Recollements for derived categories of enriched functors and triangulated categories of motives

Correspondences and stable homotopy theory

“…A genuinely local approach to $$SH(k)$$, envisioned by Voevodsky in 2001, is presented in [20]. It is based on Voevodsky's framed correspondences and the machinery of framed motives [19].…”

“…It is worth mentioning that we do not use any motivic equivalences or the $$\mathbb {A}^1$$‐relation in any of our definitions above (similarly to constructions of [20]). All constructions here are genuinely local.…”

“…Moreover, this reconstruction is given by a functor taking a symmetric $$S^1$$‐spectrum $$N$$ to its symmetric framed motive $${M}^\Sigma _{fr}(N)$$ introduced in this paper (see Definition 4.9). Another application gives yet another genuinely local model of stable motivic homotopy theory $$SH(k)$$ (in addition to [20]) and, more generally, a local model for the category of $$E$$‐modules in $$SH(k)$$, where $$E$$ is a symmetric Thom ring spectrum. See Theorem 6.7 and Corollary 6.9 for details.…”

Rational enriched motivic spaces

“…We recover the triangulated category of big A-motives DM A out of Grothendieck categories of enriched functors [B, Shv(A)] in the sense of [1], where B is either the Shv(A)-category C of the powers G ×n m or the Shv(A)-category Sm of all smooth k-schemes. To this end, we use homological algebra of enriched Grothendieck categories developed in [20,21].…”

“…We refer the reader to Section 3.1 for details. The relations are also counterparts of the axioms (2)-( 5) for special motivic Γ-spaces in the sense of [25] and framed spectral functors in the sense of [24,Section 6].…”