scholarly journals Cohomology of Presheaves of Monoids

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 116
Author(s):  
Pilar Carrasco ◽  
Antonio M. Cegarra
Keyword(s):  
Topological Space ◽  
Small Category ◽  
Mac Lane ◽  
Cohomology Of Groups

The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category with values in groupoids with abelian isotropy groups. The paper also includes a cohomological classification for extensions of presheaves of monoids, which is useful to the study of H -extensions of presheaves of regular monoids. The results apply directly in several settings such as presheaves of monoids on a topological space, simplicial monoids, presheaves of simplicial monoids on a topological space, monoids or simplicial monoids on which a fixed monoid or group acts, and so forth.

Symmetric Powers of Motives

2019 ◽  
pp. 232-252
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Homotopy Group ◽  
Orbit Space ◽  
Symmetric Power ◽  
Homotopy Groups ◽  
Integral Homology ◽  
Symmetric Powers ◽  
Mac Lane ◽  
Abelian Monoid

This chapter develops the basic theory of symmetric powers of smooth varieties. The constructions in this chapter are based on an analogy with the corresponding symmetric power constructions in topology. If 𝐾 is a set (or even a topological space) then the symmetric power 𝑆𝑚𝐾 is defined to be the orbit space 𝐾𝑚/Σ‎𝑚, where Σ‎𝑚 is the symmetric group. If 𝐾 is pointed, there is an inclusion 𝑆𝑚𝐾 ⊂ 𝑆𝑚+1𝐾 and 𝑆∞𝐾 = ∪𝑆𝑚𝐾 is the free abelian monoid on 𝐾 − {*}. When 𝐾 is a connected topological space, the Dold–Thom theorem says that ̃𝐻*(𝐾, ℤ) agrees with the homotopy groups π‎ *(𝑆∞𝐾). In particular, the spaces 𝑆∞(𝑆 𝑛) have only one homotopy group (𝑛 ≥ 1) and hence are the Eilenberg–Mac Lane spaces 𝐾(ℤ, 𝑛) which classify integral homology.

Finiteness conditions for CW complexes. Il

1966 ◽  
Vol 295 (1441) ◽  
pp. 129-139 ◽  
Keyword(s):  
Topological Space ◽  
Finite Number ◽  
Simple Form ◽  
Finite Dimension ◽  
Homotopy Equivalent ◽  
Finite Complex ◽  
Finiteness Conditions ◽  
Cw Complexes ◽  
Number Of Cells

A CW complex is a topological space which is built up in an inductive way by a process of attaching cells. Spaces homotopy equivalent to CW complexes play a fundamental role in topology. In the previous paper with the same title we gave criteria (in terms of more-or-less standard invariants of the space) for a CW complex to be homotopy equivalent to one of finite dimension, or to one with a finite number of cells in each dimension, or to a finite complex. This paper contains some simplification of these results. In addition, algebraic machinery is developed which provides a rough classification of CW complexes homotopy equivalent to a given one (the existence clause of the classification is the interesting one). The results would take a particularly simple form if a certain (rather implausible) conjecture could be established.

scholarly journals CLASSIFICATION OF CERTAIN CONTINUOUS FIELDS OF KIRCHBERG ALGEBRAS

2014 ◽  
Vol 25 (01) ◽  
pp. 1450004
Author(s):  
RASMUS BENTMANN
Keyword(s):  
Topological Space ◽  
Finite Dimensional ◽  
K Theory ◽  
Continuous Fields ◽  
Universal Coefficient Theorem

We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational K-theory groups satisfying the universal coefficient theorem. We provide a range result for fields in this class with finite-dimensional K-theory. There are versions of both results for unital continuous fields.

scholarly journals Mazur spaces

1981 ◽  
Vol 4 (1) ◽  
pp. 39-53 ◽  
Author(s):  
Albert Wilansky
Keyword(s):  
Topological Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Locally Convex Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
New Classification ◽  
Open Question ◽  
Locally Convex

A Mazur space is a locally convex topological vector spaceXsuch that everyfϵXsis continuous whereXsis the set of sequentially continuous linear functionals onX;Xsis studied whenXis of the formC(H),Ha topological space, and whenXis the weak*dual of a locally convex space. This leads to a new classification of compactT2spacesH, those for which the weak*dual ofC(H)is a Mazur space. An open question about Banach spaces with weak*sequentially compact dual ball is settled: the dual space need not be Mazur.

