Dependencies

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 \(F:\operatorname {BK}^\bullet (\mathcal{C}) \to K^\bullet (\mathcal{C})\).

An \(A_\infty \)-algebra is an \(A_\infty \)-algebra structure together with a proof of the Stasheff identities.

For an \(A_\infty \)-algebra, every algebraic Stasheff sum is equal to zero.

An \(A_\infty \)-algebra canonically determines a one-object \(A_\infty \)-precategory.

Let \(A\) be a graded \(R\)-module. An \(A_\infty \)-algebra structure on \(A\) consists of multilinear operations

for every positive arity \(n\) and every choice of input degrees.

An \(A_\infty \)-algebra structure satisfies the Stasheff identities when every one of its Stasheff sums is zero.

The Stasheff sum of an \(A_\infty \)-algebra is the Stasheff sum of its associated one-object \(A_\infty \)-precategory.

Any \(A_\infty \)-algebra structure determines a one-object \(A_\infty \)-precategory, with object type and constant hom-module equal to the underlying graded module.

An \(A_\infty \)-category is an \(A_\infty \)-precategory together with a proof that its operations satisfy the Stasheff identities.

For an \(A_\infty \)-category, every Stasheff sum computed from the structure maps is equal to zero.

An \(A_\infty \)-precategory over \(R\) with object type \(\mathrm{Obj}\) consists of a graded \(R\)-linear quiver together with operations

of the prescribed multilinear type for every positive arity \(n\).

A chain in an \(A_\infty \)-precategory consists of a positive length \(n\), a string of \(n+1\) objects, and a degree assigned to each of the \(n\) composable morphism slots.

For a chain and an index \(i\), is the module of the \(i\)-th composable morphism in that chain.

For a chain, this is the graded hom-module that receives the corresponding multilinear operation.

The source of a chain is its initial object.

The target of a chain is its final object.

An \(A_\infty \)-precategory satisfies the Stasheff identities when its structure maps define an object-indexed system satisfying .

The Stasheff sum of an \(A_\infty \)-precategory is the indexed Stasheff sum formed from its structure maps.

For a string of objects and prescribed degrees, is the \(i\)-th graded hom-module in that composable string.

For a commutative ring \(R\), a graded \(R\)-module is a \(\beta \)-indexed family of \(R\)-modules, implemented as a graded object in ModuleCat R.

A grading on a type \(\beta \) consists of an additive commutative group structure on \(\beta \), together with homomorphisms

such that \(\sigma (\phi (n))\) is the parity class of \(n\) for every integer \(n\).

An object-indexed family of multilinear operations satisfies the Stasheff identities if every indexed Stasheff sum vanishes.

The arity-\(n\) Stasheff sum is the signed sum of all valid composites of an inner operation followed by an outer operation.

Fix object-indexed operations \(m_n\). For valid indices \((r,s)\), is the term obtained by first applying the inner \(s\)-ary operation in positions \(r,\dots ,r+s-1\) and then applying the outer operation to the collapsed string.

Given input degrees \((d_0,\dots ,d_{n-1})\), the output degree of the \(n\)-ary operation is

where the final summand is interpreted using the grading shift .

For a composable string of objects and degrees, this is the graded hom-module in which the corresponding multilinear operation takes values.

The type of parities is \(\mathbb {Z}/2\mathbb {Z}\), implemented as ZMod 2.

An \(R\)-linear graded quiver on an object type \(\mathrm{Obj}\) assigns to each pair of objects \((X,Y)\) a graded \(R\)-module of morphisms from \(X\) to \(Y\).

For a grading index \(\beta \), is the image of \(n \in \mathbb {Z}\) under the distinguished map \(\phi : \mathbb {Z} \to \beta \).

The integer sign attached to \((r,s)\) is \((-1)\) raised to the parity recorded by .

For a valid pair \((r,s)\), this is the parity

computed in .

The common degree of the arity-\(n\) Stasheff relation is

in the grading index.