scholarly journals Completions in abstract homotopy theory

1970 ◽  
Vol 147 (2) ◽  
pp. 573-573 ◽  
Author(s):  
Alex Heller
Keyword(s):  
Homotopy Theory ◽  
Abstract Homotopy Theory

scholarly journals Algebraic K-theory and abstract homotopy theory

2011 ◽  
Vol 226 (4) ◽  
pp. 3760-3812 ◽  
Author(s):  
Andrew J. Blumberg ◽  
Michael A. Mandell
Keyword(s):  
Homotopy Theory ◽  
K Theory ◽  
Abstract Homotopy Theory

Daniel Quillen, the father of abstract homotopy theory

2013 ◽  
Vol 11 (3) ◽  
pp. 479-491
Author(s):  
Jean-Claude Thomas ◽  
Micheline Vigué-Poirrier
Keyword(s):  
Homotopy Theory ◽  
Short Paper ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Abstract Homotopy Theory

AbstractIn this short paper we try to describe the fundamental contribution of Quillenin the development of abstract homotopy theory and we explain how he uses this theory to lay the foundations of rational homotopy theory.

scholarly journals Chern–Weil forms and abstract homotopy theory

2013 ◽  
Vol 50 (3) ◽  
pp. 431-468 ◽  
Author(s):  
Daniel S. Freed ◽  
Michael J. Hopkins
Keyword(s):  
Homotopy Theory ◽  
Abstract Homotopy Theory

scholarly journals Structuralism, Invariance, and Univalence

2018 ◽  
Author(s):  
Steve Awodey
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Mathematical Practice ◽  
New Approach ◽  
Constructive Type Theory ◽  
Homotopy Type Theory ◽  
Computational Implementation ◽  
Geometric Content ◽  
Abstract Homotopy Theory ◽  
New System

The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the univalence axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. This powerful addition to homotopy type theory gives the new system of foundations a distinctly structural character.

RELATIONAL TOPOLOGY AS THE FOUNDATION FOR QUANTUM GRAVITY

2005 ◽  
Vol 20 (17n18) ◽  
pp. 1261-1269 ◽  
Author(s):  
LOUIS CRANE
Keyword(s):  
General Relativity ◽  
Algebraic Geometry ◽  
Quantum Theory ◽  
Homotopy Theory ◽  
Categorical Approach ◽  
New Approach ◽  
Spin Foam Models ◽  
Spin Foam ◽  
Theory Of Gravitation ◽  
Abstract Homotopy Theory

We propose a new approach to the quantum theory of gravitation, in which the point sets of regions of spacetime are replaced by objects in a category. The objects can be constructed as limits of spin foam models, thus directly connecting to approximations to general relativity which have already been studied, and leading to a mathematically natural interpretation of the meaning of the triangulations in the spin foam models. The physical motivation for our proposal is a connection between ideas about the limited amount of information which can flow from one region to another in General Relativity, an analysis of the problem of the infinities in QFT, and the relational or categorical approach to topology in abstract homotopy theory and algebraic geometry.

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