L19 Abstract homotopy theory
... A left Quillen functor P : C → D determines a derived functor LP : ho C → ho D as follows. For any object X in C the morphism ∅ → X may be factored as a cofibration ∅ → QX followed by a trivial fibration QX → X. By the definition of a model category, Q is a functor. The left derived functor LP is de ...
... A left Quillen functor P : C → D determines a derived functor LP : ho C → ho D as follows. For any object X in C the morphism ∅ → X may be factored as a cofibration ∅ → QX followed by a trivial fibration QX → X. By the definition of a model category, Q is a functor. The left derived functor LP is de ...
![THE HIGHER HOMOTOPY GROUPS 1. Definitions Let I = [0,1] be](http://s1.studyres.com/store/data/001160219_1-57248e6267c8b98f385fc9eb1b30c0a7-300x300.png)
THE HIGHER HOMOTOPY GROUPS 1. Definitions Let I = [0,1] be
... section 4. Nonetheless, here are some examples with some partial proofs. 1. πn (Rm ) ∼ = 0 for any n, m ∈ N. This follows from the fact that Rm is homotopy equivalent to a point (and clearly a point has trivial homotopy groups). 2. πn (S n+k ) ∼ = 0 for all k ≥ 1. To prove this, we shall rely on a t ...
... section 4. Nonetheless, here are some examples with some partial proofs. 1. πn (Rm ) ∼ = 0 for any n, m ∈ N. This follows from the fact that Rm is homotopy equivalent to a point (and clearly a point has trivial homotopy groups). 2. πn (S n+k ) ∼ = 0 for all k ≥ 1. To prove this, we shall rely on a t ...
The superjump in Martin-Löf type theory
... Universes of types were introduced into constructive type theory by MartinLöf [11]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say C. The universe then ’reflects’ C. Several gadgets for generating u ...
... Universes of types were introduced into constructive type theory by MartinLöf [11]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say C. The universe then ’reflects’ C. Several gadgets for generating u ...
![Propositions as [Types] - Research Showcase @ CMU](http://s1.studyres.com/store/data/005730189_1-e85fa7d3c7cfa08d9a3b8e96a27d7888-300x300.png)
Propositions as [Types] - Research Showcase @ CMU
... by the fact that in general the result of successive pullbacks along arrows g : B → C and f : A → B is only isomorphic to the pullback along the composition g ◦ f , whereas for a completely water-tight interpretation equality is required. There are several standard ways of resolving this problem, mo ...
... by the fact that in general the result of successive pullbacks along arrows g : B → C and f : A → B is only isomorphic to the pullback along the composition g ◦ f , whereas for a completely water-tight interpretation equality is required. There are several standard ways of resolving this problem, mo ...
CLASSIFYING SPACES OF MONOIDS – APPLICATIONS IN
... theory have recently found applications in geometric combinatorics [W-Z-Ž], [Z-Ž]. It is believed that this is just the beginning and that more and more applications will follow. We give two examples of how this theory can be applied in problems of relevance for combinatorics. In Proposition 9 we ...
... theory have recently found applications in geometric combinatorics [W-Z-Ž], [Z-Ž]. It is believed that this is just the beginning and that more and more applications will follow. We give two examples of how this theory can be applied in problems of relevance for combinatorics. In Proposition 9 we ...
Fibrations handout
... homology and cohomology of F , and I want to define that. ΩB is an “H-space”: that is, it’s a topological group up to homotopy. What I mean is this: there is a product ΩB × ΩB → ΩB, defined by concatenating loops, which is associative and commutative up to homotopy, and which has an identity element ...
... homology and cohomology of F , and I want to define that. ΩB is an “H-space”: that is, it’s a topological group up to homotopy. What I mean is this: there is a product ΩB × ΩB → ΩB, defined by concatenating loops, which is associative and commutative up to homotopy, and which has an identity element ...
Chapter 5 Homotopy Theory
... • Let f : |K| → X be a triangulation of X, where K is a simplicial complex and |K| is a polyhedron |K|. • Then ...
... • Let f : |K| → X be a triangulation of X, where K is a simplicial complex and |K| is a polyhedron |K|. • Then ...
Semi-constr. theories - Stanford Mathematics
... given foundational frameworkthere is no difference in terms of prooftheoretical strength from the associated system based on full classical logic; in that respect they are equally justified. On the other hand, the semiconstructive systems in general have a further advantage that they admit more for ...
... given foundational frameworkthere is no difference in terms of prooftheoretical strength from the associated system based on full classical logic; in that respect they are equally justified. On the other hand, the semiconstructive systems in general have a further advantage that they admit more for ...
Lecture 02 - UWO Math Dept
... Lemma 5.1 is needed for the following result, while Lemma 5.2 is needed for almost all calculations of homotopy groups. The proof of Lemma 5.1 is sketched below, and the proof of Lemma 5.2 is an exercise. A map i : A → B is said to be a cofibration if it has the LLP wrt all trivial Serre fibrations. ...
... Lemma 5.1 is needed for the following result, while Lemma 5.2 is needed for almost all calculations of homotopy groups. The proof of Lemma 5.1 is sketched below, and the proof of Lemma 5.2 is an exercise. A map i : A → B is said to be a cofibration if it has the LLP wrt all trivial Serre fibrations. ...
Homotopy
... take g = f −1 . The converse is far from true, in general. The previous definition leads to a basic notion in algebraic topology. Definition 2.3. Two spaces X and Y are said to be homotopy equivalent (written X ' Y ) if there is a homotopy equivalence f : X → Y . Remark 2.4. By Remark 2.2, X∼ = Y =⇒ ...
