Previdi's delooping conjecture for exact categories

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Let A\mathcal{A} be an exact category. Write S(A)S(\mathcal{A}) for the geometric realization of the simplicial category iS∙(A)iS_{\bullet}(\mathcal{A}) given by Waldhausen's S∙S_{\bullet}-construction. The category A\mathcal{A} is partially abelian if it and its opposite have pullbacks of admissible monomorphisms with common target. Write lim⁡⟷A\displaystyle\lim_{\longleftrightarrow}\mathcal{A} for Beilinson's category of generalized Tate objects associated to A\mathcal{A}. The homotopy groups of the loop space of S(A)S(\mathcal{A}) are the algebraic KK-theory groups of A\mathcal{A}. Previdi's delooping conjecture. If A\mathcal{A} is partially abelian, then S(A)S(\mathcal{A}) is delooped by S(lim⁡⟷A)S(\displaystyle\lim_{\longleftrightarrow}\mathcal{A}). This conjecture proposes a delooping relation between the Waldhausen space of an exact category and that of its category of generalized Tate objects; the paper's abstract says that it proves a modified version using non-connective KK-theory spectra, so the precise original formulation should be checked against that modification.

References

Primary source

Sho Saito, “On Previdi's delooping conjecture for K-theory”, arXiv:1203.0831 (2013).

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