19,202 research outputs found
Second-Order Type Isomorphisms Through Game Semantics
The characterization of second-order type isomorphisms is a purely
syntactical problem that we propose to study under the enlightenment of game
semantics. We study this question in the case of second-order
λ-calculus, which can be seen as an extension of system F to
classical logic, and for which we define a categorical framework: control
hyperdoctrines. Our game model of λ-calculus is based on polymorphic
arenas (closely related to Hughes' hyperforests) which evolve during the play
(following the ideas of Murawski-Ong). We show that type isomorphisms coincide
with the "equality" on arenas associated with types. Finally we deduce the
equational characterization of type isomorphisms from this equality. We also
recover from the same model Roberto Di Cosmo's characterization of type
isomorphisms for system F. This approach leads to a geometrical comprehension
on the question of second order type isomorphisms, which can be easily extended
to some other polymorphic calculi including additional programming features.Comment: accepted by Annals of Pure and Applied Logic, Special Issue on Game
Semantic
An isomorphism of unitals, and an isomorphism of classical groups
An isomorphism between two hermitian unitals is proved, and used to treat
isomorphisms of classical groups that are related to the isomorphism between
certain simple real Lie algebras of types A and D (and rank 3)
Chern-Dirac bundles on non-K\"ahler Hermitian manifolds
We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on
Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac
operators, with Levi-Civita connection replaced by Chern connection. We then
show that the tensor product of canonical and the anticanonical spinor bundles,
called V-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of
harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar
construction establishes isomorphisms between other types of harmonic spinors
and Bott-Chern, Aeppli and twisted cohomology.Comment: 23 pages; some minor changes, accepted for publication in Rocky
Mountain J. Mat
Types and forgetfulness in categorical linguistics and quantum mechanics
The role of types in categorical models of meaning is investigated. A general
scheme for how typed models of meaning may be used to compare sentences,
regardless of their grammatical structure is described, and a toy example is
used as an illustration. Taking as a starting point the question of whether the
evaluation of such a type system 'loses information', we consider the
parametrized typing associated with connectives from this viewpoint.
The answer to this question implies that, within full categorical models of
meaning, the objects associated with types must exhibit a simple but subtle
categorical property known as self-similarity. We investigate the category
theory behind this, with explicit reference to typed systems, and their
monoidal closed structure. We then demonstrate close connections between such
self-similar structures and dagger Frobenius algebras. In particular, we
demonstrate that the categorical structures implied by the polymorphically
typed connectives give rise to a (lax unitless) form of the special forms of
Frobenius algebras known as classical structures, used heavily in abstract
categorical approaches to quantum mechanics.Comment: 37 pages, 4 figure
Etale homotopy types of moduli stacks of algebraic curves with symmetries
Using the machinery of etale homotopy theory a' la Artin-Mazur we determine
the etale homotopy types of moduli stacks over \bar{\Q} parametrizing
families of algebraic curves of genus g greater than 1 endowed with an action
of a finite group G of automorphisms, which comes with a fixed embedding in the
mapping class group, such that in the associated complex analytic situation the
action of G is precisely the differentiable action induced by this specified
embedding of G in the mapping class group.Comment: 27 page
An Intuitionistic Formula Hierarchy Based on High-School Identities
We revisit the notion of intuitionistic equivalence and formal proof
representations by adopting the view of formulas as exponential polynomials.
After observing that most of the invertible proof rules of intuitionistic
(minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms
corresponding to the high-school identities, we show that one can obtain a more
compact variant of a proof system, consisting of non-invertible proof rules
only, and where the invertible proof rules have been replaced by a formula
normalisation procedure.
Moreover, for certain proof systems such as the G4ip sequent calculus of
Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the
non-invertible proof rules as strict inequalities between exponential
polynomials; a careful combinatorial treatment is given in order to establish
this fact.
Finally, we extend the exponential polynomial analogy to the first-order
quantifiers, showing that it gives rise to an intuitionistic hierarchy of
formulas, resembling the classical arithmetical hierarchy, and the first one
that classifies formulas while preserving isomorphism
Game semantics for first-order logic
We refine HO/N game semantics with an additional notion of pointer
(mu-pointers) and extend it to first-order classical logic with completeness
results. We use a Church style extension of Parigot's lambda-mu-calculus to
represent proofs of first-order classical logic. We present some relations with
Krivine's classical realizability and applications to type isomorphisms
Affine Hecke algebras of type D and generalisations of quiver Hecke algebras
We define and study cyclotomic quotients of affine Hecke algebras of type D.
We establish an isomorphism between (direct sums of blocks of) these cyclotomic
quotients and a generalisation of cyclotomic quiver Hecke algebras which are a
family of Z-graded algebras closely related to algebras introduced by Shan,
Varagnolo and Vasserot. To achieve this, we first complete the study of
cyclotomic quotients of affine Hecke algebras of type B by considering the
situation when a deformation parameter p squares to 1. We then relate the two
generalisations of quiver Hecke algebras showing that the one for type D can be
seen as fixed point subalgebras of their analogues for type B, and we carefully
study how far this relation remains valid for cyclotomic quotients. This allows
us to obtain the desired isomorphism. This isomorphism completes the family of
isomorphisms relating affine Hecke algebras of classical types to
(generalisations of) quiver Hecke algebras, originating in the famous result of
Brundan and Kleshchev for the type A.Comment: 26 page
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