335,328 research outputs found
Restriction categories III: colimits, partial limits, and extensivity
A restriction category is an abstract formulation for a category of partial
maps, defined in terms of certain specified idempotents called the restriction
idempotents. All categories of partial maps are restriction categories;
conversely, a restriction category is a category of partial maps if and only if
the restriction idempotents split. Restriction categories facilitate reasoning
about partial maps as they have a purely algebraic formulation.
In this paper we consider colimits and limits in restriction categories. As
the notion of restriction category is not self-dual, we should not expect
colimits and limits in restriction categories to behave in the same manner. The
notion of colimit in the restriction context is quite straightforward, but
limits are more delicate. The suitable notion of limit turns out to be a kind
of lax limit, satisfying certain extra properties.
Of particular interest is the behaviour of the coproduct both by itself and
with respect to partial products. We explore various conditions under which the
coproducts are ``extensive'' in the sense that the total category (of the
related partial map category) becomes an extensive category. When partial
limits are present, they become ordinary limits in the total category. Thus,
when the coproducts are extensive we obtain as the total category a lextensive
category. This provides, in particular, a description of the extensive
completion of a distributive category.Comment: 39 page
Total Maps of Turing Categories
AbstractWe give a complete characterization of those categories which can arise as the subcategory of total maps of a Turing category. A Turing category provides an abstract categorical setting for studying computability: its (partial) maps may be described, equivalently, as the computable maps of a partial combinatory algebra. The characterization, thus, tells one what categories can be the total functions for partial combinatory algebras. It also provides a particularly easy criterion for determining whether functions, belonging to a given complexity class, can be viewed as the class of total computable functions for some abstract notion of computability
Majorisation ordering of measures invariant under transformations of the interval
PhDMajorisation is a partial ordering that can be applied to the set of probability measures
on the unit interval I = [0, 1). Its defining property is that one measure Ī¼
majorises another measure , written Ī¼ , if
R
I fdĪ¼
R
I fd for every convex
real-valued function f : I ! R.
This means that studying the majorisation of MT , the set of measures invariant
under a transformation T : I ! I, can give us insight into finding the maximising
and minimising T-invariant measures for convex and concave f.
In this thesis I look at the majorisation ordering of MT for four categories of
transformations T: concave unimodal maps, the doubling map T : x 7! 2x (mod 1),
the family of shifted doubling maps T : x 7! 2x + (mod 1), and the family of
orientation-reversing weakly-expanding maps
Residually many BV homeomorphisms map a null set onto a set of full measure
Let be the unit square in . We prove that in a
suitable complete metric space of homeomorphisms with
, the generical homeomorphism (in the sense of Baire
categories) maps a null set in a set of full measure and vice versa. Moreover
we observe that, for , in the most reasonable complete metric space
for such problem, the family of homemomorphisms satisfying the above
property is of first category, instead
Formalizing restriction categories
Restriction categories are an abstract axiomatic framework by Cockett and Lack for reasoning about (generalizations of the idea of) partiality of functions. In a restriction category, every map defines an endomap on its domain, the corresponding partial identity map. Restriction categories cover a number of examples of different flavors and are sound and complete with respect to the more synthetic and concrete partial map categories. A partial map category is based on a given category (of total maps) and a map in it is a map from a subobject of the domain. In this paper, we report on an Agda formalization of the first chapters of the theory of restriction categories, including the challenging completeness result. We explain the mathematics formalized, comment on the design decisions we made for the formalization, and illustrate them at work
An Operational Understanding of Bisimulation from Open Maps
AbstractModels can be given to a range of programming languages combining concurrent and functional features in which presheaf categories are used as the semantic domains (instead of the more usual complete partial orders). Once this is done the languages inherit a notion of bisimulation from the āopenā maps associated with the presheaf categories. However, although there are methodological and mathematical arguments for favouring semantics using presheaf categoriesāin particular, there is a ādomain theoryā based on presheaf categories which systematises bisimulation at higher-orderāit is as yet far from a routine matter to read off an āoperational characterisationā; by this I mean an equivalent coinductive definition of bisimulation between terms based on the operational semantics. I hope to illustrate the issues on a little process-passing language. This is joint work with Gian Luca Cattani
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