# nLab twisted complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

Let $C$ be a differential graded category.

A twisted complex $E$ in $C$ is

• a graded set $\left\{{E}_{i}{\right\}}_{i\in ℤ}$ of objects of $C$, such that only finitely many ${E}_{i}$ are not the zero object;

• a set of morphisms $\left\{{q}_{ij}:{E}_{i}\to {E}_{j}{\right\}}_{i,j\in ℤ}$ such that

• $\mathrm{deg}\left({q}_{ij}\right)=i-j+1$;

• $\forall i,j:\phantom{\rule{thickmathspace}{0ex}}d{q}_{ij}+{\sum }_{k}{q}_{kj}\circ {q}_{ik}=0$.

The differential graded category $\mathrm{PreTr}\left(C\right)$ of twisted complexes in $C$ has as objects twisted complexes and

$\mathrm{PreTr}\left(C\right)\left(\left({E}_{•},q\right),\left(E{\prime }_{•},q\prime \right){\right)}^{k}=\coprod _{l+j-i=k}C\left({E}_{i},E{\prime }_{j}{\right)}^{l}$PreTr(C)((E_\bullet, q), (E'_\bullet, q'))^k = \coprod_{l + j - i = k} C(E_i, E'_j)^l

with differential given on $f\in C\left({E}_{i},E{\prime }_{j}{\right)}^{l}$ given by

$df={d}_{C}f+\sum _{m}\left({q}_{jm}\circ f+\left(-1{\right)}^{l\left(i-m+1\right)}f\circ {q}_{mi}\right)\phantom{\rule{thinmathspace}{0ex}}.$d f = d_C f + \sum_m (q_{j m}\circ f + (-1)^{l(i-m+1)} f \circ q_{m i}) \,.

The construction of categories of twisted complexes is functorial in that for $F:C\to C\prime$ a dg-functor, there is a dg-functor

$\mathrm{PreTr}\left(F\right):\mathrm{PreTr}\left(C\right)\to \mathrm{PreTr}\left(C\prime \right)\phantom{\rule{thinmathspace}{0ex}}.$PreTr(F) : PreTr(C) \to PreTr(C') \,.

etc.

## Properties

Passing from a dg-category to its category of twisted complexes is a step towards enhancing it to a pretriangulated dg-category.

Revised on March 12, 2013 20:28:03 by Ingo Blechschmidt (137.250.162.16)