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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 1st 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI polished and expanded a bit the section _[chain complex - definition - in components](http://ncatlab.org/nlab/show/chain+complex#InComponents)_

   I polished and expanded a bit the section chain complex - definition - in components

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 27th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexunder "Examples" ([here](https://ncatlab.org/nlab/show/chain+complex#Examples)) I have added mentioning dgc-algebras and sdcg-algebras as commutative monoids in categories of chain complexes of (super-)vector spaces <a href="https://ncatlab.org/nlab/revision/diff/chain+complex/30">diff</a>, <a href="https://ncatlab.org/nlab/revision/chain+complex/30">v30</a>, <a href="https://ncatlab.org/nlab/show/chain+complex">current</a>

   under “Examples” (here) I have added mentioning dgc-algebras and sdcg-algebras as commutative monoids in categories of chain complexes of (super-)vector spaces

   diff, v30, current

3. • CommentRowNumber3.
   • CommentAuthorHurkyl
   • CommentTimeDec 27th 2021
   • (edited Dec 27th 2021)
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexIs there any source that talks about chain complexes in $\infty$-categories? I.e. something that looks like sequences in a suitable $\infty$-category with vanishing conditions? I mean to contrast against doing things like: * Only ever looking at chain complexes in 1-categories; e.g. complexes in a homotopy category, or a model 1-category of chain complexes. * Switching over to some different formulation of the theory; e.g. simplicial objects, filtered objects, or gap sequences, as in Lurie's Higher Algebra I like to think of chain complexes as a recipe for constructing objects as a "total object" (e.g. by iterated cofibers), and while I sort of have a start on how to encode such things (e.g. functors out of an iterated join of copies of the discrete category $\{0, 1\}$ that send all of the $0$'s to zero objects), I'm hoping to avoid having to devise the theory from scratch.

   Is there any source that talks about chain complexes in $\infty$-categories? I.e. something that looks like sequences in a suitable $\infty$-category with vanishing conditions?

   I mean to contrast against doing things like:

   • Only ever looking at chain complexes in 1-categories; e.g. complexes in a homotopy category, or a model 1-category of chain complexes.
   • Switching over to some different formulation of the theory; e.g. simplicial objects, filtered objects, or gap sequences, as in Lurie’s Higher Algebra

   I like to think of chain complexes as a recipe for constructing objects as a “total object” (e.g. by iterated cofibers), and while I sort of have a start on how to encode such things (e.g. functors out of an iterated join of copies of the discrete category $\{0, 1\}$ that send all of the $0$’s to zero objects), I’m hoping to avoid having to devise the theory from scratch.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeDec 28th 2021
   • (edited Dec 28th 2021)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexRe #3: Yes, apart from Remark 1.2.2.3 in Lurie's Higher Algebra, see also <https://arxiv.org/abs/1912.06368v1>.

   Re #3: Yes, apart from Remark 1.2.2.3 in Lurie’s Higher Algebra, see also https://arxiv.org/abs/1912.06368v1.

5. • CommentRowNumber5.
   • CommentAuthorHurkyl
   • CommentTimeDec 28th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexRemark 1.2.2.3 is of the "chain complex in the homotopy category" type. I had only really interpreted I suppose if there was an answer like "one right thing to do is to consider sequences $\mathbb{Z} \to \mathcal{C}$ such that composing with $\mathcal{C} \to \mathrm{h} \mathcal{C}$ gives a complex" it would be a positive answer to my question, but the details of such an implication are not obvious to me. The paper you linked looks good, though. Corollary 4.1.2, in particular, is a statement I had hoped would be true, but seemed daunting to work out myself in the generality I had hoped for. Thanks!

   Remark 1.2.2.3 is of the “chain complex in the homotopy category” type. I had only really interpreted

   I suppose if there was an answer like “one right thing to do is to consider sequences $\mathbb{Z} \to \mathcal{C}$ such that composing with $\mathcal{C} \to \mathrm{h} \mathcal{C}$ gives a complex” it would be a positive answer to my question, but the details of such an implication are not obvious to me.

   The paper you linked looks good, though. Corollary 4.1.2, in particular, is a statement I had hoped would be true, but seemed daunting to work out myself in the generality I had hoped for. Thanks!