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 $\{q_{i j} : E_i \to E_j \}_{i,j \in \mathbb{Z}}$ such that

  • $deg(q_{i j}) = i-j+1$;

  • $\forall i,j : \; d q_{i j} + \sum_{k} q_{k j}\circ q_{i k} = 0$.

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

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

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

$$d f = d_C f + \sum_m (q_{j m}\circ f + (-1)^{l(i-m+1)} f \circ q_{m i}) \,.$$

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

$$PreTr(F) : PreTr(C) \to PreTr(C') \,.$$

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
• Alexei Bondal, Mikhail Kapranov, Enhanced triangulated categories, Матем. Сборник, Том 181 (1990), No.5, 669–683 (Russian); transl. in USSR Math. USSR Sbornik, vol. 70 (1991), No. 1, pp. 93–107, (MR91g:18010) (Bondal-Kapranov Enhanced triangulated categories pdf)
• Bernhard Keller, A brief introduction to $A_\infty$-algebras (pdf)
• Mikhail Kapranov, Svyatoslav Pimenov, Derived varieties of complexes and Kostant’s theorem for gl(m|n),

  arxiv/1504.00339