Embeddings vs. Homeomorphisms (Lecture 13)
... Rn to itself: Theorem 1 is equivalent to the assertion that the inclusion between these topological spaces is a weak homotopy equivalence. Remark 3. We can also define simplicial sets which parametrize PL embeddings and PL homeomorphisms from Rn to itself. Theorem 1 continues to hold in this case, u ...
... Rn to itself: Theorem 1 is equivalent to the assertion that the inclusion between these topological spaces is a weak homotopy equivalence. Remark 3. We can also define simplicial sets which parametrize PL embeddings and PL homeomorphisms from Rn to itself. Theorem 1 continues to hold in this case, u ...
23 Introduction to homotopy theory
... overview I want to explain certain features and constructions with these categories which will become relevant in the seminar. One needs 1-categories or model categories in order to capture theses objects on a technical level. In this overview I want to explain how one can work with spaces and spect ...
... overview I want to explain certain features and constructions with these categories which will become relevant in the seminar. One needs 1-categories or model categories in order to capture theses objects on a technical level. In this overview I want to explain how one can work with spaces and spect ...
Higher algebra and topological quantum field theory
... 4. (EQUIV) on each set Mori (A), there is an equivalence relation ∼ compatible with the source and target maps. The induced equivalence relation on Mori (x, y) is also denoted ∼. 5. (COMP) for any 0 < i ≤ n and any three i − 1-morphisms u, v and w sharing the same source and target, there is a well ...
... 4. (EQUIV) on each set Mori (A), there is an equivalence relation ∼ compatible with the source and target maps. The induced equivalence relation on Mori (x, y) is also denoted ∼. 5. (COMP) for any 0 < i ≤ n and any three i − 1-morphisms u, v and w sharing the same source and target, there is a well ...
Classifying Spaces - School of Mathematics and Statistics
... • If H is a closed subgroup of G then BH −→ BG is a fibration with fibre G/H. • If N is a closed normal subgroup of G then BG −→ B(G/H) is a fibration with fibre BH. ...
... • If H is a closed subgroup of G then BH −→ BG is a fibration with fibre G/H. • If N is a closed normal subgroup of G then BG −→ B(G/H) is a fibration with fibre BH. ...
Posets and homotopy
... points. In more recent literature they are called beat points. A point x ∈ P is called an up-beat point under y iff y is the unique minimal element of {z : z > x}. Down-beat points are defined dually. Removing beat points of P , one by one, we arrive at a space P C that has no beat points. It is cal ...
... points. In more recent literature they are called beat points. A point x ∈ P is called an up-beat point under y iff y is the unique minimal element of {z : z > x}. Down-beat points are defined dually. Removing beat points of P , one by one, we arrive at a space P C that has no beat points. It is cal ...
THE WEAK HOMOTOPY EQUIVALENCE OF Sn AND A SPACE
... has not seen homotopy groups may become quickly lost in the forest of jargon-filled mathematics that’s quickly approaching. Fear not, reader! Though I will not go into homotopy groups (it would take far too much space and would be bothersome for those readers who are already familiar with the topic) ...
... has not seen homotopy groups may become quickly lost in the forest of jargon-filled mathematics that’s quickly approaching. Fear not, reader! Though I will not go into homotopy groups (it would take far too much space and would be bothersome for those readers who are already familiar with the topic) ...
Dependent Types In Lambda Cube
... If we have ∀x : N at.P (x), we can read it (intuitionistic way) as ”I have a method for constructing an object of the type P (x) using any given object x of the type N at”. So here, it is more like a generalization of the ordinary function type. And that is the reason, why the term ”Dependent functi ...
... If we have ∀x : N at.P (x), we can read it (intuitionistic way) as ”I have a method for constructing an object of the type P (x) using any given object x of the type N at”. So here, it is more like a generalization of the ordinary function type. And that is the reason, why the term ”Dependent functi ...
