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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 -categories? I.e. something that looks like sequences in a suitable -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 that send all of the ’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 such that composing with 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!