Introduction to symmetric spectra I
... all stable homotopy equivalences in SpN , we obtain a certain category called the stable homotopy category. It is equivalent to the s.h. category originally introduced by Boardman; his construction was written up by Vogt [Vo70]. This category has been extensively studied by algebraic topologists, a ...
... all stable homotopy equivalences in SpN , we obtain a certain category called the stable homotopy category. It is equivalent to the s.h. category originally introduced by Boardman; his construction was written up by Vogt [Vo70]. This category has been extensively studied by algebraic topologists, a ...
E∞-Comodules and Topological Manifolds A Dissertation presented
... in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homological obstructions among a collection of chain maps he constructed. Using lat ...
... in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homological obstructions among a collection of chain maps he constructed. Using lat ...
Proofs in theories
... These course notes are organized in four parts. In Chapters 1, 2, and 3, we shall present the basic notions of proof, theory and model used in these course notes. When presenting the notion of proof we emphasize the notion of constructivity and that of cut. When we present the notion of theory, we e ...
... These course notes are organized in four parts. In Chapters 1, 2, and 3, we shall present the basic notions of proof, theory and model used in these course notes. When presenting the notion of proof we emphasize the notion of constructivity and that of cut. When we present the notion of theory, we e ...
SIMPLICIAL APPROXIMATION Introduction
... information about the homotopy types of realizations of simplicial sets. The other is simplicial approximation. Simplicial approximation theory is a part of the classical literature [1],[2], but it was never developed in a way that was systematic enough to lead to results about model structures. Tha ...
... information about the homotopy types of realizations of simplicial sets. The other is simplicial approximation. Simplicial approximation theory is a part of the classical literature [1],[2], but it was never developed in a way that was systematic enough to lead to results about model structures. Tha ...
An Introduction to Simplicial Sets
... In this section, we define simplicial sets without providing motivation, and we describe the combinatorial data necessary for specifying a simplicial set. We then try to build intuition by bringing in the geometric notion of simplices from algebraic topology. We first define the category ∆, a visual ...
... In this section, we define simplicial sets without providing motivation, and we describe the combinatorial data necessary for specifying a simplicial set. We then try to build intuition by bringing in the geometric notion of simplices from algebraic topology. We first define the category ∆, a visual ...
SIMPLICIAL APPROXIMATION Introduction
... information about the homotopy types of realizations of simplicial sets. The other is simplicial approximation. Simplicial approximation theory is a part of the classical literature [1],[2], but it was never developed in a way that was systematic enough to lead to results about model structures. Tha ...
... information about the homotopy types of realizations of simplicial sets. The other is simplicial approximation. Simplicial approximation theory is a part of the classical literature [1],[2], but it was never developed in a way that was systematic enough to lead to results about model structures. Tha ...
MINIMAL FINITE MODELS 1. Introduction
... Therefore, his methods of reduction are not always effective and could not be applied to prove Theorems 2.13 and 4.7 mentioned above. We think that the methods and tools that we develop in this article are, in some cases, as important as the results that we obtain. We will show that these new tools, ...
... Therefore, his methods of reduction are not always effective and could not be applied to prove Theorems 2.13 and 4.7 mentioned above. We think that the methods and tools that we develop in this article are, in some cases, as important as the results that we obtain. We will show that these new tools, ...
Forking in simple theories and CM-triviality Daniel Palacín Cruz
... examples of small theories that do not eliminate hyperimaginaries; but the question is still open for small simple theories. In addition, there is an example of a theory without the strict order property which does not eliminate hyperimaginaries [18]. It is worth remarkable a theorem due to Lascar a ...
... examples of small theories that do not eliminate hyperimaginaries; but the question is still open for small simple theories. In addition, there is an example of a theory without the strict order property which does not eliminate hyperimaginaries [18]. It is worth remarkable a theorem due to Lascar a ...
ALGEBRAIC TOPOLOGY Contents 1. Informal introduction
... topology which is proved using the ideas explained above. This is only a very rough outline and much of our course will be spent trying to fill in the details of this proof. Let Dn be the n-dimensional disc. You could think of it as a solid ball of radius one having its center at the origin of Rn , ...
