Binary detection by the ordinary rank-zero K-stack comparison

Assume The Axiom of Choice and the compatible universe closure convention of Grothendieck universe closure convention, with the separate existence and size bounds in the Statement. These are the inherited premises of the categorical and K-stack inputs in steps 1.1–2.1.

K-stack mapping spaces and extension from smooth affine schemes supplies the pointed equivalence $u^{\ast }:{\mathrm{Map}}_{\ast }(\tilde{\mathcal{K}},{\mathrm{\Omega }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}N)\simeq {\mathrm{Map}}_{\ast }(B_{\mathrm{\infty }},{\mathrm{\Omega }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}N)$ for every excisive $E$-module $N$, natural in $N$. The sources and their specified basepoints are those of that supplier.

The framework supplies presentable coefficient categories, their colimit-preserving stack action, motivic suspension and its evaluation adjunction. For an unpointed stack $T$, write $N^{T_{+}}$ for the cotensor, right adjoint to $T_{+}\otimes -$. It retains the $E$-module action by transposing the action on $N$. Evaluation gives $({\mathrm{\Omega }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}N{)}^T\simeq {\mathrm{\Omega }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}(N^{T_{+}})$. Here $T_{+}$ adjoins a disjoint point, and the superscript on spaces is cartesian internal Hom. Source-specific non-A¹ motivic spectrum framework and hypotheses

Source record 0 supplies the excision localization test: its generating maps are $R\otimes {\chi }_n$ for all motivic spectra $R$, where ${\chi }_n$ is the map from the pushout of the three other corners of the elementary blowup square to ${\mathbb{A}}^n$. Being local means that mapping these maps into the target gives equivalences.

External symmetric P¹ model, orientation criterion and fundamental stabilization supplies orientation inheritance by $E$-modules, orientable implies fundamental, and the fundamental coefficient-stabilization equivalence. Its hypotheses are the already assumed compact generation, preservation of filtered colimits by $(-{)}^{{\mathbb{P}}^1}$, and multiplicativity of the ambient coefficient category $\mathcal{V}={\mathrm{S}\mathrm{t}}_S$; the K-stack supplier retains these hypotheses and includes this coefficient category. The cotensor in [F2] stays in that same category. Consequently every $E$-module $N$, including $N^{T_{+}}$, has a lift $\widehat{N}$ to motivic spectra with coefficients in $\mathrm{S}\mathrm{p}({\mathrm{S}\mathrm{t}}_S)$, with $N\simeq {\mathrm{\Omega }}^{\mathrm{\infty }}\widehat{N}$. Here ${\mathrm{\Omega }}^{\mathrm{\infty }}$ is ordinary coefficient evaluation, applied levelwise, distinct from ${\mathrm{\Omega }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}$. Thus ${\mathrm{\Omega }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}N\simeq {\mathrm{\Omega }}^{\mathrm{\infty }}({\mathrm{\Omega }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}\widehat{N})$ is a group-like infinite-loop object of ${\mathrm{S}\mathrm{t}}_S$, with zero, addition and inverse. Ordinary spectral conventions are those of Stable spectra and exact triangles; the passage from these motivic modules to spectral coefficients uses the fundamental-stabilization supplier. No compactness or filtered-colimit preservation of $(-{)}^{T_{+}}$ is required.