TheoremStatement: AI-adaptedProof: Literature-sourced

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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.

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External stable mapping and derived linear target interfaces

Statement

Work in a fixed hierarchy of universes, enlarging for Yoneda applications so the source categories are small there; all mapping spaces and diagrams under discussion are small in the chosen universe. An infinity-category retains mapping spaces and coherent higher composition. A coherent diagram is a functor from its indexing infinity-category; a limit is characterized by its mapping-space universal property, and a colimit by the dual property. A stable infinity-category $C$ has a zero object, finite limits and colimits, with pushout squares exactly pullback squares. Its opposite is stable; its homotopy category $h C$ , with morphisms $π_{0} {Map}_{C}$ , is triangulated by suspension and cofiber triangles. A cofiber of $A \to B$ is a pushout $B ∐_{A} 0$ ; a fiber of $V \to W$ is a pullback $V \times_{W} 0$ .

For an object $E$ of an infinity-category $D$ , the functor ${Map}_{D} (E, -)$ preserves existing small limits. In particular ${Map}_{C} (-, E)$ is left-exact on $C^{o p}$ .

For any stable infinity-category $D$ , every left-exact functor $F : D \to S$ has an essentially unique left-exact lift $\hat{F} : D \to S p$ with $Ω^{\infty} \hat{F} ≃ F$ . Here $S$ denotes spaces, left-exact means preserving finite limits, and $S p$ denotes spectra, equivalently coherent towers $Y_{n} ≃ Ω Y_{n + 1}$ . The category $S p$ is stable, so its terminal object is zero.

For a commutative ring $R$ , let $H R$ be its Eilenberg-Mac Lane ring spectrum. The derived infinity-category $D (R)$ of complexes of $R$ -modules localized at quasi-isomorphisms is equivalent to ${Mod}_{H R}$ . The forgetful functor ${Mod}_{H R} \to S p$ detects equivalences and computes finite limits. The notation $D (R)$ means $h D (R)$ .

Statement references

Triangulated category

Source contract

Given: The stated universes, a stable infinity-category, and, only for the linear target, a commutative ring $R$ .

[F1] The stable-category and spectrum-lifting interfaces are imported from the first source record. The conventional meaning of triangulated category is supplied by Triangulated category.

[F2] The forgetful limit criterion and conservativity are imported from the second source record.

[F3] The derived-module comparison is imported from the third source record.

[F4] The corepresentable mapping-space limit interface is the Yoneda and pointwise-limit consequence in the fourth source record.

External input

Import [F1], [F2], [F3] and [F4] with the hypotheses and reading limits in their source records. Infinity-categories, mapping spaces and coherent diagrams, including the mapping-space universal properties of limits and colimits, are literature inputs. No construction of these foundations is claimed here. In particular this input neither supplies an enhancement of an arbitrary triangulated category nor represents any specified cohomology theory by mapping objects.

Depends on

Used by · 9

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Jacob Lurie, Higher Algebra (September 18, 2017), stable categories and spectrum-valued lifts · Definitions 1.1.1.1 and 1.1.1.4, pp.17–18; Definitions 1.1.1.6 and 1.1.1.9 and Remark 1.1.1.13, pp.19–20; Theorem 1.1.2.14 and Remark 1.1.2.15, p.27; Proposition 1.1.3.4, pp.31–32; Corollary 1.4.2.17, p.146; Corollary 1.4.2.23, Proposition 1.4.2.24 and Remark 1.4.2.25, p.147; Definition 1.4.3.1, p.150. Printed page numbers; zero-based PDF index is printed page minus one.

Stable infinity-categories have finite limits and colimits, with pushout squares exactly pullback squares; their opposites are stable and their homotopy categories are triangulated. A finite-limit-preserving space-valued functor on a stable infinity-category lifts, through Omega-infinity, to a finite-limit-preserving spectrum-valued functor. Spectrum objects have the coherent loop-tower description. The infinity-category of spectra is stable, so its terminal object is its zero object.

Hypotheses: Fix universes so mapping spaces and diagrams under discussion are small. The source of the lifting functor is stable, and the target before stabilization is spaces, which has finite limits. The theorem on the homotopy category uses its canonical suspension and cofiber triangles.

Reading limits: Read the listed definitions and theorem/corollary statements in the actual PDF text, and excerpts of the proof of Proposition 1.1.3.4. The proofs of spectral lifting, the loop-tower comparison and triangulation are imported, not reconstructed. No enhancement of an arbitrary triangulated category is inferred.

