scholarly journals Abstract homotopy theory and generalized sheaf cohomology

1973 ◽  
Vol 186 ◽  
pp. 419-419 ◽  
Author(s):  
Kenneth S. Brown
Keyword(s):  
Homotopy Theory ◽  
Sheaf Cohomology ◽  
Abstract Homotopy Theory

On the homotopy theory of sheaves of simplicial groupoids

1996 ◽  
Vol 120 (2) ◽  
pp. 263-290 ◽  
Author(s):  
André Joyal ◽  
Myles Tierney
Keyword(s):  
Homotopy Theory ◽  
Characteristic Classes ◽  
Spectral Sequences ◽  
Classifying Spaces ◽  
Sheaf Cohomology ◽  
Natural Context ◽  
Simplicial Sheaves ◽  
Cohomology Operations

The aim of this paper is to contribute to the foundations of homotopy theory for simplicial sheaves, as we believe this is the natural context for the development of non-abelian, as well as extraordinary, sheaf cohomology.In [11] we began constructing a theory of classifying spaces for sheaves of simplicial groupoids, and that study is continued here. Such a theory is essential for the development of basic tools such as Postnikov systems, Atiyah-Hirzebruch spectral sequences, characteristic classes, and cohomology operations in extraordinary cohomology of sheaves. Thus, in some sense, we are continuing the program initiated by Illusie[7], Brown[2], and Brown and Gersten[3], though our basic homotopy theory of simplicial sheaves is different from theirs. In fact, the homotopy theory we use is the global one of [10]. As a result, there is some similarity between our theory and the theory of Jardine[8], which is also partially based on [10]

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.

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