Stable homotopy theory of higher categories

Abstract

Stable homotopy theory is governed by the principle that inverting the operation of forming loop spaces produces representing objects for homology theories. We show that this principle is not specific to topology but reflects an intrinsic structural feature of higher category theory: inverting the operation of forming endomorphism $(\infty,\infty)$-categories leads to a stable homotopy theory of higher categories, in which higher categories play the role of spaces and categorical spectra represent homology theories of higher categories. This theory necessarily requires enrichment in the Gray tensor product, reflecting its genuinely higher-categorical nature. Classical stable homotopy theory is recovered by passing to classifying spaces. Its fundamental mechanisms arise as shadows of richer categorical phenomena: stabilization is governed by a categorical Freudenthal suspension theorem, whose classical counterpart arises by passing to classifying spaces. Our main result is a categorical Brown representability theorem classifying homology theories of higher categories by categorical spectra. As a consequence, categorical homology theories give rise to categorical analogues of long exact sequences and to homological algebra of higher categories. As guiding example we study categorical homology, the categorical homology theory whose coeffients are the natural numbers, and prove a Hurewicz theorem and Eilenberg-Zilber theorem.