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Results tagged with homotopy-theory
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user 118688
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
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Understanding the definition of left homotopy as given in Quillen’s Homotopical algebra book
Given two topological spaces $X,Y$, and two maps $f,g:X\rightarrow Y$, there is a notion of homotopy between $f$ and $g$. It is given by a continuous map $H:X\times I\rightarrow Y$ such that the compo …
- 30.3k
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On fundamental groupoid of fundamental groupoid
Given a topological space $X$, we have the notion of the fundamental groupoid $\Pi_1(X)$.
Here, the fundamental groupoid $\Pi_1(X)$ is made into a topological groupoid giving a topology on the morph …
- 12.3k
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References for Homotopy transfer problem
I am trying to read Algebra+homotopy=operad by Bruno Vallette.
Consider the following set up :
chain complexes $(A,d_A),(H,d_H)$,
a degree $1$ morphism of chain complexes $h:(A,d_A)\rightarrow (A,d_A …
- 6,023
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Homotopy for functors
I am reading this paper on Homotopy for functors by Ming-Jung
Lee.
The author gives a definition (at the beginning of section $3$) as follows :
Let $\varphi,\varphi':\Lambda\rightarrow \Gamma$ …
- 1,215
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Roadmap to understand gerbe in the sense of Lurie’s Higher Topos Theory
Definition $7.2.2.20$ : Let $\mathfrak{X} $ be an $\infty$-topos. An $n$-gerbe on $\mathfrak{X}$ is an object in $\mathfrak{X}$ which is $n$-connective and $n$-truncated.
Above is the definition …
- 6,023
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answer
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Motivation for classifying vector bundles
The statement I am familiar with regarding classification of vector bundles is :
Given a paracompact space $X$. The set of isomorphism classes of rank $n$ vector bundles over $X$ is in bijective c …
- 666
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Homotopy for functors
The author means there is a zigzag of natural transformations. That is, "a natural transformation between $\varphi_i$ and $\varphi_{i+1}$" is intended to be nonspecific as to the direction of the tr …
- 66.8k