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Derived free sheaf chains and unbounded Eilenberg targets

Statement

Assume the Axiom of Choice and compatible Grothendieck universes containing $ω$ . Let $X$ be any topological space whose opens form a small site in these bounds. Simplicial presheaves are contravariant functors from opens to simplicial sets. A local weak equivalence induces weak homotopy equivalences on every stalk. Let $H_{X}$ denote their infinity-localization at these equivalences, the stalkwise local homotopy convention. Write $D_{X}$ for the infinity-localization of unbounded cochain complexes of abelian sheaves at cohomology-sheaf isomorphisms. Its homotopy category is $D (Sh (X, Z))$ .

For $Y \in H_{X}$ , let $L_{X} (Y)$ be degreewise abelian sheafification of $N Z [Y]$ , in cochain degree $- n$ for normalized chain degree $n$ . Here $N (A)_{n} = ⋂_{i < n} \ker d_{i}$ , with differential $(- 1)^{n} d_{n}$ . This construction is independent up to equivalence of the simplicial representative. Abelian sheafification is exact; the stalk of $L_{X} (Y)$ is $N Z [Y_{x}]$ . A map of these complexes is a quasi-isomorphism exactly when it is so at every point.

For any unbounded $C \in D_{X}$ , there is a natural derived adjunction. For an open $U$ , let $h_{U}$ be its representable simplicial presheaf: $h_{U} (V) = {*}$ when $V \subseteq U$ and $h_{U} (V) = \emptyset$ otherwise, viewed as discrete simplicial sets. Throughout, $E_{X} (C) (U) : = {Map}_{H_{X}} (h_{U}, E_{X} (C))$ denotes derived evaluation. For $V \subseteq U$ , restriction is precomposition with $h_{V} \to h_{U}$ . Then ${Map}_{H_{X}} (Y, E_{X} (C)) ≃ {Map}_{D_{X}} (L_{X} (Y), C), E_{X} (C) (U) ≃ D K (τ_{\leq 0} R Γ (U, C)) .$ In this formula $R Γ (U, C) = R Hom (Z [U], C)$ is a derived complex of abelian groups; $Z [U]$ is the free abelian sheaf on the represented open. The good cochain truncation $τ_{\leq 0}$ keeps negative degrees, replaces degree zero by the kernel of its differential, and kills positive degrees; $D K$ first reindexes to nonnegative chains and then applies inverse Dold–Kan and forgets the abelian structure. Truncation is applied to derived sections, never to underived sections in place of a resolution. Equivalently $E_{X} (C) (U) ≃ Ω^{\infty} H (R Γ (U, C))$ . Here $D (A b)$ is the infinity-localization of unbounded cochain complexes of abelian groups at quasi-isomorphisms. For any such complex $B$ , define its Eilenberg–Mac Lane spectrum $H (B)$ by the coherent tower of pointed spaces $H (B)_{n} = {Map}_{D (A b)} (Z [0], B [n])$ for $n \geq 0$ , based at zero. The cochain shift has $B [n]^{p} = B^{p + n}$ and differential $(- 1)^{n} d_{B}$ ; the tower equivalences $H (B)_{n} ≃ Ω H (B)_{n + 1}$ are induced by $B [n] ≃ Ω B [n + 1]$ in the stable derived category. A spectrum here means such a coherent tower, and $Ω^{\infty}$ evaluates its zeroth space. This gives $Ω^{\infty} H (B) ≃ D K (τ_{\leq 0} B)$ naturally on unbounded derived complexes, with the Dold–Kan comparison established in the proof.

For pointed $Y$ and target basepoint zero, replace $L_{X} (Y)$ by ${\tilde{L}}_{X} (Y) = cofib (Z \to L_{X} (Y))$ , where the arrow is induced by the given basepoint. The same adjunction holds for pointed mapping spaces. Consequently a (reduced, in the pointed case) chain quasi-isomorphism induces an equivalence of these mapping spaces for every unbounded coefficient $C$ . No dimension, noetherian, or coefficient-continuity hypothesis is required. The space convention is the stalkwise localization, without identifying ordinary sheaves with hypercomplete sheaves.

Facts & Assumptions

[A1]

Choice is retained for the imported resolutions and model structures. Universes contain the space, its open-set site and $ω$ , with larger universes for categories; existence of these bounds is assumed separately. The Axiom of Choice Grothendieck universe closure convention

[F1]

The fixed-ring normalized Dold–Kan equivalence is Dold-Kan equivalence for simplicial modules with explicit inverse. Its finite kernel and finite degeneracy-sum formulas apply sectionwise, commute with restrictions, and extend to abelian sheaves. This supplies the indicated normalization and inverse convention.

