# Tagged Questions

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### Homology of mapping telescope

It is stated here http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf that if $X$ is an increasing union of the type $X=\bigcup_{i \in I}X_i$ (where $X_i \subset X_{i+1}$), then we have an ...
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### Homological algebra (homotopical approach)

I have gone through a couple of courses in homological algebra, in the context of derived functors, abelian categories,... Now I would like to watch it from another perspective: my main interest is ...
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### chain homotopy equivalence between mapping cone complexes

Given continuous maps $f_i : X_i \to Y_i$ ($i=1, 2$) we may consider the singular chain cocomplexes $$C^n(Y_i) \oplus C^{n-1}(X_i)$$ with boundary operator:  (u^n, v^{n-1}) \mapsto (-\delta u^n, ...
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### $[X,F] \to [X,E] \to [X,B]$ is exact sequence of pointed sets

how to show: if $F \to E \to B$ is a fibration then for any space $X$ the sequence $[X,F] \to [X,E] \to [X,B]$ is exact sequence of pointed sets. any hints, thanx.
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### Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic) [closed]

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
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### Equivalence and complexes homotopically-minimal

Let $A$ and $B$ be two finite-dimensional algebras over a field $k$ and $G\colon \mathcal{K}^{-}(\mathcal{P}_A) \to \mathcal{K}^{-}(\mathcal{P}_B)$ be an (triangulated) equivalence. By [Krause-05], a ...
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### Cylinder object in the model category of chain complexes

Let $\text{Ch}⁺(R)$ be the category of non-negative chain complexes of $R$-modules where $R$ is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex ...
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### How are injective model structures cofibrantly generated?

I have a question about the injective model structure on functor categories. As background : If $\mathcal{M}$ is a combinatorial model category and $\mathcal{C}$ is a small category, then there are ...