... topology which is proved using the ideas explained above. This is only a very rough outline and much of our course will be spent trying to fill in the details of this proof. Let Dn be the n-dimensional disc. You could think of it as a solid ball of radius one having its center at the origin of Rn , ...
Homology Theory - Section de mathématiques
... Homology theory has been around for about 115 years. It’s founding father was the french mathematician Henri Poincaré who gave a somewhat fuzzy definition of what “homology” should be in 1895. Thirty years later, it was realised by Emmy Noether that abelian groups were the right context to study hom ...
... Homology theory has been around for about 115 years. It’s founding father was the french mathematician Henri Poincaré who gave a somewhat fuzzy definition of what “homology” should be in 1895. Thirty years later, it was realised by Emmy Noether that abelian groups were the right context to study hom ...
Spectra for commutative algebraists.
... 0. Introduction. This article grew out of a short series of talks given as part of the MSRI emphasis year on commutative algebra. The purpose is to explain to commutative algebraists what spectra (in the sense of homotopy theory) are, why they were originally defined, and how they can be useful for ...
... 0. Introduction. This article grew out of a short series of talks given as part of the MSRI emphasis year on commutative algebra. The purpose is to explain to commutative algebraists what spectra (in the sense of homotopy theory) are, why they were originally defined, and how they can be useful for ...
Spectra for commutative algebraists.
... 0. Introduction. This article grew out of a short series of talks given as part of the MSRI emphasis year on commutative algebra. The purpose is to explain to commutative algebraists what spectra (in the sense of homotopy theory) are, why they were originally defined, and how they can be useful for ...
... 0. Introduction. This article grew out of a short series of talks given as part of the MSRI emphasis year on commutative algebra. The purpose is to explain to commutative algebraists what spectra (in the sense of homotopy theory) are, why they were originally defined, and how they can be useful for ...
homotopy types of topological stacks
... This theorem implies that every diagram of topological stacks has a natural weak homotopy type as a diagram of topological spaces. Furthermore, the transformation ϕ relates the given diagram of stacks with its weak homotopy type, thus allowing one to transport homotopical information back and forth ...
... This theorem implies that every diagram of topological stacks has a natural weak homotopy type as a diagram of topological spaces. Furthermore, the transformation ϕ relates the given diagram of stacks with its weak homotopy type, thus allowing one to transport homotopical information back and forth ...
minimum models: reasoning and automation
... system serves the purpose of pointing out where the main difficulty resides when building a proof system for minimum model reasoning that is sound, complete and effective, for unrestricted state descriptions and general theories. The report discusses these difficulties in detail and then goes to des ...
... system serves the purpose of pointing out where the main difficulty resides when building a proof system for minimum model reasoning that is sound, complete and effective, for unrestricted state descriptions and general theories. The report discusses these difficulties in detail and then goes to des ...
Chapter 1 LOCALES AND TOPOSES AS SPACES
... Since Tarski, 1938 it has been known that topologies – by which we mean specifically the lattices of open sets for topological spaces – can provide models for intuitionistic propositional logic. For discrete topologies, i.e. powersets, this is no surprise. Classical propositional logic can be embedd ...
... Since Tarski, 1938 it has been known that topologies – by which we mean specifically the lattices of open sets for topological spaces – can provide models for intuitionistic propositional logic. For discrete topologies, i.e. powersets, this is no surprise. Classical propositional logic can be embedd ...
¾ - Hopf Topology Archive
... topology. In algebra they often arise as the stable category of a Frobenius category ([Hel68, 4.4], [GM03, IV.3 Exercise 8]). In topology they usually appear as a full triangulated subcategory of the homotopy category of a Quillen stable model category [Hov99, 7.1]. The triangulated categories which ...
... topology. In algebra they often arise as the stable category of a Frobenius category ([Hel68, 4.4], [GM03, IV.3 Exercise 8]). In topology they usually appear as a full triangulated subcategory of the homotopy category of a Quillen stable model category [Hov99, 7.1]. The triangulated categories which ...
