Algebraic Topology
... rather broad coverage of the subject. The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology. In a sense, the book could have been written thirty or forty years ago since virtually everything in it is at least that old. However, the passage of the i ...
... rather broad coverage of the subject. The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology. In a sense, the book could have been written thirty or forty years ago since virtually everything in it is at least that old. However, the passage of the i ...
Algebraic models for rational G
... are sheaves of graded Q–modules with some additional structure. These sheaves are over the space of closed subgroups of G denoted by Sub(G) and have the property that the fibre over a closed subgroup H gives information about the H–fixed points. The proof of this result goes by constructing a zig–za ...
... are sheaves of graded Q–modules with some additional structure. These sheaves are over the space of closed subgroups of G denoted by Sub(G) and have the property that the fibre over a closed subgroup H gives information about the H–fixed points. The proof of this result goes by constructing a zig–za ...
The Simplicial Lusternik
... of contiguity). As in the case of finite T0 -space, the core of a simplicial complex is unique up to isomorphism and two simplicial complexes have the same strong homotopy type if and only if their cores are isomorphic. Finally, we will present some results that show some properties that are preserv ...
... of contiguity). As in the case of finite T0 -space, the core of a simplicial complex is unique up to isomorphism and two simplicial complexes have the same strong homotopy type if and only if their cores are isomorphic. Finally, we will present some results that show some properties that are preserv ...
Motivic Homotopy Theory
... sSet, where the enrichment is required to be compatible with the model structure. Such categories are very useful as they carry a natural notion of a simplicial mapping space between any two objects. Finally the third section is devoted to the heavy machinery of ...
... sSet, where the enrichment is required to be compatible with the model structure. Such categories are very useful as they carry a natural notion of a simplicial mapping space between any two objects. Finally the third section is devoted to the heavy machinery of ...
SYMMETRIC SPECTRA Contents Introduction 2 1
... At approximately the same time as the third author discovered symmetric spectra, the team of Elmendorf, Kriz, Mandell, and May [EKMM97] also constructed a symmetric monoidal category of spectra, called S-modules. Some generalizations of symmetric spectra appear in [MMSS1]. These many new symmetric m ...
... At approximately the same time as the third author discovered symmetric spectra, the team of Elmendorf, Kriz, Mandell, and May [EKMM97] also constructed a symmetric monoidal category of spectra, called S-modules. Some generalizations of symmetric spectra appear in [MMSS1]. These many new symmetric m ...
CW-complexes in the category of exterior spaces
... Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque ...
... Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque ...
Introduction to Combinatorial Homotopy Theory
... more appropriate. In this framework of combinatorial topology, the sensible topological spaces can be combinatorially defined, and also installed and processed on a computer. This is valid even for very complicated or abstract spaces such as classifying spaces, functional spaces; the various importa ...
... more appropriate. In this framework of combinatorial topology, the sensible topological spaces can be combinatorially defined, and also installed and processed on a computer. This is valid even for very complicated or abstract spaces such as classifying spaces, functional spaces; the various importa ...
on the ubiquity of simplicial objects
... on simplicial objects in abelian categories. The main result of this section is the Dold-Kan Theorem (Theorem 3.1.1) which gives a correspondence between simplicial objects and (nonnegative) chain complexes. We will see the construction of Eilenberg-MacLane spaces as an application of this theorem. ...
... on simplicial objects in abelian categories. The main result of this section is the Dold-Kan Theorem (Theorem 3.1.1) which gives a correspondence between simplicial objects and (nonnegative) chain complexes. We will see the construction of Eilenberg-MacLane spaces as an application of this theorem. ...
Finite spaces and larger contexts JP May
... When I first started talking about finite spaces, in the summer of 2003, my interest had nothing at all to do with my own areas of research, which seemed entirely disjoint. However, it has gradually become apparent that finite spaces can be integrated seamlessly into a global picture of how posets, ...
... When I first started talking about finite spaces, in the summer of 2003, my interest had nothing at all to do with my own areas of research, which seemed entirely disjoint. However, it has gradually become apparent that finite spaces can be integrated seamlessly into a global picture of how posets, ...
Topolog´ıa Algebraica de Espacios Topológicos Finitos y Aplicaciones
... particular, some of the results that we obtain, to analize two important open conjectures of algebraic and geometric topology: Quillen’s conjecture on the poset of p-subgroups of a group and the geometric Andrews-Curtis conjecture. Homotopy types of finite spaces can be described through elemental m ...
... particular, some of the results that we obtain, to analize two important open conjectures of algebraic and geometric topology: Quillen’s conjecture on the poset of p-subgroups of a group and the geometric Andrews-Curtis conjecture. Homotopy types of finite spaces can be described through elemental m ...
Galois Extensions of Structured Ring Spectra
... Furthermore, the global model S → M U is an S[BU ]-Hopf–Galois extension of commutative S-algebras, with coaction β : M U → M U ∧ S[BU ] given by the Thom diagonal. For each element g ∈ Gn its Galois action on En can be directly recovered from this S[BU ]-coaction, up to the adjunction of some roots ...
