hence also a $B^0$-module. The choices of Hamiltonians are equivalent to defining a $B$-module structure on $E$ which is compatible with $B^0$, i.e., all homomorphisms in the category of $B^0$-algebras between $B$ and the endomorphisms of $E$ as a $B^0$-module [@Rowen2008]. This is the definition of the “${{\mathrm{Hom}}}$" functor, ${{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,{{\mathrm{End}}}_{B^0}{E})$ and we thus associate it with the algebra extension $$\label{MapstoDef}
(B^0\hookrightarrow B)\mapsto \lim_{\dim E\to\infty}{{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,{{\mathrm{End}}}_{B^0}{E}),$$ where we take the stable limit [@kitaev2009periodic; @kennedy2016bott]. The known results for the Clifford algebra extension problems are thus formulated as $$\label{MapstoR}
{{\mathrm{Cl}}}_{p+1,q}\mapsto {\mathcal{R}}_{p-q+2},$$ where ${{\mathcal{R}}_q={\mathcal{R}}_{q+8}}$. Here, we omit the explicit extension ${({{\mathrm{Cl}}}_{p,q}\hookrightarrow{{\mathrm{Cl}}}_{p+1,q})}$ since the Clifford algebras always satisfy ${{{\mathrm{Cl}}}_{p+1,q}^0={{\mathrm{Cl}}}_{p,q}}$. This classification scheme is reviewed in Appendix \[app:class-space\]; see Fig. \[fig:diag1\].

The eightfold Bott periodicity structure [@atiyah1964clifford] in Table \[tab:per\] follows from the relations [@cliff] $${{\mathrm{Cl}}}_{p+1,q+1}\cong M_2({{\mathrm{Cl}}}_{p,q}),{{\;\;\:}}{{\mathrm{Cl}}}_{p+8,q}\cong M_{16}({{\mathrm{Cl}}}_{p,q}).$$ And indeed, the algebraic structure of Eq. (\[MapstoDef\]) is invariant under the Morita equivalence of $B$ with $M_n(B)$. The complex Clifford algebras [@cliff], ${{\mathrm{Cl}}}_{q}$, are given by the complexification, ${{\mathbb{C}}\otimes_{\mathbb{R}}{{\mathrm{Cl}}}_{p,q}={{\mathrm{Cl}}}_{p+q}}$, and satisfy $$\label{MapstoC}
{{\mathrm{Cl}}}_{q+1}\mapsto {\mathcal{C}}_{q},$$ where ${{\mathcal{C}}_q={\mathcal{C}}_{q+2}}$. Useful identities include, $$\label{RCH}
{\mathbb{H}}\otimes_{\mathbb{R}}{\mathbb{H}}\cong M_4({\mathbb{R}}),{{\;\;\:}}{\mathbb{C}}\otimes_{\mathbb{R}}{\mathbb{H}}\cong M_2({\mathbb{C}}),{{\;\;\:}}{\mathbb{C}}\otimes_{\mathbb{R}}{\mathbb{C}}\cong{\mathbb{C}}^{{\oplus 2}}.$$ A particular consequence is that ${{\mathbb{C}}\otimes_{\mathbb{R}}{{\mathrm{Cl}}}_q\cong{{\mathrm{Cl}}}_q^{{\oplus 2}}}$, which enables one to directly read off the classifying spaces of the complex AZ classes (A 