Bounded Functional Interpretation
... urging a shift of attention from the obtaining of precise witnesses to the obtaining of bounds for the witnesses. One of the main advantages of working with the extraction of bounds is that the non-computable mathematical objects whose existence is claimed by various ineffective principles can somet ...
... urging a shift of attention from the obtaining of precise witnesses to the obtaining of bounds for the witnesses. One of the main advantages of working with the extraction of bounds is that the non-computable mathematical objects whose existence is claimed by various ineffective principles can somet ...
Pointed spaces - home.uni
... Example 2.2. U : Gp → Set the underlying set functor, which associates to a group its underlying set. Remark 2.3. The functor U forgets the group structure of G. For this reason, a functors of that type is often called a forgetful functor. Example 2.4. F : Set → Gp the free group functor. Remark 2.5 ...
... Example 2.2. U : Gp → Set the underlying set functor, which associates to a group its underlying set. Remark 2.3. The functor U forgets the group structure of G. For this reason, a functors of that type is often called a forgetful functor. Example 2.4. F : Set → Gp the free group functor. Remark 2.5 ...
First-Order Logic with Dependent Types
... First-Order Logic with Dependent Types Florian Rabe florian@cs.cmu.edu ...
... First-Order Logic with Dependent Types Florian Rabe florian@cs.cmu.edu ...
Subset Types and Partial Functions
... deal with the case when X 'A Y contains free variables. Note that variables occurring free in a predicate P in a subset type A|P are considered part of the free variables of that type. The occurrence of X that is replaced cannot be beneath a λ-binding of any symbol x which occurs free in X 'A Y and ...
... deal with the case when X 'A Y contains free variables. Note that variables occurring free in a predicate P in a subset type A|P are considered part of the free variables of that type. The occurrence of X that is replaced cannot be beneath a λ-binding of any symbol x which occurs free in X 'A Y and ...
- Free Documents
... Thierry Coquand Equality and dependent type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nicola Gambino The Univalence Axiom and Function Extensionality . . . . . . . . . . . . . . . . . . Peter Aczel Logicenriched type theory . . . . . . . . . . . . . . . . . . . . . . . . ...
... Thierry Coquand Equality and dependent type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nicola Gambino The Univalence Axiom and Function Extensionality . . . . . . . . . . . . . . . . . . Peter Aczel Logicenriched type theory . . . . . . . . . . . . . . . . . . . . . . . . ...
PDF (216 KB)
... proofs of squash–stable propositions into constructive ones. This principle is valid in a standard type theory semantics if we consider it in the classical metatheory. Therefore this principle does not destroy the constructive nature of type theory in the sense that we can always extract a witness t ...
... proofs of squash–stable propositions into constructive ones. This principle is valid in a standard type theory semantics if we consider it in the classical metatheory. Therefore this principle does not destroy the constructive nature of type theory in the sense that we can always extract a witness t ...
The homotopy category is a homotopy category. Arne Str¢m
... satisfying the conditions of Proposition 7 with respect to and define ...
... satisfying the conditions of Proposition 7 with respect to and define ...
Category Theory for Dummies (I)
... – Objects are vertices. – Arrows are paths (sequences of edges). ...
... – Objects are vertices. – Arrows are paths (sequences of edges). ...
SAM III General Topology
... written as π1 (X , x), is the group of automorphisms of x in the fundamental groupoid Π1 X of the space. Show that in a path-connected space, fundamental groups at any two distinct points are isomorphic to each other. Deduce this from a general observation about an abstract groupoid. ...
... written as π1 (X , x), is the group of automorphisms of x in the fundamental groupoid Π1 X of the space. Show that in a path-connected space, fundamental groups at any two distinct points are isomorphic to each other. Deduce this from a general observation about an abstract groupoid. ...
