Quasi-uniform and syntopogenous structures on categories

Philosophiae Doctor - PhDIn a category C with a proper (E; M)-factorization system for morphisms, we further investigate categorical topogenous structures and demonstrate their prominent role played in providing a uni ed approach to the theory of closure, interior and neighbourhood operators. We then introduce and study an abstract notion of C asz ar's syntopogenous structure which provides a convenient setting to investigate a quasi-uniformity on a category. We demonstrate that a quasi-uniformity is a family of categorical closure operators. In particular, it is shown that every idempotent closure operator is a base for a quasi-uniformity. This leads us to prove that for any idempotent closure operator c (interior i) on C there is at least a transitive quasi-uniformity U on C compatible with c (i). Various notions of completeness of objects and precompactness with respect to the quasi-uniformity de ned in a natural way are studied. The great relationship between quasi-uniformities and closure operators in a category inspires the investigation of categorical quasi-uniform structures induced by functors. We introduce the continuity of a C-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) and utilize it to investigate the quasiuniformities induced by pointed and copointed endofunctors. Amongst other things, it is shown that every quasi-uniformity on a re ective subcategory of C can be lifted to a coarsest quasi-uniformity on C for which every re ection morphism is continuous. The notion of continuity of functors between categories endowed with xed quasi-uniform structures is also introduced and used to describe the quasi-uniform structures induced by an M- bration and a functor having a right adjoint

Topogenous structures and related families of morphisms

In a category $\mathcal{C}$ with a proper $(\mathcal{E}, \mathcal{M})$-factorization system, we study the notions of strict, co-strict, initial and final morphisms with respect to a topogenous order. Besides showing that they allow simultaneous study of four classes of morphisms obtained separately with respect to closure, interior and neighbourhood operators, the initial and final morphisms lead us to the study of topogenous structures induced by pointed and co-pointed endofunctors. We also lift the topogenous structures along an $\mathcal{M}$-fibration. This permits one to obtain the lifting of interior and neighbourhood operators along an $\mathcal{M}$-fibration and includes the lifting of closure operators found in the literature. A number of examples presented at the end of the paper demonstrates our results.Comment: arXiv admin note: text overlap with arXiv:2302.0275

Topogenous structures on categories

Magister Scientiae - MScAlthough the interior operators correspond to a special class of neighbourhood operators, the closure operators are not nicely related to the latter. We introduce and study the notion of topogenous orders on a category which provides a basis for categorical study of topology. We show that they are equivalent to the categorical neighbourhood operators and house the closure and interior operators. The natural notion of strict morphism with respect to a topogenous order is shown to capture the known ones in the settings of closure, interior and neighbourhood operators

Wetland resource evaluation and the NRA's role in its conservation. Classification of British wetlands

This is the Wetland resource evaluation and the NRA's role in its conservation: Classification of British wetlands report produced by the National Rivers Authority in 1995. This R&D document provides a clear classification for wetlands in England and Wales. The classification incorporates many of the existing ideas on the subject but avoids some of the problems associated with other classifications. A two-layered 'hydrotopographical' classification is proposed. The first layer identifies situation-types, i.e. the position the wetland occupies in the landscape, with special emphasis upon the principal sources of water. The second layer identifies hydrotopographical elements, i.e. units with distinctive water supply and, sometimes, distinctive topography in response to this. This system is seen as an independent, basic, classification upon which it is possible to superimpose additional, independent classifications based on other features (e.g. base-status, fertility, vegetation, management etc.). Some proposals for such additional classifications are provided

Wetland resource evaluation and the NRA's role in its conservation. 1. Resource assessment

This is the Wetland resource evaluation and the NRA's role in its conservation: Resource assessment report produced by the National Rivers Authority in 1995. This R&D document provides a strategy for the assessment of the wetland resource of England and Wales. As a first step the report defines wetlands in their UK context. The following working definition is suggested: Wetland is land that has (or had until modified) a water level predominantly at, near, or up to 1.5 m above the ground surface for sufficient time during the year to allow hydrological processes to be a major influence on the soils and biota. These processes may be expressed in certain features, such as characteristic soils and vegetation. The report also summarises a hydrotopographical classification of wetlands. The report then develops a strategy for the establishment of a wetland resource Inventory based on a geographical information system (GIS) as a means of storing and manipulating site data from across England and Wales

