Twisting cochains and arbitrary dgcategories
Having recently been thinking about twisting cochains (a major part of my thesis) a bit more, I think I better understand one reason why they are very useful (and why they were first introduced by Bondal and Kapranov), and that’s in ‘fixing’ a small quirk about dgcategories that I didn’t quite understand way back when I wrote this post about derived, dg, and $A_\infty$categories and their role in ‘homotopy things’.
This isn’t a long post and could probably instead be a tweet but then this blog would be a veritable ghost town.
First of all, for the actual definitions of twisting/twisted cochains/complexes (the nomenclature varies wildly and seemingly inconsistently),^{1} I will shamelessly refer the interested reader to some notes I wrote a while back. (update: these notes have been subsumed into my PhD thesis)
Secondly, the ‘quirk’ of dgcategories about which I’m talking^{2} is that, for a lot of people^{3}, it is the (pre)triangulated structure that is interesting. This means that (as far as I am aware)^{4} an arbitrary dgcategory lacks some sort of homotopic interpretation because it has no structure corresponding to stability ‘upstairs’. Twisting cochains then, as they were introduced by Bondal and Kapranov^{5}, are a sort of solution to this problem, in that (to quote from where else but the nLab) “passing from a dgcategory to its category of twisted complexes is a step towards enhancing it to a pretriangulated dgcategory”.^{6} In essence, they give us the ‘smallest’ ‘bigger’ dgcategory in which we have shifts and functorial cones.
Really I am just parroting back the reasons why these things were initially invented, but it’s something that I hadn’t fully appreciated, since I’ve been working with specific types of twisted complexes (ones that somehow correspond to projective/free things and concentrated in a single degree) that really arise in what appears (to me) to be a completely different manner: namely in the setting of (O’Brian), Toledo, and Tong^{7} where they are (to be vague) thought of as resolutions of coherent sheaves, or firstorder perturbations of certain bicomplexes by flat connections.
I really have no geometric/homotopic intuition as to why this specific case of twisted complexes corresponds thusly, and haven’t been able to find any references at all. Any ideas?
Footnotes

Although for me, at least, I (tend to) use twisted complex to refer to the concept of Bondal and Kapranov, and twisting cochain to refer to the concept of (O’Brian), Toledo, and Tong. ↩

Having hidden this part in the main post and not the excerpt makes me feel like I’m writing the mathematical equivalent of clickbait journalism. Next will come posts with titles such as “Nine functors that you wouldn’t believe have derived counterparts — number six will shock you!” and “You Will Laugh And Then Cry When You See What This Child Did With The Grothendieck Construction”. I apologise in advance. ↩

[weasel words] [citation needed] ↩

which is, admittedly, best measured on the Planck scale. ↩

A. I. Bondal, M. M. Kapranov, “Enhanced triangulated categories”, Mat. Sb., 181:5 (1990), 669–683; Math. USSRSb., 70:1 (1991), 93–107. ↩

A bunch of papers, but in particular e.g. D. Toledo and Y. L. L. Tong, “Duality and Intersection Theory in Complex Manifolds. I.”, Math. Ann., 237 (1978), 41—77. ↩