Math 8301, Manifolds and Topology Homework 3
... Show that if α and β are any paths in X, the definition (α · β)(t) = m(α(t), β(t)) is homotopy invariant, in the sense that [α] ∗ [β] = [α ∗ β] is well-defined on homotopy classes of paths. 4. Show that the product of the previous problem satisfies an interchange law (α · β) ∗ (γ · δ) = (α ∗ γ) · (β ...
... Show that if α and β are any paths in X, the definition (α · β)(t) = m(α(t), β(t)) is homotopy invariant, in the sense that [α] ∗ [β] = [α ∗ β] is well-defined on homotopy classes of paths. 4. Show that the product of the previous problem satisfies an interchange law (α · β) ∗ (γ · δ) = (α ∗ γ) · (β ...
An introduction to equivariant homotopy theory Groups Consider
... Theorem. Let G be a compact Lie group. The homotopy theory of rational G-spectra is modeled by differential graded modules over a rational DGA whose homology is isomorphic to the homotopy of the rational stable orbit category OS G. Proof. Q - G -spectra 'Q Mod-(HQ ∧ OS G) 'Q d.g. Mod- Θ(HQ ∧ OS G) ...
... Theorem. Let G be a compact Lie group. The homotopy theory of rational G-spectra is modeled by differential graded modules over a rational DGA whose homology is isomorphic to the homotopy of the rational stable orbit category OS G. Proof. Q - G -spectra 'Q Mod-(HQ ∧ OS G) 'Q d.g. Mod- Θ(HQ ∧ OS G) ...
Notes
... SL or construct. We need to use a weaker form of or defined by Gödel and Kolmogorov. They use ∼∼ (α | ∼ α) for α | ∼ α where ∼ α is defined to be α → void. ...
... SL or construct. We need to use a weaker form of or defined by Gödel and Kolmogorov. They use ∼∼ (α | ∼ α) for α | ∼ α where ∼ α is defined to be α → void. ...
Lecture 10 Notes
... philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important factors in the technology of knowledge creation. For PC we have a clear computational semantics for understanding the logical ...
... philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important factors in the technology of knowledge creation. For PC we have a clear computational semantics for understanding the logical ...
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... Definition Suppose that X and Y are topological spaces and f : X → Y is a continuous map. If there exists a continuous map g : Y → X such that f ◦ g ' idY (i.e. f ◦ g is homotopic to the identity mapping on Y ), and g ◦ f ' idX , then f is a homotopy equivalence. This homotopy equivalence is sometim ...
... Definition Suppose that X and Y are topological spaces and f : X → Y is a continuous map. If there exists a continuous map g : Y → X such that f ◦ g ' idY (i.e. f ◦ g is homotopic to the identity mapping on Y ), and g ◦ f ' idX , then f is a homotopy equivalence. This homotopy equivalence is sometim ...
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... the inverse of (a morphism represented by) a path being (the homotopy class of) the “reverse” path. Notice that for x ∈ X, the group of automorphisms of x is the fundamental group of X with basepoint x, HomΠ1 (X) (x, x) = π1 (X, x) . Definition 2. Let f : X → Y be a continuous function between two t ...
... the inverse of (a morphism represented by) a path being (the homotopy class of) the “reverse” path. Notice that for x ∈ X, the group of automorphisms of x is the fundamental group of X with basepoint x, HomΠ1 (X) (x, x) = π1 (X, x) . Definition 2. Let f : X → Y be a continuous function between two t ...
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... 3. We will in due course explore the notion of constructive “truth” or evidence. We will see that we can’t decide whether there is evidence for a given proposition, i.e. whether a programming task is solvable. 4. The notion of evidence/truth depends on a theory of types and programs which we are gra ...
... 3. We will in due course explore the notion of constructive “truth” or evidence. We will see that we can’t decide whether there is evidence for a given proposition, i.e. whether a programming task is solvable. 4. The notion of evidence/truth depends on a theory of types and programs which we are gra ...
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.