Covariant functors in categories of topological spaces

This survey is devoted to the properties of certain concrete covariant functors-normal and almost normal functors-in the category of compacta, as well as the algebraic theory of covariant functors, and the connections between the theory of functors with absolute extensors and manifolds.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 28, pp. 47–95, 1990.     

Closed structures on categories of topological spaces

The category of all topological spaces and continuous maps and its full subcategory of all T_o-spaces admit (up to isomorphism) precisely one structure of symmetric monoidal closed category (see [2]). In this paper we shall prove the same result for any epireflective subcategory of the category of topological spaces (particularly e.g. for the categories of Hausdorff spaces, regular spaces, Tychonoff spaces).     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

Strong movable categories and strong movability of topological spaces

The paper is devoted to one of the important notions of the shape theory: that of strong movability, which was primarily introduced by K. Borsuk for metrizable compacts. A strong movability criterion is proved for topological spaces, which in particular reveals a new, categorical approach to the strong movability.     

The absence of bases from certain separable linear topological spaces

We give examples of separable linear topological spaces without Shauder-type bases. We prove that every linear set X of dimension ₀2₀ can be provided with a separable locally convex topology for which there is no Shauder-type basis.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 583–589, November, 1972.In conclusion the author wishes to express his gratitude to M. M. Dragilev for his attention to the paper.     

Stratifying derived categories of cochains on certain spaces

In recent years, Benson, Iyengar and Krause have developed a theory of stratification for compactly generated triangulated categories with an action of a graded commutative Noetherian ring. Stratification implies a classification of localizing and thick subcategories in terms of subsets of the prime ideal spectrum of the given ring. In this paper two stratification results are presented: one for the derived category of a commutative ring-spectrum with polynomial homotopy and another for the derived category of cochains on certain spaces. We also give the stratification of cochains on a space a topological content.     

Relations between certain classes of effectively topological spaces

Relations between properties of effectively topological spaces such as the existence of a countable regular base, a computable intersection, effective regularity, normality etc., are studied; an arithmetic (constructive) analog of Uryson's Theorem on the metrization of topological spaces with denumerable bases is established; it is shown that the conditions in the analog are independent.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 483–495, April, 1969.The author is deeply grateful to V. A. Uspenskii for proposing the topic of investigation and for his great attention to the work.     

Injectivity of topological categories

Injective objects in concrete categories frequently turn out to be objects with particularly pleasant properties. Often some form of completeness provides a characterization of injectivity in such a category, with injective hulls achieved through certain standard completion processes. Several results during the past decade have shown that certain specific topological categories are precisely the injective objects in various natural quasicategories of concrete categories, with injective hulls obtained via certain sieve constructions. When the base category is trivial, some of these results specialize to classical results in certain categories of ordered structures; e.g., the injectives in posets characterized as complete lattices, with injective hulls the MacNeille completions, and the injectives in semilattices characterized as locales, with injective hulls the locale hulls.This paper contains two main results. The first provides a characterization of injective objects in a setting sufficiently general as to include the above mentioned characterizations as well as many others. The second theorem gives a characterization of those objects that have injective hulls, and provides a construction of the hulls as well. Corollaries of this theorem yield numerous known injective hull constructions. The second theorem uses a much stronger hypothesis than the first. That this hypothesis is indispensible follows from a result of E. Nelson on the non-existive of injective hulls of certain -semilattices.In Memory of Evelyn NelsonPresented by F.E.J. Linton.This research was partially sponsored by the U.S. National Science Foundation Grant DCR-8604080 and by support from the National Academies of Sciences of Czechoslovakia and the United States.     

Universal topological algebra needs closed topological categories

It is shown that a development of universal topological algebra, based in the obvious way on the category of topological spaces, leads in general to a pathological situation. The pathology disappears when the base category is changed to a cartesian closed topological category or to a topological category endowed with a compatible closed symmetric monoidal structure, provided that in the latter case, the algebraic operations are expressed in terms of monoidal powers rather than the usual cartesian powers. With such base categories, universal topological algebra becomes virtually as well-behaved as ordinary (setbased) universal algebra.     

Cyclic homology of certain topological spaces formal in the sense of Sullivan

Translated from Matematicheskie Zametki, Vol. 50, No. 6, pp. 131–141, December, 1991.     

The embeddability ordering of topological spaces

For K a set of topological spaces and X,Y∈K, the notation Xh_⊆Y means that X embeds homeomorphically into Y; and X∼Y means Xh_⊆Yh_⊆X. With , the equivalence relation ∼ on K induces a partial order h_? well-defined on K/∼ as follows: if Xh_⊆Y.For posets (P,P_?) and (Q,Q_?), the notation (P,P_?)?(Q,Q_?) means: there is an injection such that p₀P_?p₁ in P if and only if h(p₀)Q_?h(p₁) in Q. For κ an infinite cardinal, a poset (Q,Q_?) is a κ-universal poset if every poset (P,P_?) with |P|?κ satisfies (P,P_?)?(Q,Q_?).The authors prove two theorems which improve and extend results from the extensive relevant literature.

Theorem 2.2. There is a zero-dimensional Hausdorff space S with|S|=κsuch that(P(S)/∼,h_?)is a κ-universal poset.     

The structure of certain rigid tensor categories

We consider rigid tensor categories over a field of characteristic zero in which some exterior power of each object is zero. To cite this article: P. O'Sullivan, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Recursive topological spaces

The research was supported by the Russian Foundation for Fundamental Research (Grant 093-01-01506).