Recursos de colección
Project Euclid (Hosted at Cornell University Library) (191.996 recursos)
Homology, Homotopy and Applications
Homology, Homotopy and Applications
Casas, José Manuel; Casado, Rafael F.; Khmaladze, Emzar; Ladra, Manuel
We construct a pair of adjoint functors between the categories
of crossed modules of Lie and associative algebras, which
extends the classical one between the categories of Lie and associative
algebras. This result is used to establish an equivalence
of categories of modules over a Lie crossed module and its universal
enveloping crossed module.
Fiorenza, Domenico; Rogers, Christopher L.; Schreiber, Urs
To any manifold equipped with a higher degree closed form, one can associate an $L_\infty$-algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means of an explicit
homotopy equivalence, we interpret this $L_\infty$-algebra in terms of infinitesimal autoequivalences of higher prequantum bundles. By truncating the connection data on the prequantum bundle, we produce
analogues of the (higher) Lie algebras of sections of the Atiyah Lie algebroid and of the Courant Lie 2-algebroid. We also exhibit the $L_\infty$-cocycle that realizes the $L_\infty$-algebra of local observables
as a Kirillov-Kostant-Souriau-type $L_\infty$-extension of the Hamiltonian vector fields. When restricted along a...
Crans, Alissa S.; Wagemann, Friedrich
We generalize the notion of a crossed module of groups to
that of a crossed module of racks. We investigate the relation
to categorified racks, namely strict 2-racks, and trunk-like objects
in the category of racks, generalizing the relation between
crossed modules of groups and strict 2-groups. Then we explore
topological applications. We show that by applying the rackspace
functor, a crossed module of racks gives rise to a covering.
Our main result shows how the fundamental racks associated to
links upstairs and downstairs in a covering fit together to form
a crossed module of racks.
Patel, D.; Ravindra, G. V.
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero, and let $Y \subset X$ be a smooth ample hyperplane section. The Weak Lefschetz conjecture for Chow groups states
that the natural restriction map $\mathrm{CH}^p (X)_{\mathbb{Q}} \to \mathrm{CH}^p (Y)_{\mathbb{Q}}$ is an isomorphism for all $p \lt \dim (Y) / 2$. In this note, we revisit a strategy introduced by Grothendieck
to attack this problem by using the Bloch-Quillen formula to factor this morphism through a continuous $\mathrm{K}$-cohomology group on the formal completion of $X$ along $Y$. This splits the conjecture
into two smaller conjectures: one consisting of...
Stevenson, Greg
Let $S$ be a commutative ring with topologically noetherian spectrum, and let $R$ be the absolutely flat approximation of $S$. We prove that subsets of the spectrum of $R$ parametrise the localising subcategories
of $\mathsf{D}(R)$. Moreover, we prove the telescope conjecture holds for $\mathsf{D}(R)$. We also consider unbounded derived categories of absolutely flat rings that are not semi-artinian and exhibit a localising
subcategory that is not a Bousfield class and a cohomological Bousfield class that is not a Bousfield class.
Geisser, Thomas
The purpose of this paper is to give homological descent theorems
for motivic homology theories (for example, Suslin homology)
and motivic Borel-Moore homology theories (for example,
higher Chow groups) for certain hypercoverings.
Hohloch, Sonja
We generalize Cohen & Jones & Segal's flow category, whose objects are the critical points of a Morse function and whose morphisms are the Morse moduli spaces between the critical points
to an $n$-category. The $n$-category construction involves repeatedly doing Morse theory on Morse moduli spaces for which we have to construct a class of suitable Morse functions. It turns out to be
an 'almost strict' $n$-category, i.e. it is a strict $n$-category 'up to canonical isomorphisms'.
Álvarez, J. N. Alonso; Vilaboa, J. M. Fernández; Rodríguez, R. González
In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra $H$. We show that the second cohomology group classifies completely weak crossed products, having a common preunit,
of $H$ with a commutative left $H$-module algebra $A$.
Munson, Brian A.
Goodwillie’s proof of the Blakers-Massey Theorem for n- cubes relies on a lemma whose proof invokes transversality. The
rest of his proof follows from general facts about cubes of spaces
and connectivities of maps. We present a purely homotopytheoretic
proof of this lemma. The methods are elementary,
using a generalization and modification of an argument originally
due to Puppe used to prove the Blakers-Massey Theorem
for squares.
Bohmann, Anna Marie
For any compact Lie group $G$, there are several well-established definitions of a $G$-equivariant spectrum. In this paper, we develop the definition of a global orthogonal spectrum. Loosely speaking,
this is a coherent choice of orthogonal $G$-spectrum for each compact Lie group $G$. We use the framework of enriched indexed categories to make this precise. We also consider equivariant $K$-theory
and $\operatorname{Spin}^c$-cobordism from this perspective, and we show that the Atiyah-Bott-Shapiro orientation extends to the global context.
