The span functor from double ∞-categories to ∞-categories admits a right adjoint given by squares, yielding new proofs of equivalences among the Q-, S-, cobordism, and squares models of algebraic K-theory.
On the Q construction for exact quasicategories
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abstract
We prove that the K-theory of an exact quasicategory can be computed via a higher categorical variant of the Q construction. This construction yields a quasicategory whose weak homotopy type is a delooping of the K-theory space. We show that the direct sum endows this homotopy type with the structure of a infinite loop space, which agrees with the canonical one. Finally, we prove a proto-devissage result, which gives a necessary and sufficient condition for a "nilimmersion" of stable quasicategories to be a K-theory equivalence. In particular, we prove that a well-known conjecture of Ausoni and Rognes is equivalent to the weak contractibility of a particular quasicategory.
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math.CT 1years
2026 1verdicts
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The span-squares adjunction
The span functor from double ∞-categories to ∞-categories admits a right adjoint given by squares, yielding new proofs of equivalences among the Q-, S-, cobordism, and squares models of algebraic K-theory.