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Nonconnective K theory in degree zero

PDF p. 12: Bundle classes in spectral K zero

Statement

For a small idempotent-complete stable infinity-category C, the canonical map from the abelian group generated by equivalence classes of objects, subject to [Y]=[X]+[Z] for every cofiber sequence X→Y→Z, to π₀𝕂(C) is an isomorphism. Here 𝕂 is the nonconnective algebraic K-theory functor of Blumberg–Gepner–Tabuada. The isomorphism sends the generator of an object to its K-theory class.

Idempotent complete means that every idempotent endomorphism splits. A cofiber sequence is a sequence X→Y→Z in which Z is the pushout of X→Y along X→0. The group is the cofiber Grothendieck group.

External input

This is the established degree-zero comparison recalled in Kasprowski–Winges, §2.2, applied to an idempotent-complete category. Their §2.1 identifies the nonconnective functor with the BGT localizing invariant. Only the passages specified in sources.references were read; no local reconstruction of this theorem is claimed.

Facts & Assumptions

Given: A small idempotent-complete stable infinity-category C and the cited nonconnective functor.

Goal: Identify π₀𝕂(C) with the cofiber Grothendieck group, with the stated generator map.

[F1]

The external input identifies π₀𝕂(C) with the cofiber Grothendieck group of the idempotent completion of C, naturally in C, by the object-class map.

Proof

technique · Apply the imported comparison under its stated hypotheses
1.1F1assume-hyp

Since C is idempotent complete, its canonical inclusion into its idempotent completion is an equivalence. Thus the cofiber Grothendieck group appearing in [F1] is the group presented using the objects and cofiber sequences of C.

2.1F1step 1.1∎

Apply [F1] to C and use step 1.1. The resulting natural isomorphism sends each object generator to its K-theory class, as asserted.

Depends on

Used by · 8

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Algebraic K-theory of stable ∞-categories via binary complexes · Section 2.2, paragraph immediately preceding Lemma 2.12

For a small stable infinity-category C, its cofiber Grothendieck group after idempotent completion is naturally isomorphic to π₀ of its nonconnective K-theory spectrum; in particular the completion can be omitted when C is idempotent complete.

Hypotheses: C is a small stable infinity-category; the stated specialization assumes C is idempotent complete.

Reading limits: Read the source title, §2.1 identification of its nonconnective functor with the BGT functor, and the stated paragraph in §2.2. The underlying proofs of the recalled comparison were not reconstructed or independently read in full.

A universal characterization of higher algebraic K-theory · Definition 9.6 and Remark 9.33

Definition 9.6 constructs the nonconnective K-theory functor; Remark 9.33 records its degree-zero comparison with connective K-theory in the stated Waldhausen setting.

Hypotheses: The source's small stable infinity-category and Waldhausen hypotheses apply in the respective passages.

Reading limits: Read the cited definition and remark only; this reference identifies the functor and supplies context, not a claimed local proof of the general comparison.

Sources