6,762 research outputs found
A Convenient Category of Domains
We motivate and define a category of "topological domains",
whose objects are certain topological spaces, generalising
the usual -continuous dcppos of domain theory.
Our category supports all the standard constructions of domain theory,
including the solution of recursive domain equations. It also
supports the construction of free algebras for (in)equational
theories, provides a model of parametric polymorphism,
and can be used as the basis for a theory of computability.
This answers a question of Gordon Plotkin, who asked
whether it was possible to construct a category of domains
combining such properties
Towards a Convenient Category of Topological Domains
We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models
Approximation systems for functions in topological and in metric spaces
A notable feature of the TTE approach to computability is the representation
of the argument values and the corresponding function values by means of
infinitistic names. Two ways to eliminate the using of such names in certain
cases are indicated in the paper. The first one is intended for the case of
topological spaces with selected indexed denumerable bases. Suppose a partial
function is given from one such space into another one whose selected base has
a recursively enumerable index set, and suppose that the intersection of base
open sets in the first space is computable in the sense of Weihrauch-Grubba.
Then the ordinary TTE computability of the function is characterized by the
existence of an appropriate recursively enumerable relation between indices of
base sets containing the argument value and indices of base sets containing the
corresponding function value.This result can be regarded as an improvement of a
result of Korovina and Kudinov. The second way is applicable to metric spaces
with selected indexed denumerable dense subsets. If a partial function is given
from one such space into another one, then, under a semi-computability
assumption concerning these spaces, the ordinary TTE computability of the
function is characterized by the existence of an appropriate recursively
enumerable set of quadruples. Any of them consists of an index of element from
the selected dense subset in the first space, a natural number encoding a
rational bound for the distance between this element and the argument value, an
index of element from the selected dense subset in the second space and a
natural number encoding a rational bound for the distance between this element
and the function value. One of the examples in the paper indicates that the
computability of real functions can be characterized in a simple way by using
the first way of elimination of the infinitistic names.Comment: 21 pages, published in Logical Methods in Computer Scienc
Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads
This is the first of two papers in which we prove that a cell model of the
moduli space of curves with marked points and tangent vectors at the marked
points acts on the Hochschild co--chains of a Frobenius algebra. We also prove
that a there is dg--PROP action of a version of Sullivan Chord diagrams which
acts on the normalized Hochschild co-chains of a Frobenius algebra. These
actions lift to operadic correlation functions on the co--cycles. In
particular, the PROP action gives an action on the homology of a loop space of
a compact simply--connected manifold.
In this first part, we set up the topological operads/PROPs and their cell
models. The main theorems of this part are that there is a cell model operad
for the moduli space of genus curves with punctures and a tangent
vector at each of these punctures and that there exists a CW complex whose
chains are isomorphic to a certain type of Sullivan Chord diagrams and that
they form a PROP. Furthermore there exist weak versions of these structures on
the topological level which all lie inside an all encompassing cyclic
(rational) operad.Comment: 50 pages, 7 figures. Newer version has minor changes. Some material
shifted. Typos and small things correcte
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