MGL-modules over a scheme are equivalent to motivic spectra with finite syntomic transfers, and the infinite P^1-loop space of MGL is the A^1-homotopy type of the moduli stack of finite syntomic schemes.
Some characterizations of acyclic maps
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abstract
We discuss two categorical characterizations of the class of acyclic maps between (path-connected) spaces. The first one is in terms of the higher categorical notion of an epimorphism. The second one employs the notion of a balanced map, that is, a map whose homotopy pullbacks define also homotopy pushouts. We also identify the modality in the homotopy theory of spaces that is defined by the class of acyclic maps, and discuss the content of the generalized Blakers-Massey theorem for this modality.
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Modules over algebraic cobordism
MGL-modules over a scheme are equivalent to motivic spectra with finite syntomic transfers, and the infinite P^1-loop space of MGL is the A^1-homotopy type of the moduli stack of finite syntomic schemes.