Rational Enriched Motivic Spaces

Enriched motivic $\mathcal A$-spaces are introduced and studied in this paper, where $\mathcal A$ is an additive category of correspondences. They are linear counterparts of motivic $\Gamma$-spaces. It is shown that rational special enriched motivic $\widetilde{\mathrm{Cor}}$-spaces recover connective motivic bispectra with rational coefficients, where $\widetilde{\mathrm{Cor}}$ is the category of Milnor--Witt correspondences.