Grigor'yan, A.; Muranov, Yu. V.; Yau, Shing-Tung
The cohomology of digraphs was introduced for the first time by Dimakis and Müller-Hoissen. Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices
with relations that follow naturally from the structure of the set of edges. A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing
the (co)homology groups were developed. The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers. The digraph
$G_B$ of the partially ordered...
Groth, Moritz; Ponto, Kate; Shulman, Michael
We show that stable derivators, like stable model cate-
gories, admit Mayer-Vietoris sequences arising from cocartesian
squares. Along the way we characterize homotopy exact squares
and give a detection result for colimiting diagrams in derivators.
As an application, we show that a derivator is stable if and only
if its suspension functor is an equivalence.
Ellis, Steven P.; Klein, Arno
In multivariate data analysis dependence beyond pair-wise can
be important. With many variables, however, the number of simple
summaries of even third-order dependence can be unmanageably
large.
¶ “Concurrence topology” is an apparently new method for describing
high-order dependence among up to dozens of dichotomous (i.e.,
binary) variables (e.g., seventh-order dependence in 32 variables).
This method generally produces summaries of dependence of manageable
size. (But computing time can be lengthy.) For time series,
this method can be applied in both the time and Fourier domains.
¶ Write each observation as a vector of 0’s and 1’s. A “concurrence”
is a group of variables all labeled “1” in the same observation. The
collection...
Powell, Geoffrey M. L.
The connective $ku$-(co)homology of elementary abelian $2$-groups is determined as a functor of the elementary abelian $2$-group, using the action of the Milnor operations $Q_0, Q_1$ on mod $2$ group
cohomology, the Atiyah-Segal theorem for $KU$-cohomology, together with an analysis of the functorial structure of the integral group ring; the functorial structure then reduces calculations to the rank 1 case.
¶ These results are used to analyse the local cohomology spectral sequence calculating $ku$-homology, via a functorial version of local duality for Koszul complexes, giving a conceptual explanation of results of Bruner and
Greenlees.
Abuhlail, Jawad Y.
Basic homological lemmas well known for modules over rings
and, more generally, in the context of abelian categories, have
been extended to many other concrete and abstract-categorical
contexts by various authors. We propose a new such extension,
specifically for commutative monoids and semimodules; these
two contexts are equivalent since the forgetful functors from
varieties of semimodules to the variety of commutative monoids
preserve all limits and colimits.
Fialowski, Alice; Mukherjee, Goutam; Naolekar, Anita
We develop deformation theory of algebras over quadratic
operads where the parameter space is a commutative local algebra.
We also give a construction of a distinguised deformation of
an algebra over a quadratic operad with a complete local algebra
as its base—the so-called versal deformation—which induces all
other deformations of the given algebra.
Grier, Michael; Nelson, Sam
We define invariants of unoriented knots and links by enhancing the integral kei counting invariant $\Phi_X^{\mathbb{Z}}(K)$ for a finite kei $X$ using representations of the kei algebra,
$\mathbb{Z}_K[X]$, a quotient of the quandle algebra $\mathbb{Z}[X]$ defined by Andruskiewitsch and Graña. We give an example that demonstrates that the enhanced invariant is stronger
than the unenhanced kei counting invariant. As an application, we use a quandle module over the Takasaki kei on $\mathbb{Z}_3$ which is not a $\mathbb{Z}_K[X]$-module to detect the non-invertibility of a virtual knot.
Kucharz, Wojciech; Maciejewski, Łukasz
Let $Y$ be a real algebraic variety. We are interested in determining the supremum, $\beta(Y)$, of all nonnegative integers $n$ with the following property: For every $n$-dimensional compact connected
nonsingular real algebraic variety $X$, every continuous map from $X$ into $Y$ is homotopic to a regular map. We give an upper bound for $\beta(Y)$, based on a construction involving complexification
of real algebraic varieties. In some cases, we obtain the exact value of $\beta(Y)$.
Iriye, Kouyemon; Kishimoto, Daisuke
A generalized Postnikov tower (GPT) is defined as a tower of
principal fibrations with the classifying maps into generalized
Eilenberg–Mac Lane spaces. We study fundamental properties
of GPT’s such as their existence, localization and length. We
further consider the distribution of torsion in a GPT of a finite
complex, motivated by the result of McGibbon and Neisendorfer. We also give an algebraic description of the length of a
GPT of a rational space.
Kahl, Thomas
Precubical sets are used to model concurrent systems. We
introduce the homology graph of a precubical set, which is a
directed graph whose nodes are the homology classes of the
precubical set. We show that the homology graph is invariant
under weak morphisms that are homeomorphisms.