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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 1st 2012

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 27th 2018

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

• CommentRowNumber3.
• CommentAuthorHurkyl
• CommentTimeDec 27th 2021
• (edited Dec 27th 2021)

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.

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeDec 27th 2021
• (edited Dec 27th 2021)

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

• CommentRowNumber5.
• CommentAuthorHurkyl
• CommentTimeDec 27th 2021

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!