... take g = f −1 . The converse is far from true, in general. The previous definition leads to a basic notion in algebraic topology. Definition 2.3. Two spaces X and Y are said to be homotopy equivalent (written X ' Y ) if there is a homotopy equivalence f : X → Y . Remark 2.4. By Remark 2.2, X∼ = Y =⇒ ...
The Anti-Foundation Axiom in Constructive Set Theories
... subsection, it is pivotal to observe that, unlike Aczel’s type of iterative sets, a strong system type is not required to have an inductive structure, i.e. there need not be an elimination rule for it. The strength of a type theory ML1 with the type constructors Π, Σ, +, I, N, N0 , N1 and one univer ...
... subsection, it is pivotal to observe that, unlike Aczel’s type of iterative sets, a strong system type is not required to have an inductive structure, i.e. there need not be an elimination rule for it. The strength of a type theory ML1 with the type constructors Π, Σ, +, I, N, N0 , N1 and one univer ...
Midterm 1 solutions
... Let φ : X → Y be a continuous map. Another continuous map ψ : Y → X is called homotopy inverse for φ if ψ ◦ φ ' IdX and φ ◦ ψ ' IdY . If there exists a homotopy inverse for φ, then φ is called homotopy equivalence. In that case, we say that X is homotopy equivalent to Y (or X has the same homotopy t ...
... Let φ : X → Y be a continuous map. Another continuous map ψ : Y → X is called homotopy inverse for φ if ψ ◦ φ ' IdX and φ ◦ ψ ' IdY . If there exists a homotopy inverse for φ, then φ is called homotopy equivalence. In that case, we say that X is homotopy equivalent to Y (or X has the same homotopy t ...
Notes on wedges and joins
... simplex living in Rn with the centre at the origin. Then |∆| × [0, 1] lives in Rn × R. For each point x ∈ |∆| × [0, 1] let r(x) be the x intersection point of |K| with the ray originating at ~0 × 2 ∈ Rn × R and going through x. Then we can define the desired extension by r(x) F (x) = F̄ (r(x)). Note ...
... simplex living in Rn with the centre at the origin. Then |∆| × [0, 1] lives in Rn × R. For each point x ∈ |∆| × [0, 1] let r(x) be the x intersection point of |K| with the ray originating at ~0 × 2 ∈ Rn × R and going through x. Then we can define the desired extension by r(x) F (x) = F̄ (r(x)). Note ...
MATH 6280 - CLASS 1 Contents 1. Introduction 1 1.1. Homotopy
... homotopy groups better it the Freudenthal Suspension Theorem. Definition 1.15. A space is n–connected if πq Y = 0 for q ≤ n. Theorem 1.16 (Freudenthal Suspension Theorem). Let n ≥ 2. If Y is n–connected and X is a CW complex of dimension q, then the natural map [X, Y ] → [ΣX, ΣY ] is an isomorphism ...
... homotopy groups better it the Freudenthal Suspension Theorem. Definition 1.15. A space is n–connected if πq Y = 0 for q ≤ n. Theorem 1.16 (Freudenthal Suspension Theorem). Let n ≥ 2. If Y is n–connected and X is a CW complex of dimension q, then the natural map [X, Y ] → [ΣX, ΣY ] is an isomorphism ...
2. Homeomorphisms and homotopy equivalent spaces. (14 October
... 2. Homeomorphisms and homotopy equivalent spaces. (14 October) Problem 1. Prove that the following spaces are homeomorphic: a) the set of lines in Rn+1 passing through the origin (this topological space is called real projective space and denoted by RP n ); b) the set of hyperplanes in Rn+1 passing ...
... 2. Homeomorphisms and homotopy equivalent spaces. (14 October) Problem 1. Prove that the following spaces are homeomorphic: a) the set of lines in Rn+1 passing through the origin (this topological space is called real projective space and denoted by RP n ); b) the set of hyperplanes in Rn+1 passing ...
Lecture XI - Homotopies of maps. Deformation retracts.
... topological spaces. This would be particularly useful in the second part of the course. It also leads to a powerful notion of deformation retracts which is often useful in deciding whether two spaces have the same fundamental group. Homotopy of maps is a useful coarsening of the notion of homeomorph ...
... topological spaces. This would be particularly useful in the second part of the course. It also leads to a powerful notion of deformation retracts which is often useful in deciding whether two spaces have the same fundamental group. Homotopy of maps is a useful coarsening of the notion of homeomorph ...
SPECTRA ACCORDING TO LURIE 0.1. This introduction is probably
... and we can call the resulting functor Σ. 2.3. The above construction produces disastrous results if taken in a nonhomotopical sense. Clearly, taking the ordinary limit (or colimit) in spaces will not produce the loop space of X, nor the reduced suspension of X, in any ordinary sense. If, however, by ...
... and we can call the resulting functor Σ. 2.3. The above construction produces disastrous results if taken in a nonhomotopical sense. Clearly, taking the ordinary limit (or colimit) in spaces will not produce the loop space of X, nor the reduced suspension of X, in any ordinary sense. If, however, by ...
A HIGHER-ORDER FINE-GRAINED LOGIC FOR INTENSIONAL
... Abstract. This paper describes a higher-order logic with fine-grained intensionality (FIL). Unlike traditional Montogovian type theory, intensionality is treated as basic, rather than derived through possible worlds. This allows for fine-grained intensionality without impossible worlds. Possible wor ...
... Abstract. This paper describes a higher-order logic with fine-grained intensionality (FIL). Unlike traditional Montogovian type theory, intensionality is treated as basic, rather than derived through possible worlds. This allows for fine-grained intensionality without impossible worlds. Possible wor ...
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.