888 research outputs found
Thin Games with Symmetry and Concurrent Hyland-Ong Games
We build a cartesian closed category, called Cho, based on event structures.
It allows an interpretation of higher-order stateful concurrent programs that
is refined and precise: on the one hand it is conservative with respect to
standard Hyland-Ong games when interpreting purely functional programs as
innocent strategies, while on the other hand it is much more expressive. The
interpretation of programs constructs compositionally a representation of their
execution that exhibits causal dependencies and remembers the points of
non-deterministic branching.The construction is in two stages. First, we build
a compact closed category Tcg. It is a variant of Rideau and Winskel's category
CG, with the difference that games and strategies in Tcg are equipped with
symmetry to express that certain events are essentially the same. This is
analogous to the underlying category of AJM games enriching simple games with
an equivalence relations on plays. Building on this category, we construct the
cartesian closed category Cho as having as objects the standard arenas of
Hyland-Ong games, with strategies, represented by certain events structures,
playing on games with symmetry obtained as expanded forms of these arenas.To
illustrate and give an operational light on these constructions, we interpret
(a close variant of) Idealized Parallel Algol in Cho
On the dihedral Euler characteristics of Selmer groups of abelian varieties
This note shows how to use the framework of Euler characteristic formulae to
study Selmer groups of abelian varieties in certain dihedral or anticyclotomic
extensions of CM fields via Iwasawa main conjectures, and in particular how to
verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this
setting. When the Selmer group is cotorsion with respect to the associated
Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch
and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with
respect to the associated Iwasawa algebra, we give a conjectural description of
the Euler characteristic of the cotorsion submodule, and explain how to deduce
inequalities from the associated main conjecture divisibilities of Perrin-Riou
and Howard.Comment: 26 pages. Previous discussion of two-variable setting removed, and
discussion of the indefinite setting modified accordingly. To appear in the
HIM "Arithmetic and Geometry" conference proceeding
Deformation of the Hopf algebra of plane posets
We describe and study a four parameters deformation of the two products and
the coproduct of the Hopf algebra of plane posets. We obtain a family of
braided Hopf algebras, generally self-dual. We also prove that in a particular
case (when the second parameter goes to zero and the first and third parameters
are equal), this deformation is isomorphic, as a self-dual braided Hopf
algebra, to a deformation of the Hopf algebra of free quasi-symmetric
functions.Comment: 28 pages. Second versio
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
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