Alphabeta Math
TheoremStatement: Literature-sourcedProof: Literature-sourced
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

External nonconnective lax symmetric monoidal K-theory

PDF p. 3: Nonconnective algebraic K theory: enhanced prerequisite interface

Statement

Assume AC and fixed universes. Let Cat_perf be the infinity-category of small idempotent-complete stable infinity-categories and exact functors. It is symmetric monoidal with tensor characterized by exact functors in each variable. The nonconnective algebraic K-theory functor 𝕂:Cat_perf→Sp exists and is lax symmetric monoidal. It is the functor denoted IK, not connective K, in BGT: 𝕂(A)≃Map(Uloc(Sp^ω),Uloc(A)). In particular it carries coherent commutative algebra categories to E∞ ring spectra, with the corresponding functorial algebra and module operations induced by its lax structure.

Source contract

Given: Small stable idempotent-complete infinity-categories, exact functors, their exact tensor and AC.

Goal: Import the precise nonconnective multiplicative functor used for schemes and supported categories.

[F1] BGT, A universal characterization, §2.2, §3.1, Definition 9.6 and Theorem 9.8, supplies the domain, nonconnective invariant and tensor universal property.

[F2] BGT, Uniqueness of the multiplicative cyclotomic trace, Proposition 5.9 and Corollary 5.10 explicitly include nonconnective IK in the lax symmetric monoidal assertion.

External input

Import [F1] and [F2], with the recorded limits. The target Sp and smash-product notation are those of Stable spectra and exact triangles. This input does not assert that every restriction sequence of schemes is a localization sequence; it does not identify π₀ with a separately defined degree-zero group. Those application/comparison theorems remain separate suppliers.

Depends on

Used by · 2

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

Blumberg–Gepner–Tabuada, A universal characterization of higher algebraic K-theory · §2.2 (Cat_perf domain), §3.1 (tensor universal property); Definition 9.6 and Theorem 9.8, PDF pp. 58–59

Nonconnective algebraic K-theory 𝕂 is a spectrum-valued functor on small stable infinity-categories, in particular small idempotent-complete ones, and is represented on localizing motives by Map(U_loc(Sp^ω),U_loc(−)). Cat_perf has the tensor universal property for functors exact separately in each variable.

Hypotheses: Small stable infinity-categories and exact functors; we apply only to idempotent-complete categories. Smallness is relative to a fixed universe. The functor called IK in the paper is nonconnective; it is distinct from the connective functor K there.

Reading limits: Read the domain definitions in §2.2, §3.1 tensor construction/universal property, Definition 9.6, Theorem 9.8 and its announced reduction, and the beginning of §9.3. The long proof of Theorem 9.8, suspension constructions and classical comparison theorems are imported, not reconstructed. No claim to have independently established localization for schemes or the degree-zero comparison.

Blumberg–Gepner–Tabuada, Uniqueness of the multiplicative cyclotomic trace · Proposition 5.9 and Corollary 5.10, PDF p. 23; Theorem 1.13, PDF p. 5

Both connective K and nonconnective IK:Cat_perf→Sp are lax symmetric monoidal; hence a coherent E∞ algebra category gives an E∞ ring spectrum. The nonconnective functor is Map(U_loc(Sp^ω),U_loc(−)), not the connective U_add version.

Hypotheses: Cat_perf consists of small idempotent-complete stable infinity-categories and exact functors, with its exact tensor product. Lax symmetric monoidal structure supplies coherent external products, units, and transport of algebra/module operations.

Reading limits: Read Proposition 5.9 and its complete displayed proof, Corollary 5.10, Theorem 1.13, and the explicit U_add/U_loc distinction near equation (1.3). The proof states that the nonconnective argument is identical; that is the imported nonconnective assertion. Did not reconstruct motivic monoidal localization or a point-set K-theory machine.

Sources