Quasi-uniform structures determined by closure operators

We demonstrate a one-to-one correspondence between idempotent closure operators and the so-called saturated quasi-uniform structures on a category C. Not only this result allows to obtain a categorical counterpart P of the Császár-Pervin quasi-uniformity P, that we characterize as a transitive quasi-uniformity compatible with an idempotent interior operator, but also permits to describe those topogenous orders that are induced by a transitive quasi-uniformity on C. The categorical counterpart P⁎ of P−1 is characterized as a transitive quasi-uniformity compatible with an idempotent closure operator. When applied to other categories outside topology P allows, among other things, to generate a family of idempotent closure operators on Grp, the category of groups and group homomorphisms, determined by the normal closure

Interior operators and their applications

Philosophiae Doctor - PhDCategorical closure operators were introduced by Dikranjan and Giuli in [DG87] and then developed by these authors and Tholen in [DGT89]. These operators have played an important role in the development of Categorical Topology by introducing topological concepts, such as connectedness, separatedness and compactness, in an arbitrary category and they provide a uni ed approach to various mathematical notions. Motivated by the theory of these operators, the categorical notion of interior operators was introduced by Vorster in [Vor00]. While there is a notational symmetry between categorical closure and interior operators, a detailed analysis shows that the two operators are not categorically dual to each other, that is: it is not true in general that whatever one does with respect to closure operators may be done relative to interior operators. Indeed, the continuity condition of categorical closure operators can be expressed in terms of images or equivalently, preimages, in the same way as the usual topological closure describes continuity in terms of images or preimages along continuous maps. However, unlike the case of categorical closure operators, the continuity condition of categorical interior operators can not be described in terms of images. Consequently, the general theory of categorical interior operators is not equivalent to the one of closure operators. Moreover, the categorical dual closure operator introduced in [DT15] does not lead to interior operators. As a consequence, the study of categorical interior operators in their own right is interesting

Integration of the kenzo system within sagemath for new algebraic topology computations

This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before

Phytosociological studies on rich fen systems in England and Wales

The thesis presents an attempt to detect, define and characterise, on the basis of their floristic composition, the principal types of rich fen vegetation in lowland England and Wales. Some 1,000 samples were taken from a wide variety of rich fen systems throughout England and Wales and from one area in S. Scotland, by a subjective procedure based upon stand selection. A computer-based system was developed to handle these data combining both numerical and traditional (Zurich-Montpellier) methods of analysis. The dataware processed by normal and inverse Information Analysis in. conjunction with a computer-assisted hand-sorting routine leading to the production of structured species-sample tables. One hundred and eight noda were identified and described and compared with related units recognised from Britain and N.W. Europe. Sixty eight were arranged into 11 Associations of which 4 represent new syntaxa. The remainder were placed into sociations or left as noda of uncertain status. The communities were all classified into higher syntaxa essentially following the scheme advocated by WESTHOFF & DEN HELD (1969). The Classes and Alliances used to contain the communities are: Phragmitetea Phragmition reedswamp communities Magnocaricion tall growing sedge and reed fen vegetation of topogenous mire Parvocaricetea Caricion davallianae low-growing sedge vegetation of calcareous mire Molinio-Arrhenatheretea Calthion palustric fen meadow vegetation Molinion caerulea Molinia-dominated fen grassland Filipendulion species-poor, tall herbaceous vegetation of eutrophic mires Franguletea Salicion cinereae fen scrub communities Alnetea glutinosae Alnion glutinosae alder carr vegetation The new associations described are the Peucedano- Calamagrostietum canescentis of the Magnocaricion; the Schoeneto-Juncetum subnodulosi of the Caricion davallianae; and the Cropido-Salicetum pentandrae and the Betulo-Dryopteridetum cristatae, both of the Franguletea. A short account of the occurance of rich fen systems in lowland England and Wales is given. A brief discussion of the rich fen cirlce of vegetation in lowland England and Wales is also given