Separated and Connected Maps

Using on the one hand closure operators in the sense of Dikranjan and Giuli and on the other hand left- and right-constant subcategories in the sense of Herrlich, Preuß, Arhangel'skii and Wiegandt, we apply two categorical concepts of connectedness and separation/disconnectedness to comma categories in order to introduce these notions for morphisms of a category and to study their factorization behaviour. While at the object level in categories with enough points the first approach exceeds the second considerably, as far as generality is concerned, the two approaches become quite distinct at the morphism level. In fact, left- and right-constant subcategories lead to a straight generalization of Collins' concordant and dissonant maps in the category $$\mathcal{T}op$$ of topological spaces. By contrast, closure operators are neither able to describe these types of maps in $$\mathcal{T}op$$, nor the more classical monotone and light maps of Eilenberg and Whyburn, although they give all sorts of interesting and closely related types of maps. As a by-product we obtain a negative solution to the ten-year-old problem whether the Giuli–Hušek Diagonal Theorem holds true in every decent category, and exhibit a counter-example in the category of topological spaces over the 1-sphere

Dual closure operators and their applications

Departing from a suitable categorical description of closure operators, this paper dualizes this notion and introduces some basic properties of dual closure operators. Usually these operators act on quotients rather than subobjects, and much attention is being paid here to their key examples in algebra and topology, which include the formation of monotone quotients (Eilenberg-Whyburn) and concordant quotients (Coffins). In fair categorical generality, these constructions are shown to be factors of the fundamental correspondence that relates connectecinesses and disconnectednesses in topology, as well as torsion classes and torsion-free classes in algebra. Depending on a given cogenerator, the paper also establishes a non-trivial correspondence between closure operators and dual closure operators in the category of R-modules. Dual closure operators must be carefully distinguished from interior operators that have been studied by other author

Global Fukaya category and quantum Novikov conjecture I

Conceptually, the goal here is a construction which functorially translates a Hamiltonian fibre bundle to a certain ``derived vector bundle'' over the same space, with fiber an $A _{\infty}$ category. This ``derived vector bundle'' must remember the continuity of the original bundle. Concretely, using Floer-Fukaya theory for a monotone $(M, \omega)$ we construct a natural continuous map \begin{equation*} BHam (M, \omega) \to (\mathcal{S}, NFuk (M)), \end{equation*} with $(\mathcal{S}, NFuk (M))$ denoting the $NFuk (M)$ component of the ``space'' of $\infty$-categories, where $NFuk (M)$ is the $A _{\infty}$-nerve of the Fukaya category $Fuk (M)$. This construction is very closely related to the theory of the Seidel homomorphism and the quantum Chern classes of the author, and this map is intended to be the deepest expression of their underlying geometric theory. In part II the above map is shown to be non trivial by an explicit calculation. In particular we arrive at a new non-trivial ``quantum'' invariant of any smooth manifold and a ``quantum'' Novikov conjecture.Comment: v5, 41 pages. This adds significant detail and fixes some language issue

The monotone-light factorization for categories via preordered and ordered sets

Doutoramento em MatemáticaNeste trabalho provamos que as adjunções Cat ? Preord e Cat ? Ord da categoria de todas as categorias na categoria das pré-ordens e na das ordens, respectivamente, determinam ambas distintos sistemas de factorização “monotone-light” em Cat. Caracterizamos para as duas adjunções acima os morfismos de cobertura trivial, os de cobertura, os verticais, os verticais estáveis, os separáveis, os puramente inseparáveis, os normais e os dissonantes. Daqui se segue que os sistemas de factorização “monotone-light”, concordante-dissonante e inseparável-separável em Cat coincidem para a adjunção Cat ? Preord.It is shown that the reflections Cat ? Preord and Cat ? Ord of the category of all categories into the category of preorders and orders, respectively, determine both distinct monotone-light factorization systems on Cat. We give explicit descriptions of trivial coverings, coverings, vertical, stablyvertical, separable, purely inseparable, normal and dissonant morphisms with respect to those two reflections. It follows that the monotone-light, concordantdissonant and inseparable-separable factorizations on Cat do coincide in the reflection Cat ? Preord

Generalising the concordant-dissonant factorisation

Includes bibliographical references (leaves 74-76).The "concordant-dissonant factorization of an arbitrary continuous function" was introduced by Collins in [C 1971]. He defined that a continuous map is concordant if each of its fibres is contained in a quasi-component of its domain and that a continuous map is dissonant iff each of its fibres and each quasi-component of its domain have at most one element in common

Rank-based linkage I: triplet comparisons and oriented simplicial complexes

Rank-based linkage is a new tool for summarizing a collection $S$ of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on $S$. Rank-based linkage is applied to the $K$-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected $K$-nearest neighbor graph on $S$. In $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, \sigma)$ where $\sigma(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, for varying $t$. Rank-based linkage is a functor from a category of out-ordered digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart`` the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Open combinatorial problems are presented in the last section.Comment: 37 pages, 12 figure