Proof, Sets, and Logic - Boise State University
... needs to modify the definitions of weak set picture and of the relation E on weak set pictures. July 12, 2009: I cleaned up the section on isomorphism types of wellfounded extensional relations (it is now a single section, not Old Version and New Version). Some of the stated Theorems will have proof ...
... needs to modify the definitions of weak set picture and of the relation E on weak set pictures. July 12, 2009: I cleaned up the section on isomorphism types of wellfounded extensional relations (it is now a single section, not Old Version and New Version). Some of the stated Theorems will have proof ...
SUBDIVISIONS OF SMALL CATEGORIES Let A be a
... posets can be identified with those categories A with at most one arrow between any two objects by defining x ≤ y if there is an arrow x −→ y between the objects x and y of A . In particular, we write [n] for the poset {0 < 1 · · · < n}. If we think of [n] as a finite space, then a continuous map f ...
... posets can be identified with those categories A with at most one arrow between any two objects by defining x ≤ y if there is an arrow x −→ y between the objects x and y of A . In particular, we write [n] for the poset {0 < 1 · · · < n}. If we think of [n] as a finite space, then a continuous map f ...
Subdivide.pdf
... posets can be identified with those categories A with at most one arrow between any two objects by defining x ≤ y if there is an arrow x −→ y between the objects x and y of A . In particular, we write [n] for the poset {0 < 1 · · · < n}. If we think of [n] as a finite space, then a continuous map f ...
... posets can be identified with those categories A with at most one arrow between any two objects by defining x ≤ y if there is an arrow x −→ y between the objects x and y of A . In particular, we write [n] for the poset {0 < 1 · · · < n}. If we think of [n] as a finite space, then a continuous map f ...
BP as a multiplicative Thom spectrum
... The most important chapter - besides the last one of course - is the one about spectra. The theory of spectra forms the background of the question that will be discussed. However, the theory of spectra is very complex. There are different models of spectra and the first encounter might be a little b ...
... The most important chapter - besides the last one of course - is the one about spectra. The theory of spectra forms the background of the question that will be discussed. However, the theory of spectra is very complex. There are different models of spectra and the first encounter might be a little b ...
Proof, Sets, and Logic - Department of Mathematics
... perhaps additional discussion and examples) then the following Section 4 on untyped set theory needs to be passed through with the idea that its relation to the theory of well-founded extensional relations should be exploited at every turn. Notice that the admission of urelements gets even stronger ...
... perhaps additional discussion and examples) then the following Section 4 on untyped set theory needs to be passed through with the idea that its relation to the theory of well-founded extensional relations should be exploited at every turn. Notice that the admission of urelements gets even stronger ...
Chiron: A Set Theory with Types, Undefinedness, Quotation, and
... we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may also have, but is not required to have, a proof system.) By this definition, a theory in a logic—such as Zermelo-Fraenkel (zf) set theory in first-order ord ...
... we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may also have, but is not required to have, a proof system.) By this definition, a theory in a logic—such as Zermelo-Fraenkel (zf) set theory in first-order ord ...
DIFFERENTIABLE GROUP ACTIONS ON HOMOTOPY SPHERES. II
... is useful to treat a slightly more general class of actions that we call ultrasemifree. The precise definition is in §4, the main feature being the existence of a preferred closed normal subgroup H C G, properly containing the principal isotropy subgroup (which is {1}), where G acts freely off the f ...
... is useful to treat a slightly more general class of actions that we call ultrasemifree. The precise definition is in §4, the main feature being the existence of a preferred closed normal subgroup H C G, properly containing the principal isotropy subgroup (which is {1}), where G acts freely off the f ...
Thesis Proposal: A Logical Foundation for Session-based
... Over the years, computation systems have evolved from monolithic single-threaded machines to concurrent and distributed environments with multiple communicating threads of execution, for which writing correct programs becomes substantially harder than in the more traditional sequential setting. Thes ...
... Over the years, computation systems have evolved from monolithic single-threaded machines to concurrent and distributed environments with multiple communicating threads of execution, for which writing correct programs becomes substantially harder than in the more traditional sequential setting. Thes ...
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.