In the work A Motivic Snaith Decomposition, Viktor Kleen extends the notion of motivic stable homotopy categories $\mathbf{SH}$ to smooth ind-schemes over a base $S$ (colimit of smooth $S$-schemes) by simply putting $$\mathbf{SH}(X) = \operatorname{holim}_i \mathbf{SH}(X_i)$$ if $X = \operatorname{colim}X_i$ is a filtered diagram of smooth schemes $X_i$ over $S$. The formalism of six operations also follows naturally. My question is that, if I understand this correctly, this definition should hold for all ind-schemes over $S$, not necesarilly smooth ones?
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