Orbit Projections as Fibrations
... If p : E → B is a Hurewicz (Serre) fibration and B is path connected (and all fibers are CW complexes), then any two fibers belong to the same homotopy type. 2.2. Compact transformation groups. Let G be a compact Lie group and let M be a G-manifold, i.e., M is a paracompact Hausdorff smooth manifold ...
... If p : E → B is a Hurewicz (Serre) fibration and B is path connected (and all fibers are CW complexes), then any two fibers belong to the same homotopy type. 2.2. Compact transformation groups. Let G be a compact Lie group and let M be a G-manifold, i.e., M is a paracompact Hausdorff smooth manifold ...
S1-Equivariant K-Theory of CP1
... If f is a homotopy equivalence, then f ∗ is a group isomorphism, and id∗X = idKG (X ) . Thus, KG is a homotopy invariant contravariant functor from the category of compact Hausdorff G -spaces to the category of abelian groups. ...
... If f is a homotopy equivalence, then f ∗ is a group isomorphism, and id∗X = idKG (X ) . Thus, KG is a homotopy invariant contravariant functor from the category of compact Hausdorff G -spaces to the category of abelian groups. ...
connections to higher type Recursion Theory, Proof-Theory
... Church's Thesis, provided that its use is not mathematically misleading. Namely, the philosophical point raised by the Thesis is surely crucial, but do we really need it when working out results ? In case a new system for general computations is proposed, it is then better to check carefully whether ...
... Church's Thesis, provided that its use is not mathematically misleading. Namely, the philosophical point raised by the Thesis is surely crucial, but do we really need it when working out results ? In case a new system for general computations is proposed, it is then better to check carefully whether ...
Geometry and Topology, Lecture 4 The fundamental group and
... Ignoring the base point issue, the fundamental group is a functor π1 : {topological spaces} → {groups}. π1 (X , x) is the geometrically defined group of ‘homotopy’ classes [ω] of ‘loops at x ∈ X ’, continuous maps ω : S 1 → X such that ω(1) = x ∈ X . A continuous map f : X → Y induces a morphism of ...
... Ignoring the base point issue, the fundamental group is a functor π1 : {topological spaces} → {groups}. π1 (X , x) is the geometrically defined group of ‘homotopy’ classes [ω] of ‘loops at x ∈ X ’, continuous maps ω : S 1 → X such that ω(1) = x ∈ X . A continuous map f : X → Y induces a morphism of ...
The Arithmetic Square (Lecture 32)
... Definition 2. Let f : X → Y be a map of topological spaces. We say that f is a rational homotopy equivalence if it induces an isomorphism on rational cohomology H∗ (Y ; Q) → H∗ (X; Q) (this is equivalent to the assertion that f induces an isomorphism on rational homology). We say that a space Z is r ...
... Definition 2. Let f : X → Y be a map of topological spaces. We say that f is a rational homotopy equivalence if it induces an isomorphism on rational cohomology H∗ (Y ; Q) → H∗ (X; Q) (this is equivalent to the assertion that f induces an isomorphism on rational homology). We say that a space Z is r ...
12 Towards a Theory of Document Structure
... If the document is built using a WYSIWYG structure editor, one is tempted to think that there is little need for abstract syntax. Only the concrete syntax of the document is presented and the editing is done by changing the concrete syntax. However, the editing commands (like navigation, insertions ...
... If the document is built using a WYSIWYG structure editor, one is tempted to think that there is little need for abstract syntax. Only the concrete syntax of the document is presented and the editing is done by changing the concrete syntax. However, the editing commands (like navigation, insertions ...
7. Homotopy and the Fundamental Group
... The properties of a topological space that we have developed so far have depended on the choice of topology, the collection of open sets. Taking a different tack, we introduce a different structure, algebraic in nature, associated to a space together with a choice of base point (X, x0 ). This structur ...
... The properties of a topological space that we have developed so far have depended on the choice of topology, the collection of open sets. Taking a different tack, we introduce a different structure, algebraic in nature, associated to a space together with a choice of base point (X, x0 ). This structur ...
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.