CSS data and a three-identity degree-two tensor word

transposeMap. For all natural numbers $m,n$ and every $\mathbb F_2$-linear map $f:\mathbb F_2^n\to\mathbb F_2^m$, where $\mathbb F_2=\mathbb Z/2\mathbb Z$, this defines the linear map $f^\top:\mathbb F_2^m\to\mathbb F_2^n$ whose standard-coordinate matrix is the transpose of the standard-coordinate matrix of $f$. Explicitly, if $e_k$ is the $k$-th standard basis word, then for $y\in\mathbb F_2^m$ and $k\in\operatorname{Fin}(n)$, $(f^\top y)_k=\sum_{i\in\operatorname{Fin}(m)}f(e_k)_i\,y_i$, with all arithmetic in $\mathbb F_2$. The parameters may be zero: an empty-index sum is $0$, and if $m=0$ or $n=0$, the corresponding source or target is the zero-dimensional binary coordinate space and the definition gives the uniquely determined linear map of the stated type.

CSSSwapProductData. For every triple $n,r_X,r_Z\in\mathbb N$, this structure consists of two $\mathbb F_2$-linear maps $H_X:\mathbb F_2^n\to\mathbb F_2^{r_X}$ and $H_Z:\mathbb F_2^n\to\mathbb F_2^{r_Z}$; proofs of both equations $H_X\circ H_Z^\top=0:\mathbb F_2^{r_Z}\to\mathbb F_2^{r_X}$ and $H_Z\circ H_X^\top=0:\mathbb F_2^{r_X}\to\mathbb F_2^{r_Z}$, where each transpose is the standard-coordinate matrix transpose described by $(H^\top y)_k=\sum_i H(e_k)_i y_i$; proofs that both $H_X$ and $H_Z$ are surjective; two words $\ell_X,\ell_Z\in\mathbb F_2^n$; and proofs that $H_Z\ell_X=0$, $H_X\ell_Z=0$, and $\sum_{k\in\operatorname{Fin}(n)}(\ell_X)_k(\ell_Z)_k=1$ in $\mathbb F_2$. The last equation says that the two words overlap in an odd number of coordinates and forces both to be nonzero; together with the cycle and surjectivity requirements, any inhabitant must satisfy $r_X<n$ and $r_Z<n$, so in particular there is no inhabitant when $n=0$. Either $r_X$ or $r_Z$ may nevertheless be $0$; for such a zero-sized codomain, the corresponding map is uniquely zero and surjective, and its associated zero-composition and cycle equations are automatic.

leftComplex. For all $n,r_X,r_Z\in\mathbb N$ and every datum $D$ consisting of surjective maps $H_X:\mathbb F_2^n\to\mathbb F_2^{r_X}$, $H_Z:\mathbb F_2^n\to\mathbb F_2^{r_Z}$ satisfying $H_XH_Z^\top=0$ and $H_ZH_X^\top=0$, together with words $\ell_X,\ell_Z\in\mathbb F_2^n$ satisfying $H_Z\ell_X=0$, $H_X\ell_Z=0$, and $\ell_X\mathbin{\cdot}\ell_Z=1$, this defines a based binary chain complex $A$ in nonnegative degrees with $\dim A_0=r_X$, $\dim A_1=n$, $\dim A_2=r_Z$, and $\dim A_i=0$ for every $i\ge3$. Its degree-$i$ boundary has source $\mathbb F_2^{\dim A_i}$ and target $\mathbb F_2^{\dim A_{i-1}}$ for $i>0$, while the degree-zero target is the space of functions on the empty index type; specifically, $\partial_0=0$, $\partial_1=H_X$, $\partial_2=H_Z^\top$, and $\partial_i=0$ for $i\ge3$. It includes the assertion $\partial_i\partial_{i+1}=0$ for every $i$, whose only potentially nonzero case is $H_XH_Z^\top=0$, and records length $2$ together with $\dim A_i=0$ whenever $2<i$. Zero values of $r_X$ or $r_Z$ produce zero-dimensional endpoint groups; parameter triples admitting no such datum $D$, including $n=0$, supply no input at which this definition can be instantiated.

rightComplex. For all $n,r_X,r_Z\in\mathbb N$ and every datum $D$ consisting of surjective maps $H_X:\mathbb F_2^n\to\mathbb F_2^{r_X}$, $H_Z:\mathbb F_2^n\to\mathbb F_2^{r_Z}$ satisfying $H_XH_Z^\top=0$ and $H_ZH_X^\top=0$, together with words $\ell_X,\ell_Z\in\mathbb F_2^n$ satisfying $H_Z\ell_X=0$, $H_X\ell_Z=0$, and $\ell_X\mathbin{\cdot}\ell_Z=1$, this defines a based binary chain complex $B$ with $\dim B_0=r_Z$, $\dim B_1=n$, $\dim B_2=r_X$, and $\dim B_i=0$ for every $i\ge3$. Its boundaries are $\partial_0=0$, $\partial_1=H_Z$, $\partial_2=H_X^\top$, and $\partial_i=0$ for $i\ge3$, with the usual degree-$i$ source and degree-$(i-1)$ target for $i>0$ and the empty-index zero-dimensional target in degree $0$. It includes $\partial_i\partial_{i+1}=0$ for every $i$, whose only potentially nonzero case is $H_ZH_X^\top=0$, and records length $2$ together with vanishing dimensions above degree $2$. Zero values of $r_X$ or $r_Z$ give zero-dimensional endpoint groups; if no datum $D$ exists for a parameter triple, as when $n=0$, the definition has no input at those parameters.

threeIdentityElement. For all $n,r_X,r_Z\in\mathbb N$ and every datum $D$ consisting of the maps and words above with both surjectivity conditions, both transpose-composition equations, both cycle equations, and dot product $1$, let $A$ be the complex with dimensions $(r_X,n,r_Z)$ in degrees $0,1,2$ and let $B$ be the complex with dimensions $(r_Z,n,r_X)$. This defines a binary word on the degree-$2$ tensor-product coordinate set: an index is a decomposition $p+q=2$ with $p,q\in\{0,1,2\}$, together with $a\in\operatorname{Fin}(\dim A_p)$ and $b\in\operatorname{Fin}(\dim B_q)$, and the word’s value at that index is $1$ exactly when the underlying natural numbers represented by $a$ and $b$ are equal, and $0$ otherwise. Thus the $(p,q)=(0,2),(1,1),(2,0)$ components are respectively the Kronecker-delta arrays $I_{r_X},I_n,I_{r_Z}$; these are the only degree decompositions. The value uses none of $D$’s boundary maps or distinguished words beyond the dimensions supplied by its parameters. If $r_X=0$ or $r_Z=0$, the corresponding component has no coordinates and is empty; parameter triples with no possible datum $D$ give no instance of the definition.