[F2]

For $x \in X$ , let $N_{x}$ be the category of open neighbourhoods of $x$ , with inclusions as arrows. The stalk construction is $F_{x} = {colim}_{N_{x}^{o p}} F (U)$ , with transition maps given by restriction. This indexing category is filtered: $X$ is a neighbourhood and finite intersections give common refinements. For simplicial presheaves and presheaves of complexes, apply this construction in each degree; the face maps, degeneracies and differentials are induced by their sectionwise maps. The stalk of a presheaf at a point Sheafification preserves stalks; exactness of abelian sheaves is detected on stalks; filtered colimits of abelian groups are exact. Sheafification preserves stalks A sequence of abelian sheaves is exact exactly when it is exact on every stalk Filtered colimits of abelian groups are exact

[F3]

Source reference 0 supplies the local simplicial free/forgetful model adjunction, the Dold–Kan model equivalence, the ordinary inclusion/good-truncation adjunction and the reduced pointed construction. Source reference 4 identifies derived mapping spaces in these simplicial models. Thus the enriched free/forgetful adjunction on cofibrant sources and fibrant targets supplies a mapping-space adjunction after localization. The comparison with the connective part of the unbounded localization is proved in step 2.1; it is not inferred from a homotopy-category stabilization statement.

[F4]

Source reference 1 supplies unbounded derived mapping spaces and the truncation adjunction. Source reference 2 ensures that all abelian sheaves here form a Grothendieck category and admit the required unbounded K-injective replacements. These are applied to all abelian sheaves, not merely quasi-coherent ones.

[F5]

Filtered colimits of spaces and based loops supplies the filtered simplicial/space colimit model and commutation with finite homotopy limits. Applying this to finite sphere and homotopy data computes homotopy groups of stalks by stalks of homotopy presheaves. A morphism of sheaves is an isomorphism exactly when it is an isomorphism on every stalk then identifies the local weak equivalences of [F3] on this open-set site with the stalkwise weak equivalences used here. This checks the enough-points application without any dimension assumption.

[F6]

Source reference 3 supplies stability of the unbounded derived category of abelian groups and the coherent loop-tower description of spectra, with $Ω^{\infty}$ evaluation at level zero. Abelian groups are $Z$ -module sheaves on a point, so [F4] supplies their Grothendieck and resolution hypotheses. The specific spectrum $H (B)$ and its Dold–Kan comparison are constructed locally, using the derived mapping-space formula of [F4].

[F7]

Source reference 4 supplies the following universal property: for a relative category $(C, W)$ , its infinity-localization $q : C \to C [W^{- 1}]$ exists, and $q^{*} : Fun (C [W^{- 1}], T) \to Fun (C, T)$ is fully faithful with image the functors carrying $W$ to equivalences. Here an ordinary category is read via its nerve, and $Fun$ is the infinity-category of functors and coherent natural transformations. In particular, relative functors descend, and natural transformations and their homotopies between the descended functors descend as well. We apply this property to cochain complexes, quasi-isomorphisms, inclusion and good truncation. This is the enhanced input used in the local proof of the connective comparison.

External input

Import the local simplicial model and ordinary truncation interfaces from reference 0, the enhanced unbounded derived and t-structure interfaces from reference 1, and the Grothendieck/K-injective input from reference 2. Reference 3 supplies stability and spectrum towers for the inline construction of $H$ . Reference 4 supplies infinity-localization, its universal property on functor categories, and the simplicial-model mapping-space comparison. The site is the small open-set site and the ring is $Z$ , so references 0–2 apply under [A1]; reference 3 is applied to abelian groups and pointed spaces, and reference 4 to the indicated relative categories and simplicial models. The proof locally establishes the enhanced connective comparison and composes the adjunctions, checks stalks and constructs pointed reduction. It does not reconstruct the imported model structures or assume left completeness of an unbounded derived category.

Proof

Goal: Establish the statement under its stated hypotheses. Given: $X, Y, C$ and the site and size bounds in the Statement.

