On the Weak Homotopy Type of Étale Groupoids

1997 ◽  
pp. 147-156
Author(s):  
Ieke Moerdijk
Keyword(s):  
Homotopy Type ◽  
Weak Homotopy Type ◽  
Weak Homotopy

On simple maps

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Homotopy Type ◽  
Geometric Realization ◽  
Homotopy Equivalence ◽  
Mapping Cylinder ◽  
Simplicial Sets ◽  
Weak Homotopy ◽  
Formal Properties ◽  
Weak Homotopy Equivalence

This chapter deals with simple maps of finite simplicial sets, along with some of their formal properties. It begins with a discussion of simple maps of simplicial sets, presenting a proposition for the conditions that qualify a map of finite simplicial sets as a simple map. In particular, it considers a simple map as a weak homotopy equivalence. Weak homotopy equivalences have the 2-out-of-3 property, which combines the composition, right cancellation and left cancellation properties. The chapter proceeds by defining some relevant terms, such as Euclidean neighborhood retract, absolute neighborhood retract, Čech homotopy type, and degeneracy operator. It also describes normal subdivision of simplicial sets, geometric realization and subdivision, the reduced mapping cylinder, how to make simplicial sets non-singular, and the approximate lifting property.

The smooth case

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Homotopy Type ◽  
Unit Vector ◽  
Smooth Case ◽  
Open Set ◽  
Birational Invariant

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

scholarly journals Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

scholarly journals Homotopy type of posets and lattice complementation

1981 ◽  
Vol 30 (1) ◽  
pp. 90-100 ◽  
Author(s):  
Anders Björner
Keyword(s):  
Homotopy Type

scholarly journals Spherical posets from commuting elements

2018 ◽  
Vol 21 (4) ◽  
pp. 593-628 ◽  
Author(s):  
Cihan Okay
Keyword(s):  
Homotopy Type ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Universal Cover ◽  
Homotopy Equivalent ◽  
Classifying Space ◽  
Partially Ordered ◽  
Abelian Subgroups

AbstractIn this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where {2r\geq 4} is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

scholarly journals Spaces which Invert Weak Homotopy Equivalences

2018 ◽  
Vol 62 (2) ◽  
pp. 553-558
Author(s):  
Jonathan Ariel Barmak
Keyword(s):  
Empty Space ◽  
Weak Equivalence ◽  
Homotopy Equivalence ◽  
Homotopy Classes ◽  
Cw Complex ◽  
Cw Complexes ◽  
Weak Homotopy ◽  
Weak Homotopy Equivalence

AbstractIt is well known that if X is a CW-complex, then for every weak homotopy equivalence f : A → B, the map f* : [X, A] → [X, B] induced in homotopy classes is a bijection. In fact, up to homotopy equivalence, only CW-complexes have that property. Now, for which spaces X is f* : [B, X] → [A, X] a bijection for every weak equivalence f? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.

Sign in / Sign up

Export Citation Format

Share Document