# Twisting cochains and arbitrary dg-categories

###### 12th December, 2018

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 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 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 Kapranov^{5},
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 Tong^{7} 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] [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. 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.↩︎