1.1A1F1F2F3F5

A short exact sequence of abelian presheaves remains exact on stalks, since stalks are filtered colimits and [F2] gives their exactness. Sheafification leaves these stalks unchanged by [F2]; stalkwise exactness therefore proves exactness of abelian sheafification. Free abelian groups commute with colimits, and normalization uses finite kernels, so [F1] and exact filtered colimits in [F2] identify the stalk of $L_{X} (Y)$ with $N Z [Y_{x}]$ . Apply stalkwise exactness to the cone to detect quasi-isomorphisms. By [F5] the local weak equivalences here are those of [F3], which become free-chain quasi-isomorphisms. Hence the formula for $L_{X}$ is well-defined in the stated localizations and agrees with the derived free functor.

2.1F1F3F4F7step 1.1

Put $A = Sh (X, Z)$ and let $C_{0}$ be the ordinary category of its cochain complexes that vanish in positive degrees. Let $Q = C_{0} [W^{- 1}]$ , with $W$ the cohomology-sheaf isomorphisms. The ordinary inclusion $i : C_{0} \to Ch (A)$ has right adjoint $t = τ_{\leq 0}$ : a chain map from a complex vanishing in positive degrees factors uniquely through the kernel of the target differential in degree zero. Its unit is $i d \to t i = i d$ and its counit is the natural inclusion $ϵ_{C} : i t C \to C$ ; hence $t ϵ_{C} = {i d}_{t C}$ and $ϵ_{i A} = {i d}_{i A}$ . Both functors preserve quasi-isomorphisms, since $H^{p} (t C) = H^{p} (C)$ for $p \leq 0$ and $H^{p} (t C) = 0$ for $p > 0$ . By [F7] they induce $\overset{ˉ}{i} : Q \to D_{X}$ and $\overset{ˉ}{t} : D_{X} \to Q$ , with descended counit and identities. Full faithfulness of restriction on functor infinity-categories descends these natural transformations and their relations, not merely their effects on homotopy categories. For $A \in Q$ and $C \in D_{X}$ , apply $\overset{ˉ}{t}$ to define $Φ : {Map}_{D_{X}} (\overset{ˉ}{i} A, C) \to {Map}_{Q} (A, \overset{ˉ}{t} C)$ , using $\overset{ˉ}{t} \overset{ˉ}{i} ≃ i d$ . Define the reverse map by $Ψ (g) = ϵ_{C} \circ \overset{ˉ}{i} (g)$ . The relation $\overset{ˉ}{t} ϵ ≃ i d$ gives $Φ Ψ ≃ i d$ . Naturality of $ϵ$ and $ϵ_{\overset{ˉ}{i} A} ≃ i d$ give $Ψ Φ (f) = ϵ_{C} \circ \overset{ˉ}{i} \overset{ˉ}{t} (f) ≃ f \circ ϵ_{\overset{ˉ}{i} A} ≃ f$ . These are homotopies of maps of entire mapping spaces, inherited from natural transformations, and are natural in $A, C$ . Thus $\overset{ˉ}{i} ⊣ \overset{ˉ}{t}$ and $\overset{ˉ}{i}$ is fully faithful. Its image consists exactly of complexes with vanishing positive cohomology: $ϵ_{C}$ is a quasi-isomorphism precisely for those $C$ . This identifies $Q$ with $D_{X}^{\leq 0}$ and its coreflection with the standard cochain truncation of [F4], by the defining mapping property. Now compose [F3]'s derived free/forgetful simplicial adjunction with normalized Dold–Kan and this locally established inclusion/right-truncation adjunction. The simplicial mapping-space identification in [F3] ensures that the first adjunction is enhanced, so composition gives $L_{X} ⊣ E_{X}$ on mapping spaces. By definition, derived evaluation at $U$ is $E_{X} (C) (U) = {Map}_{H_{X}} (h_{U}, E_{X} (C))$ . Apply the adjunction to $h_{U}$ : its free chains are $Z [U]$ in degree zero, so derived evaluation is naturally ${Map}_{D_{X}} (Z [U], C)$ . Inclusions of opens induce the restriction maps on both sides by precomposition. The K-injective mapping formula of [F4] identifies this with $D K (τ_{\leq 0} R Hom (Z [U], C))$ , naturally in restriction maps. This proves the displayed derived-section formula for arbitrary unbounded $C$ , without exchanging stalks and unbounded coefficients.