... Furthermore, the global model S → M U is an S[BU ]-Hopf–Galois extension of commutative S-algebras, with coaction β : M U → M U ∧ S[BU ] given by the Thom diagonal. For each element g ∈ Gn its Galois action on En can be directly recovered from this S[BU ]-coaction, up to the adjunction of some roots ...
barmakthesis.pdf
... In the last decades, a few interesting papers on finite spaces appeared [20, 31, 38], but the subject certainly did not receive the attention it required. In 2003, Peter May writes a series of unpublished notes [24, 23, 22] in which he synthesizes the most important ideas on finite spaces until that ...
... In the last decades, a few interesting papers on finite spaces appeared [20, 31, 38], but the subject certainly did not receive the attention it required. In 2003, Peter May writes a series of unpublished notes [24, 23, 22] in which he synthesizes the most important ideas on finite spaces until that ...
FIBRATIONS OF TOPOLOGICAL STACKS Contents 1. Introduction 2
... lifting properties (which involve building up continuous maps by attaching cells or extending maps from smaller subsets by inductive steps) often can not be applied to stacks. The main technical input which allows us to circumvent this difficulty is the theory of classifying spaces for topological s ...
... lifting properties (which involve building up continuous maps by attaching cells or extending maps from smaller subsets by inductive steps) often can not be applied to stacks. The main technical input which allows us to circumvent this difficulty is the theory of classifying spaces for topological s ...
- Iranian Journal of Fuzzy Systems
... 1. Introduction and Preliminaries The notation of fuzzy sets and fuzzy set operations were introduced by Zadeh in his paper [16]. Subsequently, some basic concepts from general topology were applied to fuzzy sets, see e.g. [1, 12, 15]. Rosenfeld defined fuzzy groups in [13]. Foster introduced the no ...
... 1. Introduction and Preliminaries The notation of fuzzy sets and fuzzy set operations were introduced by Zadeh in his paper [16]. Subsequently, some basic concepts from general topology were applied to fuzzy sets, see e.g. [1, 12, 15]. Rosenfeld defined fuzzy groups in [13]. Foster introduced the no ...
Thom Spectra that Are Symmetric Spectra
... Theorem 1.4. If (X, f ) → (Y, g) is an I-equivalence of T -good I-spaces over BF , then the induced map T (f ) → T (g) is a stable equivalence of symmetric spectra. Here stable equivalence refers to the stable model structure on SpΣ defined in [16] and [27]. It is a subtle property of this model str ...
... Theorem 1.4. If (X, f ) → (Y, g) is an I-equivalence of T -good I-spaces over BF , then the induced map T (f ) → T (g) is a stable equivalence of symmetric spectra. Here stable equivalence refers to the stable model structure on SpΣ defined in [16] and [27]. It is a subtle property of this model str ...
A Steenrod Square on Khovanov Homology
... construction of such spaces has been given by [HKK]. j The space XKh (L) gives algebraic structures on Khovanov homology which are not (yet) apparent from other perspectives. Specifically, while the cohomology of a spectrum does not have a cup product, it does carry stable cohomology operations. The ...
... construction of such spaces has been given by [HKK]. j The space XKh (L) gives algebraic structures on Khovanov homology which are not (yet) apparent from other perspectives. Specifically, while the cohomology of a spectrum does not have a cup product, it does carry stable cohomology operations. The ...
Topology I - School of Mathematics
... methods was made to problems of modern physics; in fact in several instances it would have been impossible to understand the essence of the real physical phenomena in question without the aid of concepts from topology. Since it is not possible to include treatments of all of these topics in our surv ...
... methods was made to problems of modern physics; in fact in several instances it would have been impossible to understand the essence of the real physical phenomena in question without the aid of concepts from topology. Since it is not possible to include treatments of all of these topics in our surv ...
minimalrevised.pdf
... This is exactly the order of Sh−1 S 0 . Therefore X is homeomorphic to Sh−1 S 0 . Theorem 2.13. Any space with the same homotopy groups as S n has at least 2n + 2 points. Moreover, Sn S 0 is the unique space with 2n + 2 points with this property. Proof. The case n = 1 is trivial. In the other cases, ...
... This is exactly the order of Sh−1 S 0 . Therefore X is homeomorphic to Sh−1 S 0 . Theorem 2.13. Any space with the same homotopy groups as S n has at least 2n + 2 points. Moreover, Sn S 0 is the unique space with 2n + 2 points with this property. Proof. The case n = 1 is trivial. In the other cases, ...
http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf
... equivalence between C and a full subcategory of PSh(C). 1.2. Definition of a topos. Let us define site to be pair consisting of a small category C, together with a full subcategory of PSh(C), usually denoted by Sh(C), which is closed under isomorphisms (i.e., if X → Y ∈ PSh(C) is an isomorphism, the ...
... equivalence between C and a full subcategory of PSh(C). 1.2. Definition of a topos. Let us define site to be pair consisting of a small category C, together with a full subcategory of PSh(C), usually denoted by Sh(C), which is closed under isomorphisms (i.e., if X → Y ∈ PSh(C) is an isomorphism, the ...
Homotopy type theory
In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intensional type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.This includes, among other lines of work, the construction of homotopical and higher-categorical models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the formalization of each of these in computer proof assistants.There is a large overlap between the work referred to as homotopy type theory, and as the univalent foundations project. Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also sometimes corresponds to differences in viewpoint and emphasis. As such, this article may not represent the views of all researchers in the fields equally.