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Results tagged with homotopy-theory
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user 76299
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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The homotopy inverse on Quillen's $S^{-1}S$ construction
Suppose $S$ is a symmetric monoidal groupoid. Take Quillen and Grayson's $S^{-1}S$-construction, which is a symmetric monoidal category with objects given by pairs $(m,n)$ and maps given by compositio …
- 2,303
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Realizing the 0-th Postnikov truncation of a spectrum in the category of orthogonal/symmetri...
Suppose $E$ is a connective spectrum, then there exists a natural map in the stable homotopy category $\mathcal{SHC}$, $E \rightarrow P_0 E$, called the $0$-th Postnikov truncation, which is character …
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Commuting homotopy colimits and arbitrary products in spaces
I will answer my own question, in hope that it is helpful to someone.
Given a functor $X:D \rightarrow Spc$ of $\infty$-categories, we can take the unstraightening of $X$ (the appropriate generalizati …
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Does Grayson/Quillen's "pre group completion" have a universal property?
It is the classifying category for the left action of $C$ on its product $C \times C$.
Let me elaborate on this a bit further. Let $C$ be an $E_\infty$-monoid, for example represented by a symmetric m …
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Commuting homotopy colimits and arbitrary products in spaces
Let $X : D \rightarrow Spc$ be a diagram with values in the $\infty$-category of spaces and $I$ some (discrete) set, not necessarily finite. ($D$ can be a 1-category if that makes statements easier, b …
- 2,303