We construct geometric models for the $\mathbb P^1$-spectrum $M_{ \mathbb P^1}(Y)$, which computes in Garkusha-Panin's theory of framed motives \cite{ GP14} a positively motivically fibrant $\Omega_{ \mathbb P^1}$ replacement of $\Sigma_{ \mathbb P^1}^\infty Y$ for a smooth scheme $Y\in \Sm_k$ over a perfect field $k$. Namely, we get the $T$-spectrum in the category of pairs of smooth ind-schemes that defines $\mathbb P^1$-spectrum of pointed sheaves termwise motivically equivalent to $M_{ \mathbb P^1}(Y)$.
Categories: math.AG