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Results tagged with homotopy-theory
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user 128371
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.
4
votes
1
answer
436
views
detecting weak equivalences in a simplicial model category II
The question is related to the question: detecting weak equivalences in a simplicial model category
Suppose that we have a simplicial model category $M$ and denote by $M^{f}$ the full simplicial s …
- 511
5
votes
1
answer
675
views
Localization of a model category
Let $M$ be a very nice model category (cofibrantly generated, combinatorial or cellular and left proper simplicial model category). Let $f: X\rightarrow Y$ and $g: X\rightarrow Z $ be two morphisms i …
- 511
0
votes
1
answer
183
views
detecting weak equivalences in a simplicial model category
Suppose that we have a simplicial model category $M$. The simplicial enrichment will be denoted by $map_{M}$. Let $f:A\rightarrow B$ be a morphism in the category $M$ such that $A$ is cofibrant. Sup …
- 511
4
votes
1
answer
331
views
Simplicial model categories and simplicial equivalence
Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow H …
- 511
2
votes
Accepted
Simplicial model categories and simplicial equivalence
Since there is an answer to the question, I think I should write it down.
There is a simple counterexample to my question: Let $M=N=sSet$ the standard model category of simplicial sets. Let $ex^{\in …
3
votes
1
answer
185
views
Spherical objects and K-theory
My question goes as follows: given a ring $R$ (with $1\neq 0$). Define $\mathbf{Perf}_{R}$ the the category of Prefect complexes over $R$. I want to prove that the Waldhausen $K$-theory of the categor …
- 511
4
votes
1
answer
193
views
Split cofibrations up to quasi-isomorphism
$R$ a ring $(1\neq 0)$, $\mathbf{Perf}(R)$ is the category of perfect complexes (of right $R$-modules).
Suppose that $A_{\bullet}\rightarrow B_{\bullet}\rightarrow B_{\bullet}/A_{\bullet}$ a short e …
- 511