Results 11  20
of
54
Functorial calculus in monoidal bicategories
 Applied Categorial Structures
, 2002
"... The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory VMod whose objects are categories enriched in a cocomp ..."
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Cited by 7 (1 self)
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The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory VMod whose objects are categories enriched in a cocomplete symmetric monoidal
Exponentiability in Categories of Lax Algebras
, 2003
"... For a complete cartesianclosed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable BeckChevalleytype condition, it is shown that the category of lax reflexive (T , V)algebras is a quasitopos. This result encompasses many known and new examp ..."
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Cited by 6 (2 self)
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For a complete cartesianclosed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable BeckChevalleytype condition, it is shown that the category of lax reflexive (T , V)algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi. 1.
Rational combinatorics
"... We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials. ..."
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Cited by 6 (5 self)
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We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.
pForm Electromagnetism on Discrete Spacetimes
, 2006
"... We investigate pform electromagnetism—with the Maxwell and KalbRamond fields as lowestorder cases—on discrete spacetimes, including the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitab ..."
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Cited by 6 (2 self)
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We investigate pform electromagnetism—with the Maxwell and KalbRamond fields as lowestorder cases—on discrete spacetimes, including the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitable for our purpose—a chain complex equipped with an inner product on (p + 1)cochains—we study both the classical and quantum versions of the theory, with either R or U(1) as gauge group. We find results—such as a ‘pform Bohm–Aharonov effect’—that depend in interesting ways on the cohomology of spacetime. We quantize the theory via the Euclidean path integral formalism, where the natural kernels in the U(1) theory are not Gaussians but theta functions. As a special case of the general theory, we show pform electromagnetism in p + 1 dimensions has an exact solution which reduces when p = 1 to the abelian case of 2d YangMills theory as studied by Migdal and Witten. Our main result describes pform electromagnetism as a ‘chain field theory’—a theory analogous to a topological quantum field theory, but with chain complexes replacing manifolds. This makes precise a notion of time evolution in the context of discrete spacetimes of arbitrary topology.
Quantum energy inequalities and local covariance II: Categorical formulation
, 2006
"... We formulate Quantum Energy Inequalities (QEIs) in the framework of locally covariant quantum field theory developed by Brunetti, Fredenhagen and Verch, which is based on notions taken from category theory. This leads to a new viewpoint on the QEIs, and also to the identification of a new structural ..."
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Cited by 5 (4 self)
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We formulate Quantum Energy Inequalities (QEIs) in the framework of locally covariant quantum field theory developed by Brunetti, Fredenhagen and Verch, which is based on notions taken from category theory. This leads to a new viewpoint on the QEIs, and also to the identification of a new structural property of locally covariant quantum field theory, which we call Local Physical Equivalence. Covariant formulations of the numerical range and spectrum of locally covariant fields are given and investigated, and a new algebra of fields is identified, in which fields are treated independently of their realisation on particular spacetimes and manifestly covariant versions of the functional calculus may be formulated. 1
Homotopy operations and rational homotopy type”, in Algebraic Topology: Categorical decomposition techniques
 Prog. in Math. 215, Birkhäuser, BostonBasel
"... In [HS] and [F1] Halperin, Stasheff, and Félix showed how an inductivelydefined sequence of elements in the cohomology of a graded commutative algebra over the rationals can be used to distinguish among the homotopy types of all possible realizations, thus providing a collection ..."
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Cited by 4 (4 self)
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In [HS] and [F1] Halperin, Stasheff, and Félix showed how an inductivelydefined sequence of elements in the cohomology of a graded commutative algebra over the rationals can be used to distinguish among the homotopy types of all possible realizations, thus providing a collection
RotaBaxter Categories
"... We introduce RotaBaxter categories and construct examples of such structures. Mathematics Subject Classification (2000)) 2000 05A30, 18A99, 81Q30 Keywords: RotaBaxter algebras, categorification of rings, categorical integration. 1 ..."
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Cited by 4 (4 self)
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We introduce RotaBaxter categories and construct examples of such structures. Mathematics Subject Classification (2000)) 2000 05A30, 18A99, 81Q30 Keywords: RotaBaxter algebras, categorification of rings, categorical integration. 1
Lattice pform electromagnetism and chain field theory
 LOOPS ’05, ALBERT EINSTEIN INSTITUT, MAX PLANCK GESELLSCHAFT, GOLM
, 2005
"... Since Wilson’s work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as pform electromagnetism, including the KalbRamond field in str ..."
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Cited by 4 (2 self)
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Since Wilson’s work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as pform electromagnetism, including the KalbRamond field in string theory, and its nonabelian generalizations. It is desirable to discretize such ‘higher gauge theories ’ in a way analogous to lattice gauge theory, but with the fundamental geometric structures in the discretization boosted in dimension. As a step toward studying discrete versions of more general higher gauge theories, we consider the case of pform electromagnetism. We show that discrete pform electromagnetism admits a simple algebraic description in terms of chain complexes of abelian groups. Moreover, the model allows discrete spacetimes with quite general geometry, in contrast to the regular cubical lattices usually associated with lattice gauge theory. After constructing a suitable model of discrete spacetime for pform electromagnetism, we quantize the theory using the Euclidean path integral formalism. The main result is a description of pform electromagnetism as a ‘chain field theory’ — a theory analogous to topological quantum field theory, but with chain complexes replacing
A general theory of selfsimilarity II: recognition, eprint math.DS/0411345
"... www.maths.gla.ac.uk/∼tl ..."
Super, Quantum and NonCommutative Species
, 2009
"... Dedicated to the memory of GianCarlo Rota. We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and noncommutative combinatorics. Via the usual dual ..."
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Dedicated to the memory of GianCarlo Rota. We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and noncommutative combinatorics. Via the usual duality between algebra and geometry, these constructions provide categorifications for various types of affine spaces, thus our works may be regarded as a starting point towards the construction of a categorical geometry.