nLab
twisted complex

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

Let CC be a differential graded category.

A twisted complex EE in CC is

  • a graded set {E i} i\{E_i\}_{i \in \mathbb{Z}} of objects of CC, such that only finitely many E iE_i are not the zero object;

  • a set of morphisms {q ij:E iE j} i,j\{q_{i j} : E_i \to E_j \}_{i,j \in \mathbb{Z}} such that

    • deg(q ij)=ij+1deg(q_{i j}) = i-j+1;

    • i,j:dq ij+ kq kjq ik=0\forall i,j : \; d q_{i j} + \sum_{k} q_{k j}\circ q_{i k} = 0.

The differential graded category PreTr(C)PreTr(C) of twisted complexes in CC has as objects twisted complexes and

PreTr(C)((E ,q),(E ,q)) k= l+ji=kC(E i,E j) 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 fC(E i,E j) lf \in C(E_i, E'_j)^l given by

df=d Cf+ m(q jmf+(1) l(im+1)fq mi). 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:CCF : C \to C' a dg-functor, there is a dg-functor

PreTr(F):PreTr(C)PreTr(C). 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)