Jacob Lurie, Higher Algebra, module limits · Corollaries 4.2.3.2 and 4.2.3.3, printed p.527 (zero-based PDF index 526).

For modules over a fixed algebra object, the forgetful functor detects equivalences, and a cone is a limit cone precisely when its underlying cone is a limit cone. In particular finite limits of modules over an Eilenberg-Mac Lane ring spectrum are computed in spectra.

Hypotheses: A monoidal infinity-category and a left-tensored infinity-category admitting the indicated limits. Here both ambient categories are spectra and the algebra is the fixed ring spectrum HR.

Reading limits: Read the actual statements of Corollaries 4.2.3.2–3 and their surrounding explanation in the PDF text. The operadic proof of Proposition 4.2.3.1 is not reconstructed. Apply with both ambient categories equal to spectra, algebra HR, and finite diagrams; spectra admit those limits.

Jacob Lurie, Higher Algebra, discrete rings and derived module categories · Remark 7.1.2.9, printed p.1211; Remark 7.1.2.12 and Theorem 7.1.2.13, p.1212; theorem proof p.1213. Zero-based PDF indices 1210–1212.

For a commutative ring R, modules over its discrete E-infinity ring spectrum HR are equivalent to the derived infinity-category of R-modules.

Hypotheses: R is a commutative ring; the derived category is the localization of unbounded complexes of R-modules at quasi-isomorphisms, in the fixed size conventions.

Reading limits: Read the actual Remark 7.1.2.9, Remark 7.1.2.12, theorem statement and displayed proof in the PDF text. The model structures and compact-generation results invoked there are imported, not reconstructed. Cochain complexes are obtained by reversing the chain-degree convention; no motivic representability or comparison is imported.

Jacob Lurie, Higher Topos Theory, Yoneda and pointwise limits · Proposition 5.1.3.2 and its proof, printed pp.320–321 (zero-based PDF indices 337–338); Proposition 5.1.2.2 and Corollary 5.1.2.3, pp.318–319 (PDF indices 335–336), also applied to opposite categories.

The Yoneda embedding preserves existing small limits. Functor-category limits are computed pointwise. Consequently Map_D(E,-):D→S preserves existing small limits, as the composite of Yoneda with evaluation at E. Applying this to D=C^op identifies Map_C(-,E) as a left-exact functor on C^op.

Hypotheses: The categories in a Yoneda application are small in a chosen larger universe; the indexing diagrams are small and the indicated limits exist. Use only finite limits here. The passage from pointwise colimits to pointwise limits uses opposite categories.

Reading limits: Read the actual propositions, Corollary 5.1.2.3 and the displayed proof of Proposition 5.1.3.2 in the PDF text. The Cartesian-fibration, Yoneda-construction and rigidification foundations are imported. The corepresentable limit assertion is the stated pointwise specialization; no foundation is proved locally.

Sources

Jacob Lurie, Higher Algebra (September 18, 2017), stable categories and spectrum-valued lifts · Definitions 1.1.1.1 and 1.1.1.4, pp.17–18; Definitions 1.1.1.6 and 1.1.1.9 and Remark 1.1.1.13, pp.19–20; Theorem 1.1.2.14 and Remark 1.1.2.15, p.27; Proposition 1.1.3.4, pp.31–32; Corollary 1.4.2.17, p.146; Corollary 1.4.2.23, Proposition 1.4.2.24 and Remark 1.4.2.25, p.147; Definition 1.4.3.1, p.150. Printed page numbers; zero-based PDF index is printed page minus one.
Jacob Lurie, Higher Algebra, module limits · Corollaries 4.2.3.2 and 4.2.3.3, printed p.527 (zero-based PDF index 526).
Jacob Lurie, Higher Algebra, discrete rings and derived module categories · Remark 7.1.2.9, printed p.1211; Remark 7.1.2.12 and Theorem 7.1.2.13, p.1212; theorem proof p.1213. Zero-based PDF indices 1210–1212.
Jacob Lurie, Higher Topos Theory, Yoneda and pointwise limits · Proposition 5.1.3.2 and its proof, printed pp.320–321 (zero-based PDF indices 337–338); Proposition 5.1.2.2 and Corollary 5.1.2.3, pp.318–319 (PDF indices 335–336), also applied to opposite categories.