Let \(\mathcal{C}\) be a preadditive category. Then the computable additive completion of \(\mathcal{C}\), denoted \(\operatorname {CMat}(\mathcal{C})\), has

• as objects, lists of elements in \(\mathcal{C}\).

• as morphisms, dependent matrices of morphisms in \(\mathcal{C}\). Specifically, if \((A_0, A_1, \ldots , A_n), (B_0, B_1, \ldots , B_m) \in \operatorname {Ob}\big(\operatorname {Mat\_ }(\mathcal{C})\big)\), then

  \begin{multline*} \operatorname {Hom}\big((A_0, A_1, \ldots , A_n), (B_0, B_1, \ldots , B_m)\big) \\ = \left\{ \begin{bmatrix} a_{00} & \cdots & a_{0,n-1} \\ \vdots & \ddots & \vdots \\ a_{m-1,0} & \cdots & a_{m-1,n-1} \end{bmatrix} : a_{ij} \in \operatorname {Hom}(A_j, B_i) \right\} \end{multline*}

  where \(0 \le i \le m-1\) and \(0 \le j \le n-1\).

• composition is defined by matrix multiplication

• identities are given by the identity matrix

  \[ (\mathbb {1}_{(A_0, \ldots , A_m)})_{i, j} = \begin{cases} \mathbb {1}_{A_i} & \text{reinterpreted as an element of }\operatorname {Hom}(A_i, A_j) \text{ by casting along the equality if }i = j \\ 0 & \text{ if }i\ne j \end{cases} \]

  .

Given \(A = (A_0, \ldots , A_n), B=(B_0, \ldots , B_n)\in \operatorname {CMat\_ }(\mathcal{C})\), the computable biproduct of \(A\) and \(B\) is defined by \(A\boxplus _k B = (A_0, \ldots , A_n, B_0, \ldots , B_m)\)

For any \(Z\in \operatorname {CMat\_ }(\mathcal{C})\), the index type, denoted \(\iota _Z\), is the type \(\{ 0, \ldots , (\operatorname {len}(Z)-1)\} \), and the indexing function, \(X_2: \iota _Z\to \mathcal{C}\), is defined by \(X_Z(i) = Z_i\).

Let \(n\in \mathbb {N}\). A positioning of \(1\) black strand with \(n\) red strands is an element of \(\{ 0, \ldots , n+1\} \). Given positionings of \(X\) and \(Y\), the strand set \(StrandSpace_{X, Y} = \mathbb {N}\), representing the number of dots. (Note here that \(\text{StrandSpace}_{X, Y}\) does not depend on \(X\) or \(Y\), but with more black strands it might).

If \(X, Y, Z\) are positionings of \(1\) black strand and \(n\) red strands and \(a\in \text{StrandSpace}_{X, Y}\), \(b\in \text{StrandSpace}_{Y, Z}\), then their composition is given by

Fix a commutative ring \(R\). The \(\operatorname {KLRW}\) category of \(1\) black strand and \(n\) red strands, denoted \(\operatorname {KLRW}\) category of \(1\) black strand and \(n\) red strands, denoted \(\operatorname {KLRW}^R_n\), is given by

• \(\operatorname {Ob}(\operatorname {KLRW}^R_n)\) is the set of positionings of \(1\) black strand with \(n\) red strands

• \(\operatorname {Hom}(X, Y)\) is the free \(R\)-module generated by \(\text{StrandSpace}_{X, Y}\).

Composition in \(\operatorname {KLRW}^R_n\) is defined by extending the composition of elements of the strand space linearly.

Since each \(\operatorname {Hom}(X, Y)\) is an \(R\)-module and composition is \(R\)-linear, \(\operatorname {KLRW}^R_n\) is \(R\)-linear and thus preadditive.

Let \(\mathcal{C}\) be a preadditive category with a zero object. Then a bounded cochain complex of \(\mathcal{C}\) is a (\(\mathbb {Z}\)-graded) cochain complex \(A^\bullet \) of \(\mathcal{C}\) equipped with a finite set \(S\) called the support, satisfying \(\forall i\in \mathbb {Z}, i\in S \Leftrightarrow A^i\not\cong 0\). Note that \(S\) is uniquely determined by \(A^\bullet \), though included for computability.

Let \(\operatorname {BK}^\bullet (\mathcal{C})\) denote the set of bounded cochain complexes in \(\mathcal{C}\). THen there is a function \(F:\operatorname {BK}^\bullet (\mathcal{C})\to K^\bullet (\mathcal{C})\) which forgets the finiteness. Then we equip \(\operatorname {BK}^\bullet (\mathcal{C})\) with the induced category structure of \(F\) which makes \(F\) into a fully faithful functor \(F:\operatorname {BK}^\bullet (\mathcal{C}) \to K^\bullet (\mathcal{C})\).