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External localization of one marked scheme and tensor family

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

Statement

Assume AC and suitable universes. Hinich's localization L:Cat_∞^+→Cat_∞ is a functor with a natural localization map. For a marked cocartesian fibration p:(E,V)→(B,W) satisfying Definition 2.1.1, its localization is a cocartesian fibration L(E,V)→L(B,W), with fibers the localized original fibers (Proposition 2.1.4). The four hypotheses are: p is cocartesian; cocartesian lifts of W are marked; all fiber transports preserve markings; and transports over W induce equivalences of localized fibers. Markings are saturated. For a symmetric monoidal cocartesian fibration over Fin_*, if every active transport preserves its componentwise marking, Proposition 3.2.2 gives a symmetric monoidal localization, with the full universal property for symmetric monoidal functors.

The nerve of an ordinary Grothendieck opfibration is a cocartesian fibration (dual of HTT Remark 2.4.2.2). Cocartesian fibrations classify coherent diagrams, as in Hinich §2.1.5–2.1.6/HTT straightening. These facts apply to a single family over a product base, and thus retain coherence in all base variables simultaneously.

For the fiber comparison, retain these formal inputs: the symmetric monoidal relative localization theorem of BGT Proposition 3.5; HA Proposition 1.3.4.5 identifying chain-homotopy localization with the dg nerve under finite-chain-interval closure; HA Proposition 1.3.2.10 and its following paragraph giving stability for full shift/cone-closed complex subcategories; and Drew Theorem 1.3 identifying stable Verdier quotients with localization at the maps whose cones lie in the subcategory.

Source contract

Given: Marked cocartesian families satisfying all four explicit conditions, tensor-stable markings for the monoidal assertion, and the interval/stability conditions for the separate fiber comparison. Use larger universes for the categories of schemes and sheaf complexes.

Goal: Import a genuine coherent family-localization theorem, rather than only pointwise localization statements.

[F1] Hinich Definition 2.1.1 and Proposition 2.1.4, with its entire proof, supply the family theorem and its compatibility with the functorial localization map of §1.1.

[F2] Hinich Definition 3.2.1/Proposition 3.2.2 supply the symmetric monoidal universal property; HTT Remark 2.4.2.2 and family classification supply the ordinary-opfibration and coherent-diagram bridge.

[F3] The precise BGT, HA and Drew results in the frontmatter supply the formal fiber comparisons; no field-dependent dg monoidal nerve theorem is used.

External input

Import [F1]–[F3] with the stated hypotheses and reading limits. In the application, the base is Sch_qcqs^op×Fin_* marked only by equivalences; fiber markings are componentwise quasi-isomorphisms. The local helper constructs its total Grothendieck category before localization and checks every marked-cocartesian and active-tensor hypothesis. The result is one diagram in the scheme and tensor variables at once. It identifies the supplied fiber enhancements; it neither assumes nor claims a natural comparison with a previously specified pullback/tensor diagram on those fibers, because the supplied definition specifies no such diagram.

Depends on

Used by · 2

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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, Uniqueness of the multiplicative cyclotomic trace · §3, Proposition 3.5, PDF p. 13

A symmetric monoidal relative category whose weak equivalences are preserved by tensor in each variable has a symmetric monoidal infinity-localization. A weakly symmetric monoidal functor preserving weak equivalences induces a symmetric monoidal functor on these localizations.

Hypotheses: Ordinary symmetric monoidal categories C,D, weak equivalences W_C,W_D tensor-stable in each variable, and a lax symmetric monoidal functor whose unit/product comparison maps are weak equivalences. Apply to K-flat sheaf complexes and quasi-isomorphisms, with actual strong symmetric monoidal f*.

Reading limits: Read Proposition 3.5 and its full displayed proof plus preceding Lemma 3.4 context. The cited older Higher Algebra numbering is version-specific; the theorem here is imported directly. No general field-dependent dg monoidal nerve theorem is used.

Jacob Lurie, Higher Algebra · Proposition 1.3.4.5 (statement p. 109, proof pp. 114–115); Proposition 1.3.2.10 and §1.3.5

For a full subcategory of complexes closed under tensor with the finite chain interval N_*(Δ¹), ordinary infinity-localization at chain homotopy equivalences is its dg nerve. A dg category of complexes closed under the finite stable operations has stable dg nerve; the derived sheaf category is localization at quasi-isomorphisms.

Hypotheses: An additive category, full subcategory of its complexes, with closure under the chain interval. Translate chains to cochains by degree negation. K-flat complexes have this closure.

Reading limits: Read the full statement and proof of 1.3.4.5 and the relevant stable dg nerve/derived localization statements. §1.3.5 is also present in the exact completed run supplier. Did not reconstruct the model-categorical and coherent-nerve prerequisites.

Brad Drew, Verdier quotients of stable quasi-categories are localizations · Theorem 1.3, PDF pp. 6–7; Definition 1.1

If A is a stable full subcategory of a small stable infinity-category C, then localization of C at maps whose cofibers lie in A is the stable Verdier quotient C/A. In particular that ordinary infinity-localization is stable.

Hypotheses: Smallness in a chosen universe; enlarge universes to encompass the sheaf-complex categories. No field hypothesis occurs in Theorem 1.3.

Reading limits: Read Definition 1.1, Lemma 1.2 and Theorem 1.3 with their full displayed proofs. Inspected §2/Corollary 2.6 and did not apply its field-dependent dg construction. The imported theorem handles the already defined stable dg nerve, not a field-specific enhancement.

Vladimir Hinich, Dwyer–Kan localization revisited · §1.1.2–1.1.3; Definition 2.1.1, Proposition 2.1.4 and complete proof §2.1.5–2.2.6; Definition 3.2.1 and Proposition 3.2.2; PDF pp. 4–5, 16–22

Localization is an actual functor L:Cat_∞^+→Cat_∞ with a natural localization map. A marked cocartesian fibration localizes to a cocartesian fibration with the localized fibers (Proposition 2.1.4). Tensor-stable markings give a symmetric monoidal localization with the full symmetric monoidal functor universal property (Proposition 3.2.2).

Hypotheses: Definition 2.1.1 requires a cocartesian fibration, marked cocartesian lifts of marked base edges, preservation of fiber markings by every transport, and equivalence of localized fibers over marked base edges. Markings are saturated. For symmetric monoidal localization every active Fin_* transport must preserve the markings. Use equivalences as base markings, componentwise quasi-isomorphisms in the fibers, and all categories in a sufficiently large universe.

Reading limits: Read all statements and displayed proofs in §1.1.2–1.1.3, §2.1–2.2 and §3.2, including the mapping-simplex proof of fiberwise localization. Did not reconstruct the referenced straightening model structure or all earlier DK comparison results. Proposition 3.2.2 is used only under tensor-stable markings, verified below for K-flat complexes.

Jacob Lurie, Higher Topos Theory · Remark 2.4.2.2 and its dual; Proposition 3.2.5.21; Hinich §2.1.5–2.1.6

The nerve of an ordinary Grothendieck opfibration is a cocartesian fibration; such a fibration classifies a coherently functorial diagram of infinity-categories.

Hypotheses: Ordinary categories and Grothendieck opfibrations in a chosen universe. A pseudofunctor of ordinary categories gives the opfibration by the ordinary Grothendieck construction; composition is defined using its coherent natural comparison isomorphisms.

Reading limits: Read Remark 2.4.2.2, Proposition 3.2.5.21 and its displayed proof, and Hinich §2.1.5–2.1.6 for the family classification. The straightening/unstraightening foundations are imported rather than reconstructed.

Sources