3.1F1F4F6step 2.1

By [F6], $D (A b)$ is stable. The loop object of $B [n + 1]$ is the pullback $0 \times_{B [n + 1]}^{h} 0 ≃ B [n]$ . By the limit universal property, mapping out of $Z [0]$ carries this pullback to the based-loop pullback of spaces. The resulting equivalences $H (B)_{n} ≃ Ω H (B)_{n + 1}$ are natural in $B$ ; iterating the suspension-loop equivalence in the stable category supplies their coherence. Thus the stated tower defines a spectrum functor on unbounded derived complexes. For a K-injective replacement $B [n] \to I$ , evaluation at $1 \in Z$ identifies the Hom complex from $Z [0]$ to $I$ with $I$ . Since this Hom functor is exact, its derived Hom complex is naturally $B [n]$ . The mapping formula of [F4] gives $H (B)_{n} ≃ D K (τ_{\leq 0} B [n])$ . Evaluating at zero and taking $B = R Γ (U, C)$ proves the asserted spectrum comparison, naturally with respect to restriction in $U$ ; no boundedness or limit reconstruction of $B$ is used.

4.1F3F4step 2.1∎

A pointed mapping space is the homotopy fiber, over the zero map, of evaluation at the source basepoint. Under step 2.1 this is the fiber of $Map (L_{X} (Y), C) \to Map (Z, C)$ ; the cofiber universal property identifies it with $Map ({\tilde{L}}_{X} (Y), C)$ . This is also the reduced free construction of [F3]. A quasi-isomorphism becomes an equivalence in $D_{X}$ , and precomposition with an equivalence is an equivalence on mapping spaces. For a pointed map which is a quasi-isomorphism on unreduced chains, the induced map on reduced chains is likewise an equivalence by the cofiber square with identity on $Z$ . This proves both detection assertions under exactly the stated hypotheses.

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External dependencies

J. F. Jardine, Lectures on Local Homotopy Theory, Lecture 06 · Section 13, printed pp. 1–11: free/forgetful adjunction, Lemmas 13.1 and 13.3, Proposition 13.5 and Corollary 13.6; Section 14, p. 12, inclusion/good-truncation adjunction; p. 16, reduced pointed adjunction

On a small site, the free/forgetful and sheafification functors give simplicial model adjunctions for modules over a presheaf of commutative unital rings. Normalization transports local weak equivalences to homology-sheaf isomorphisms. Inclusion of nonnegative chain complexes has good truncation as right adjoint. Reduced free modules give the pointed adjunction.

Hypotheses: A fixed small Grothendieck site and a presheaf of commutative unital rings; use the constant ring Z and the open-set site of a topological space. Translate homological degree n to cochain degree -n.

Reading limits: Freshly read the named adjunctions, Lemmas 13.1 and 13.3, Proposition 13.5, Corollary 13.6, the start of Section 14 and its pointed paragraph in the live author-hosted extraction on 12 November 2026. Model existence is imported. Proposition 14.3 is not used to infer an enhanced connective comparison; that comparison is proved locally in Proof 2.1. No reconstruction of Boolean localization or full-course review is claimed.

Jacob Lurie, Higher Algebra, September 18, 2017 · Remark 1.3.1.12 and Construction 1.3.1.13, printed pp. 83–84; Definition 1.3.5.8 and Propositions 1.3.5.13–15, pp. 125–127; Proposition 1.3.5.21(1)–(2), p. 129; Proposition 1.2.1.5 and Notation 1.2.1.7, p. 38

For a Grothendieck abelian category, unbounded derived localization has a fibrant K-injective dg model. Its mapping spaces are Dold–Kan of good truncation of derived Hom. The derived category has the standard t-structure and truncation adjoints. These inputs do not alone identify the separate localization of strictly connective complexes with its connective full subcategory; Proof 2.1 supplies that identification.

Hypotheses: A Grothendieck abelian category in fixed compatible universes. Use abelian sheaves, reverse homological grading to cochains, and truncate after derived replacement. No left completeness of the t-structure is assumed.

Reading limits: Freshly read Remark 1.3.1.12, Construction 1.3.1.13, Definition 1.3.5.8, Propositions 1.3.5.13–15 and their proofs, Proposition 1.3.5.21(1)–(2), Proposition 1.2.1.5 and Notation 1.2.1.7 in the live author PDF extraction on 12 November 2026. These enhanced-derived interfaces are imported. The source uses homological indexing: its connective right truncation becomes good cochain truncation at degree zero here. No left-completeness or full-book review is asserted.

