Stable homotopy theory of higher categories with adjoints

Abstract

We demonstrate that higher categories with adjoints behave like homotopy types. By the homotopy hypothesis, homotopy types can be regarded as higher categories in which all cells are invertible. We show that replacing invertibility by the existence of higher adjoints still supports a homotopy theory that is at least as rich as classical homotopy theory. The basic invariants of this theory are the homotopy monoidal categories \[ Π_n(X,x)=\operatorname{Ho}\bigl(Ω^n(X,x)\bigr) \] associated to a based $(\infty,\infty)$-category with adjoints $(X,x)$. These invariants are monoidal categories with duals and replace iterated loops with iterated endomorphisms. The classical homotopy groups of the classifying space are recovered as the isomorphism classes of objects in the groupoid completion. We prove a Whitehead theorem for homotopy monoidal categories and show that fibrations of higher categories give rise to long oriented exact sequences of homotopy monoidal categories. To develop homological algebra of higher categories with adjoints, we prove an equivalence between skeletal sequences of categorical spectra with adjoints and categorical chain complexes, extending a result outlined by Lurie in the proof of the cobordism hypothesis.