nLab
twisted complex

Definition

Let $C$ be a differential graded category.

A twisted complex $E$ in $C$ is

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

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

  • $\mathrm{deg}(q_{ij})=i-j+1$;

  • $\forall i,j:d{q}_{ij}+{\sum }_k{q}_{kj}\circ q_{ik}=0$.

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

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

with differential given on $f\in C(E_i,E{\prime }_j{)}^l$ given by

$$df=d_{C}f+\sum _m(q_{jm}\circ f+(-1{)}^{l(i-m+1)}f\circ q_{mi}).$$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 $F:C\to C\prime$ a dg-functor, there is a dg-functor

$$\mathrm{PreTr}(F):\mathrm{PreTr}(C)\to \mathrm{PreTr}(C\prime ).$$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.

Related concepts

• Landau-Ginzburg model, pretriangulated dg-category