Results 11  20
of
142
A SURVEY OF (∞, 1)CATEGORIES
, 2006
"... Abstract. In this paper we give a summary of the comparisons between different definitions of socalled (∞,1)categories, which are considered to be models for ∞categories whose nmorphisms are all invertible for n> 1. They are also, from the viewpoint of homotopy theory, models for the homotopy ..."
Abstract

Cited by 16 (3 self)
 Add to MetaCart
Abstract. In this paper we give a summary of the comparisons between different definitions of socalled (∞,1)categories, which are considered to be models for ∞categories whose nmorphisms are all invertible for n> 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasicategories. 1.
Bialgebraic Methods and Modal Logic in Structural Operational Semantics
 Electronic Notes in Theoretical Computer Science
, 2007
"... Bialgebraic semantics, invented a decade ago by Turi and Plotkin, is an approach to formal reasoning about wellbehaved structural operational semantics (SOS). An extension of algebraic and coalgebraic methods, it abstracts from concrete notions of syntax and system behaviour, thus treating various ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
(Show Context)
Bialgebraic semantics, invented a decade ago by Turi and Plotkin, is an approach to formal reasoning about wellbehaved structural operational semantics (SOS). An extension of algebraic and coalgebraic methods, it abstracts from concrete notions of syntax and system behaviour, thus treating various kinds of operational descriptions in a uniform fashion. In this paper, bialgebraic semantics is combined with a coalgebraic approach to modal logic in a novel, general approach to proving the compositionality of process equivalences for languages defined by structural operational semantics. To prove compositionality, one provides a notion of behaviour for logical formulas, and defines an SOSlike specification of modal operators which reflects the original SOS specification of the language. This approach can be used to define SOS congruence formats as well as to prove compositionality for specific languages and equivalences. Key words: structural operational semantics, coalgebra, bialgebra, modal logic, congruence format 1
HOMOTOPY FIBER PRODUCTS OF HOMOTOPY THEORIES
, 2008
"... Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is wellbehaved with respect to translating it into the setting of more gen ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is wellbehaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have welldefined homotopy pullbacks.
The Convergence Approach to Exponentiable Maps
 352 MARIA MANUEL CLEMENTINO, DIRK HOFMANN AND WALTER
, 2000
"... Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
(Show Context)
Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the classical result for compact Hausdorff spaces. Furthermore, in generalization of the WhiteheadMichael characterization of locally compact Hausdorff spaces, we characterize exponentiable maps of Top between Hausdorff spaces as restrictions of perfect maps to open subspaces.
SKEW MONOIDALES, SKEW WARPINGS AND QUANTUM CATEGORIES
"... Abstract. Kornel Szlachányi [28] recently used the term skewmonoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skewmonoidal structures on the category of onesided Rmodules for which the lax un ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
(Show Context)
Abstract. Kornel Szlachányi [28] recently used the term skewmonoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skewmonoidal structures on the category of onesided Rmodules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudomonoids) in any monoidal bicategory M. These are skewmonoidal categories when M is Cat. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories [10] with base comonoid C in a suitably complete braided monoidal category V are precisely skew monoidales in Comod(V) with unit coming from the counit of C. Quantum groupoids (in the sense of [6] rather than [10]) are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping de ned in [3] to modify monoidal structures. 1.
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 ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 11 (7 self)
 Add to MetaCart
(Show Context)
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
Tannaka duality and convolution for duoidal categories
, 2013
"... Given a horizontal monoid M in a duoidal category F, we examine the relationship between bimonoid structures on M and monoidal structures on the category F ∗M of right Mmodules which lift the vertical monoidal structure of F. We obtain our result using a variant of the socalled Tannaka adjunction; ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
Given a horizontal monoid M in a duoidal category F, we examine the relationship between bimonoid structures on M and monoidal structures on the category F ∗M of right Mmodules which lift the vertical monoidal structure of F. We obtain our result using a variant of the socalled Tannaka adjunction; that is, an adjunction inducing the equivalence which expresses Tannaka duality. The approach taken utilizes homenriched categories rather than categories on which a monoidal category acts (“actegories”). The requirement of enrichment in F itself demands the existence of some internal homs, leading to the consideration of convolution for duoidal categories. Proving that certain homfunctors are monoidal, and so take monoids to monoids, unifies classical convolution in algebra and Day convolution for categories. Hopf bimonoids are defined leading to a lifting of closed structures on F to F ∗M. We introduce the concept of warping monoidal structures and this permits the construction of new duoidal categories.
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 ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
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.
LOCALLY COMPACT OBJECTS IN EXACT CATEGORIES
, 710
"... Abstract. We identify two categories of locally compact objects on an exact category. They correspond to the wellknown constructions of Beilinson’s ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We identify two categories of locally compact objects on an exact category. They correspond to the wellknown constructions of Beilinson’s