A 3-Dimensional Non-Abelian Cohomology of Groups With Applications to Homotopy Classification of Continuous Maps

1991 ◽  
Vol 43 (2) ◽  
pp. 265-296 ◽  
Author(s):  
Manuel Bullejos ◽  
Antonio M. Cegarra
Keyword(s):  
General Problem ◽  
Cohomology Theory ◽  
Homotopy Classification ◽  
3 Dimensional ◽  
Continuous Maps ◽  
Crossed Modules ◽  
Long Time ◽  
Cohomology Of Groups

The general problem of what should be a non-abelian cohomology, what is it supposed to do, and what should be the coefficients, form a set of interesting questions which has been around for a long time. In the particular setting of groups, a comprehensible and well motivated cohomology theory has been so far stated in dimensions ≤ 2, the coefficients for being crossed modules. The main effort to define an appropriate for groups has been done by Dedecker [16] and Van Deuren [40]; they studied the obstruction to lifting non-abelian 2-cocycles and concluded with first approach for , which requires “super crossed groups” as coefficients. However, as Dedecker said “some polishing work remains necessary” for his cohomology.

Manifolds of homotopy type K(π, 1). I

1971 ◽  
Vol 70 (3) ◽  
pp. 387-393 ◽  
Author(s):  
F. E. A. Johnson
Keyword(s):  
Topological Space ◽  
Homotopy Type ◽  
Classification Problem ◽  
Universal Covering ◽  
Type K ◽  
Simply Connected Manifolds ◽  
Simply Connected

1. Introduction: In all that follows, by a manifold we shall mean a paracompact, Hausdorff, locally Euclidean topological space. By means of the universal covering construction, the classification problem for connected manifolds splits into two parts, namely (i) classification of simply connected manifolds, (ii) classification of covering actions of groups on simply connected manifolds.

scholarly journals On a Homotopy Classification of Mappings of an (n+1) Dimensional Complex Into an Arcwise Connected Topological Space Which is Aspherical in Dimensions Less Thann (n >2)

1951 ◽  
Vol 3 ◽  
pp. 67-72 ◽  
Author(s):  
Nobuo Shimada ◽  
Hiroshi Uehara
Keyword(s):  
Topological Space ◽  
Nauk Sssr ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Continuous Mappings ◽  
Arcwise Connected ◽  
New Type

Pontrjagin classified mappings of a three dimensional sphere into anndimensional complex, where he made use of a new type of product of cocycles. By the aid of the generalized Pontrjagin’s product of cocycles Steenrod enumerated effectively all the homotopy classes of mappings of an (n+1) dimensional complex into annsphere. According to the recent issue of the Mathematical Reviews it is reported that M. M. Postnikov extended Steenrod’s case to the case where an arcwise connected topological space which is aspherical in dimensions less thann, takes place of annsphere. (Postnikov M. M., Classification of continuous mappings of an(n+1)dimensional complex into a connected topological space which is aspherical in dimensions less thann. Doklady Akad. Nauk SSSR (N.S.) 71., 1027-1028, 1950 (Russian. No. proof is given.)) But here in Japan no details are yet to hand. We intend to give a solution to this problem in case wheren>2, and also to give an application concerning the(n+ 3)-extension cocycle.

Sequence selection properties in Cp (X) with the double ideals

2021 ◽  
Vol 71 (1) ◽  
pp. 147-154
Author(s):  
Sumit Singh ◽  
Brij K. Tyagi ◽  
Manoj Bhardwaj
Keyword(s):  
Topological Space ◽  
Frequency Spectrum ◽  
Nauk Sssr ◽  
Sequence Selection ◽  
Akad Nauk Sssr ◽  
Covering Properties ◽  
Selection Principles ◽  
Normal Convergence

Abstract Recently Bukovský, Das and Šupina [Ideal quasi-normal convergence and related notions, Colloq. Math. 146 (2017), 265–281] started the study of sequence selection properties (𝓘, 𝓙-α 1) and (𝓘, 𝓙-α 4) of Cp (X) using the double ideals, where 𝓘 and 𝓙 are the proper admissible ideals of ω, which are motivated by Arkhangeľskii local αi -properties [The frequency spectrum of a topological space and the classification of spaces, Dokl. Akad. Nauk SSSR 13 (1972), 1185–1189]. In this paper, we obtain some characterizations of (𝓘, 𝓙-α 1) and (𝓘, 𝓙-α 4) properties of Cp (X) in the terms of covering properties and selection principles. Under certain conditions on ideals 𝓘 and 𝓙, we identify the minimal cardinalities of a space X for which Cp (X) does not have (𝓘, 𝓙-α 1) and (𝓘, 𝓙-α 4) properties.

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