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External enhanced quotient criterion and localizing nonconnective K

PDF p. 7: Zariski locality; PDF p. 15: Localization diagram

Statement

Consider an actual sequence of exact functors $A \to B \to C$ between small stable infinity-categories, with zero composite and fully faithful first arrow. It is exact precisely when the induced functor $h B / h A \to h C$ is an equivalence after idempotent completion. Equivalently the actual enhanced quotient $B / A \to C$ becomes an equivalence after idempotent completion. Here idempotent completion adjoins all retracts, and $h$ retains objects and homotopy classes of maps.

The BGT nonconnective functor $K$ used in the supplied K interface carries any such exact sequence to a cofiber, hence fiber, sequence of spectra. These sequences are functorial for diagrams of the given exact functors and their nullhomotopies.

Source contract

Given: Small stable categories and actual exact functors, with fixed universes; take $κ = ω$ in BGT.

Goal: Specify the model-independent enhanced localization criterion and nonconnective localizing property.

[F1] BGT Propositions 5.13–5.15 compare actual enhanced quotients and triangulated quotients. Finite-colimit preservation is exactly the $κ = ω$ hypothesis.

[F2] BGT Theorem 8.7 makes $U_{l o c}$ localizing, while Theorem 9.8 identifies $K (-)$ naturally with $Map (U_{l o c} ({S p}^{ω}), U_{l o c} (-))$ .

External input

Import [F1] and [F2] with the recorded limits. Mapping spectra in a stable category preserve finite limits, and cofiber sequences are fiber sequences; therefore the displayed description in [F2] implies the localizing property. This is the nonconnective $I K$ of BGT already chosen by Nonconnective K theory of perfect complexes. This input applies to an actual exact functor and never upgrades an unimplemented equivalence of triangulated categories to an infinity-functor.

Depends on

Used by · 1

The actual enhanced restriction sequence and its K fiber

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Andrew J. Blumberg, David Gepner and Gonçalo Tabuada, A universal characterization of higher algebraic K-theory · Definition 5.12, Proposition 5.13 and Proposition 5.14 pp. 35–36; Proposition 5.15 p. 36; Theorem 8.7 p. 55; Definition 9.6 and Theorem 9.8 p. 58; localizing invariant convention p. 4.

For an actual sequence of exact functors between small stable infinity-categories, exactness is detected by the associated triangulated sequence: zero composite, full first inclusion and Verdier quotient equivalence after idempotent completion. Nonconnective K is naturally Map(U_loc(Sp^omega),U_loc(−)); U_loc sends exact sequences to cofiber sequences.

Hypotheses: Take kappa=omega: stable categories have finite colimits and exact functors preserve them. Categories are small relative to fixed universes (or essentially small via equivalent small models); quotient equivalence is only claimed after idempotent completion.

Reading limits: Read pp. 4, 33–37, 55 and 58, including Proposition 5.15 and its full proof and Theorem 8.7/9.8 statements. The proof of 9.8 and the construction of motives and quotient comparisons are imported, not reconstructed. The supplied nonconnective functor is the same BGT IK; connective K is not substituted.

Sources

Andrew J. Blumberg, David Gepner and Gonçalo Tabuada, A universal characterization of higher algebraic K-theory · Definition 5.12, Proposition 5.13 and Proposition 5.14 pp. 35–36; Proposition 5.15 p. 36; Theorem 8.7 p. 55; Definition 9.6 and Theorem 9.8 p. 58; localizing invariant convention p. 4.