**Title:** Synthetic spectra and the cellular motivic category

**Speaker:** Piotr Pstragowski

**Speaker Info:** Northwestern University

**Brief Description:**

**Special Note**:

**Abstract:**

A basic invariant of a space or a spectrum is its homology, which often carries a structure of a comodule. The homological algebra of comodules informs our understanding of stable phenomena, for example, it forms the second page of the Adams spectral sequence.We describe how to any reasonable homology theory E one can associate a notion of a synthetic spectrum, this is a kind of a sheaf on the site of finite spectra with projective E-homology. We show the homotopy theory of synthetic spectra is in a precise sense a deformation of Hovey's stable homotopy theory of comodules, the generic fibre is given by the homotopy theory of spectra.

In the particular case of the complex bordism spectrum MU, we show that the variant of our construction recovers the p-complete cellular motivic category over Spec(C), providing context for the Ctau-philosophy of Gheorghe, Isaksen, Wang and Xu.

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