The Stacks Project, Remark 19.13.3, tag 079Q · Remark 19.13.3, both paragraphs and the displayed derived Hom formula

Module sheaves on a ringed site form a Grothendieck abelian category; every unbounded complex admits a K-injective resolution computing derived morphisms.

Hypotheses: All module sheaves on a ringed site; apply to the constant sheaf Z on the open-set site, with fixed universe bounds. No boundedness or finiteness assumption.

Reading limits: Freshly read the complete live remark, its displayed derived Hom formula and its module-sheaf example on 12 November 2026. K-injective existence is imported from the cited result, not reconstructed. The infinity-categorical enhancement comes from reference 1, not from this homotopy-category remark.

Jacob Lurie, Higher Algebra, September 18, 2017 — stability and spectrum objects · Proposition 1.3.5.9, printed pp. 125–126; Notation 1.4.2.20, p. 146; Proposition 1.4.2.24 and Remark 1.4.2.25, p. 147

The unbounded derived infinity-category of a Grothendieck abelian category is stable. Spectra are coherent towers of pointed spaces with successive loop equivalences; the infinite-loop-space functor evaluates at level zero.

Hypotheses: For stability use the Grothendieck category of abelian groups. For the tower description use pointed spaces, which admit finite limits. Work in the stated compatible universes. Cochain suspension is B[1], so B[n] is the loop object of B[n+1]. No boundedness or completeness hypothesis is imposed.

Reading limits: Freshly read Proposition 1.3.5.9 and its proof, Notation 1.4.2.20, Proposition 1.4.2.24 and Remark 1.4.2.25 in the live author PDF extraction on 12 November 2026. Stability and the spectrum-tower interface are imported; the particular tower and Dold–Kan comparison are local constructions below. No full-book review is claimed.

Jacob Lurie, Higher Algebra — infinity-localization and simplicial model mapping spaces · Definition 1.3.4.1 and Remark 1.3.4.2, printed p. 108; Definition 1.3.4.15 and Remark 1.3.4.16, p. 116; Theorem 1.3.4.20, p. 117

Infinity-localization exists and restriction on functor infinity-categories is fully faithful, with image the functors inverting the chosen arrows. For a simplicial model category, the nerve of its fibrant-cofibrant simplicial subcategory presents its infinity-localization; functorial replacement also permits localization of all objects.

Hypotheses: Categories and chosen arrows in compatible universes. For the model comparison use the simplicial local models of reference 0, with functorial factorizations, and the simplicial presheaf local model appearing there. Theorem 1.3.4.20 is not applied to the nonsimplicial unbounded complex model; use reference 1 for that model.

Reading limits: Freshly read Definition 1.3.4.1, Remark 1.3.4.2, Definition 1.3.4.15, Remark 1.3.4.16 and the statement of Theorem 1.3.4.20 in the live author PDF extraction on 12 November 2026. The localization and model comparison are imported, without reconstruction of their foundational proofs. Descent of the explicit connective inclusion/truncation and the inverse mapping-space formulas are proved locally below.

Sources

J. F. Jardine, Lectures on Local Homotopy Theory, Lecture 06 · Section 13, printed pp. 1–11: free/forgetful adjunction, Lemmas 13.1 and 13.3, Proposition 13.5 and Corollary 13.6; Section 14, p. 12, inclusion/good-truncation adjunction; p. 16, reduced pointed adjunction
Jacob Lurie, Higher Algebra, September 18, 2017 · Remark 1.3.1.12 and Construction 1.3.1.13, printed pp. 83–84; Definition 1.3.5.8 and Propositions 1.3.5.13–15, pp. 125–127; Proposition 1.3.5.21(1)–(2), p. 129; Proposition 1.2.1.5 and Notation 1.2.1.7, p. 38
The Stacks Project, Remark 19.13.3, tag 079Q · Remark 19.13.3, both paragraphs and the displayed derived Hom formula
Jacob Lurie, Higher Algebra, September 18, 2017 — stability and spectrum objects · Proposition 1.3.5.9, printed pp. 125–126; Notation 1.4.2.20, p. 146; Proposition 1.4.2.24 and Remark 1.4.2.25, p. 147
Jacob Lurie, Higher Algebra — infinity-localization and simplicial model mapping spaces · Definition 1.3.4.1 and Remark 1.3.4.2, printed p. 108; Definition 1.3.4.15 and Remark 1.3.4.16, p. 116; Theorem 1.3.4.20, p. 117