Alexander duality is applied in the context of the Zngraded local cohomology functors Hi I (−) for squarefree monomial ideals I in the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I = m is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, that Terai’s formula for the Hilbert series of H i I (S) is equivalent to Hochster’s for Hn−i m (S/I). 1
Twovector bundles and forms of elliptic cohomology
 in Topology, Geometry and Quantum Field Theory, LMS Lecture note series 308
, 2004
"... The work to be presented in this paper has been inspired by several of Professor Graeme Segal’s papers. Our search for a geometrically defined elliptic cohomology theory with associated elliptic objects obviously stems from his Bourbaki seminar [Se88]. Our readiness to form group completions of symm ..."
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Cited by 32 (4 self)
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The work to be presented in this paper has been inspired by several of Professor Graeme Segal’s papers. Our search for a geometrically defined elliptic cohomology theory with associated elliptic objects obviously stems from his Bourbaki seminar [Se88]. Our readiness to form group completions of symmetric monoidal categories
A Nonconnecting Delooping of Algebraic KTheory
"... Given a ring R, it is known that the topological space BGl(R) + is an infinite loop space. One way to construct an infinite loop structure is to consider the category F of free Rmodules, or rather its classifying space BF, as food for suitable infinite loop space machines. These machines produce co ..."
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Cited by 30 (3 self)
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Given a ring R, it is known that the topological space BGl(R) + is an infinite loop space. One way to construct an infinite loop structure is to consider the category F of free Rmodules, or rather its classifying space BF, as food for suitable infinite loop space machines. These machines produce connective spectra whose zeroth space is (BF Z × BGl(R) +. In this paper we consider categories C (F) = F, C (F),... of parameterized =1 = = =1 = free modules and bounded homomorphisms and show that the spaces (BC) =0 + = (BF)
On Inner Product In Modular Tensor Categories. II Inner Product On Conformal Blocks.
 I & II, math.QA/9508017 and qalg/9611008
, 1995
"... this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras or, equivalently, from WessZuminoWitten model of conformal field theory. We briefly recall construction of this category, first suggested by Moore and Seiberg (see [MS1,2]) and la ..."
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Cited by 28 (0 self)
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this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras or, equivalently, from WessZuminoWitten model of conformal field theory. We briefly recall construction of this category, first suggested by Moore and Seiberg (see [MS1,2]) and later refined by Kazhdan and Lusztig ([KL14]) and Finkelberg ([F]) in Section 9. In particular, spaces of homomorphisms in this category are the spaces of conformal blocks of WZW model. Thus, the general theory developed in Section 2 of [K] gives us an inner product on the space of conformal blocks, and so defined inner product is modular invariant. This definition is constructive: we show how it can be rewritten so that it only involves Drinfeld associator, or, equivalently, asymptotics of solutions of KnizhnikZamolodchikov equations. Since there are integral formulas for the solutions of KZ equations, this shows that the inner product on the space of conformal blocks can be written explicitly in terms of certain integrals. In the case g = sl 2 these integrals can be calculated (see [V]), using Selberg integral, and the answer is written in terms of \Gammafunctions. Thus, in this case we can write explicit formulas for inner product on the space of conformal blocks. These expressions are 1991 Mathematics Subject Classification. Primary 81R50, 05E35, 18D10; Secondary 57M99
Finite tensor categories
 Moscow Math. Journal
"... These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We wil ..."
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Cited by 26 (8 self)
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These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We will also assume the basics of the theory of abelian categories (for a more detailed treatment see the book [F]). If C is a category, the notation X ∈ C will mean that X is an object of C, and the set of morphisms between X, Y ∈ C will be denoted by Hom(X, Y). Throughout the notes, for simplicity we will assume that the ground field k is algebraically closed unless otherwise specified, even though in many cases this assumption will not be needed. 1. Monoidal categories 1.1. The definition of a monoidal category. A good way of thinking
Spherical categories
 Adv. Math
, 1999
"... Abstract. This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a ..."
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Cited by 24 (2 self)
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Abstract. This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following [MacLane 1963]. In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical. In the
Hilbert modules and modules over finite von Neumann algebras and applications to L²invariants
 MATH. ANN. 309, 247285 (1997)
, 1997
"... ..."