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 dg-categories 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 dg-categories about which I’m talking2 is that, for a lot of people3, it is the (pre-)triangulated structure that is interesting. This means that (as far as I am aware)4 an arbitrary dg-category 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 Kapranov5, are a sort of solution to this problem, in that (to quote from where else but the nLab) “passing from a dg-category to its category of twisted complexes is a step towards enhancing it to a pretriangulated dg-category”.6 In essence, they give us the ‘smallest’ ‘bigger’ dg-category 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 Tong7 where they are (to be vague) thought of as resolutions of coherent sheaves, or first-order 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?
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 click-bait 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]  ↩
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. USSR-Sb., 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. ↩