All Questions
Tagged with infinity-categories chain-complexes
9 questions
3
votes
1
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152
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Homotopy coherent transformation and totalization
Let $C$ be the category of chain complexes over a field $F$ and $C^\prime$ be the subcategory of chain complexes with zero differentials. If $X:I\to C$ is a functor, there is an induced "homology&...
- 12.9k
2
votes
0
answers
117
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Hypercube of chain complexes as functor from (Δ^1 )^n to ∞-category of chain complexes
A hypercube of chain complexes consists of $\mathbb{Z}$-graded vector spaces $C_\epsilon$ for $\epsilon\in\{0,1\}^n$ and maps $D_{\epsilon,\epsilon^\prime}:C_{\epsilon}\to C_{\epsilon^\prime}$ for $\...
- 673
4
votes
1
answer
257
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Homotopy totalization and chains - reference
Simple case. Take $X_{\bullet}$ a cosimplicial space. Is it true that the chain complex of its homotopy totalization is quasi-isomorphic to the homotopy totalization of its chain complex? Because of ...
- 2,152
4
votes
1
answer
230
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Gluing a manifold along its boundary, via chain complexes
Given closed oriented $n$-manifolds $M, M', M''$ and bordisms $W, W'$ with $\partial W = M \sqcup - M'$ and $\partial W' = M' \sqcup - M''$, we can collar-glue them to obtain a bordism from $M$ to $M''...
- 898
2
votes
0
answers
310
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Invariants of objects in $\operatorname{Ch}(\mathrm{Ab})$ up to chain homotopy
$\newcommand\Ab{\mathrm{Ab}}\newcommand\ab{\mathrm{ab}}\DeclareMathOperator\Ch{Ch}\DeclareMathOperator\Kom{Kom}\newcommand\ho{\mathrm{ho}}$Let $\Ab$ be the category of finitely generated abelian ...
- 12.9k
6
votes
2
answers
963
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Projective objects in the derived category of chain complexes
I have been trying to understand projective objects in the derived category of chain complexes of modules over a ring.
If we stick to the category of chain complexes, the only projective objects are ...
- 15.8k
4
votes
1
answer
191
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Homotopy coherent space maps induces homotopy coherent chain complex morphisms
It is an elementary fact that a map $f:X \to Y$ between spaces induces a chain complex morphism $f_* : C_*(X) \to C_*(Y) $. This allows one to transfer 1-category theoretic arguments from spaces to ...
- 15.8k
23
votes
3
answers
4k
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how to make the category of chain complexes into an $\infty$-category
I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes.
Has anyone ever written down ...
- 56.9k
6
votes
0
answers
180
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(Reference Request) Tensor product of chain complexes in terms of strict $\infty$-categories
(note: this question is essentially a reference request for the tensor product described at the end. the rest is context)
It is well known that the category of chain complexes (in positive degree, ...
- 42.4k