∞-TOPOS WALDHAUSEN // IDEMPOTENT COMPLETION // EXCISIVE APPROXIMATION
An ∞-topos is a presentable ∞-category satisfying descent (Giraud axioms in ∞-categorical form). To do K-theory, one identifies a Waldhausen subcategory of perfect objects and applies the S-construction. For a ring spectrum R, perfect R-modules form a stable ∞-category Perf(R).
A stable ∞-category C is idempotent complete if every idempotent e: X → X splits (admits a retract). The idempotent completion C^{idem} is the universal idempotent complete category receiving C. By Thomason, K(C) → K(C^{idem}) is an equivalence on K_n for n ≥ 1.
K-theory itself is the universal 1-excisive approximation to the identity on stable ∞-categories (Blumberg-Gepner-Tabuada). That is, K: Cat^{ex}_∞ → Sp is the best approximation to the "identity" functor that satisfies excision (takes Verdier sequences to fiber sequences).