CONES AND COMPARISONS IN IND-AFFINE HOMOTOPY THEORY

Abstract

Ind-affine schemes over an algebraically closed field k are introduced. The cone functor is then defined and characterized in the based category (ind-aff)* of ind-affine schemes. Homotopy theories, one induced from the monad related to the cone functor and the other via unirational and then singular simplices, are compared. Some homotopy groups vis-a-vis (ind-aff)* taking as our model of the circle the set of points (x,y) in k² satisfying x²+y² = 1 are determined.     

Local connectedness in merotopic spaces

Abstract

In this paper uniformly locally uniformly connected merotopic spaces are studied. It turns out that their structural behaviour is essentially similar to that one of locally connected topological spaces. The introduced concept is also investigated for spaces of functions between filter-merotopic spaces (e.g. topological spaces, proximity spaces, convergence spaces) and the relationship to other concepts of local connectedness is clarified. In particular, the category of uniformly locally uniformly connected filter-merotopic spaces is Cartesian closed.     

The cayley-dickson doubling process revisited

We introduce the category of composition triples over locally ringed spaces. In this category we find the category of composition algebras over locally ringed spaces as subcategory. We describe the Cay ley-Dickson doubling process in the category of composition triples.     

EXPONENTIAL LAW AND θ-CONTINUOUS FUNCTIONS

Abstract

The category θ-Top of topological spaces and θ-continuous functions is not Cartesian closed; but it is known that under certain local property assumptions, the exponential law in θ-Top is fulfilled. We define a functor from θ-Top to the category of H-θ-topological spaces and prove that in this category the exponential law holds without any local property assumptions. We also provide a functor from θ-Top to Katětov's category of filter-merotopic spaces, which is Cartesian closed.     

On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:

1. Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.

2. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.

3. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T₄ and the countable axiom of choice holds true.

   We also show:

4. It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.

5. It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

     

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

Free Modules over Cartesian Closed Topological Categories

The construction of free R-modules over a Cartesian closed topological category X is detailed (where R is a ring object in X), and it is shown that the insertion of generators is an embedding. This result extends the well-known construction of free groups, and more generally of free algebras over a Cartesian closed topological category. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Notes on pre-Frölicher spaces

In this work we introduce a class of Sikorski differential spaces (M, D) called pre-Fr¨olicher spaces, on which the process of yielding a Fr¨olicher structure on the underlying set M is D preserving, their category we denote by preFrl. We investigate some algebraic properties on these spaces whose subsequent geometric properties are mostly similar to those of smooth manifolds, except for the invariance of dimension, and also that preFrl naturally induces a Cartesian closed subcategory of the category Frl in which there is no discrete object. Using this Cartesian property, it is shown that the Gelfand representation is a smooth map, that the tangent as well as cotangent bundles are made smooth spaces in an unusual but more natural way via smooth curves.     

A Convenient Category of Locally Preordered Spaces

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of “locally preordered” spaces. We show that our new category is Cartesian closed that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.     

Relative Riemann-Zariski spaces

In this paper we study relative Riemann-Zariski spaces associated to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasiseparated schemes factors as a composition of an affine morphism and a proper morphism. In particular, we obtain a new proof of Nagata’s compactification theorem.     

Deitmar’s versus To?n-Vaquié’s schemes over {\mathbb{F}_{1}}

Deitmar introduced schemes over ${\mathbb {F}_{1}}$ , the so-called “field with one element”, as certain spaces with an attached sheaf of monoids, generalizing the definition of schemes as ringed spaces. On the other hand, To?n and Vaquié defined them as particular Zariski sheaves over the opposite category of monoids, generalizing the definition of schemes as functors of points. We show the equivalence between Deitmar’s and To?n-Vaquiés notions and establish an analog of the classical case of schemes over ${\mathbb {Z}}$ . This result has been assumed by the leading experts on ${\mathbb {F}_{1}}$ , but no proof was given. During the proof, we also conclude some new basic results on commutative algebra of monoids, such as a characterization of local flat epimorphisms and of flat epimorphisms of finite presentation. We also inspect the base-change functors from the category of schemes over ${\mathbb {F}_{1}}$ to the category of schemes over ${\mathbb {Z}}$ .     

A Structure Theorem for \displaystyle\mathbb{P}^{{1}} — Spec k-bimodules

Let k be an algebraically closed field. Using the Eilenberg–Watts theorem over schemes (Nyman, J Pure Appl Algebra 214:1922–1954, 2010), we determine the structure of k-linear right exact direct limit and coherence preserving functors from the category of quasi-coherent sheaves on $\mathbb{P}^{1}_{k}$ to the category of vector spaces over k. As a consequence, we characterize those functors which are integral transforms.     

On the open immersion problem

We give two examples of complex spaces on which global holomorphic functions separate points and give local coordinates and they cannot be realized as open subsets of Stein spaces. At the same time we notice that these examples are open subsets of Stein schemes, a notion introduced by Grauert (Math Z 81:377–391, 1963). In the context of complex schemes we notice that by contracting a Nori string one obtains a complex scheme and not a complex space. The covering spaces of 1-convex surfaces are divided in two categories: those that have an envelope of holomorphy and those that do not. More interesting are those in the second category and they correspond to covering spaces for singularities which in the desingularization with normal crossings contain cycles in the exceptional set.     

CARTESIAN CLOSEDNESS AND THE MCNEILLE COMPLETION OF AN INITIALLY STRUCTURED CATEGORY

Abstract

An adaptation of a theorem by Herrlich [5] shows that every initially structured category A can be fully embedded in a topological category A_C, which is, in fact, a MacNeille completion of A. It is then shown that A is Cartesian closed if and only if A_C is.

Also developed is the notion of a Cartesian closed initially structured (CCIS) hull of a category. The theory of the CCIS hull is analogous to that of the Cartesian closed topological (CCT) hull. It is proved that a category has a CCT hull-if and only if it has a CCIS hull; and this allows the list of conditions equivalent to the existence of a CCT hull to be supplemented.

Examples are given, drawn mainly from the various categories of binary relations.     

Multirectangular invariants for power Köthe spaces

Using some new linear topological invariants, isomorphisms and quasidiagonal isomorphisms are investigated on the class of first type power Köthe spaces [Proceedings of 7th Winter School in Drogobych, 1976, pp. 101-126; Turkish J. Math. 20 (1996) 237-289; Linear Topol. Spaces Complex Anal. 2 (1995) 35-44]. This is the smallest class of Köthe spaces containing all Cartesian and projective tensor products of power series spaces and closed with respect to taking of basic subspaces (closed linear hulls of subsets of the canonical basis). As an application, it is shown that isomorphic spaces from this class have, up to quasidiagonal isomorphisms, the same basic subspaces of finite (infinite) type.     

Embeddings of countable closed sets and reverse mathematics

Summary If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR ₀. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of .     

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

On Homotopy Types of Alexandroff Spaces

We generalise some results of R. E. Stong concerning finite spaces to wider subclasses of Alexandroff spaces. In particular, we characterize pairs of spaces X,Y such that the compact-open topology on C(X,Y) is Alexandroff, give a homotopy type classification of a class of infinite Alexandroff spaces and prove some results concerning cores of locally finite spaces. We also discuss a mistake found in an article of F.G. Arenas. Since the category of T ₀ Alexandroff spaces is equivalent to the category of posets, our results may lead to a deeper understanding of the notion of a core of an infinite poset.     

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan’s theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.