# Bounded section recognition from injective mapping complexes This proof concerns ordinary infinity-sheaves on the open-set site of an arbitrary topological space \(X\), with coefficients \(D_\infty(k)\), for any commutative unital ring \(k\). Every module sheaf is allowed. There is no dimension, countability, constructibility, finite-global-dimension or perfection hypothesis. The categorical floor is the ordinary module-sheaf infinity-category and its standard t-structure: in cohomological notation its \(\geq a\) part consists of objects whose section complexes on **every** open have lower bound \(a\), and its \(\leq b\) part is characterized by vanishing cohomology sheaves above \(b\). Its heart is the ordinary category of \(k\)-module sheaves. These precise properties and existence of the t-structure are the specialization of DAG VIII, Proposition 2.1.3 and Remark 2.1.5. Ordinary spectral sheafification, the differential graded nerve and coefficient spectra are also part of this categorical floor. The proof does not assume a general derived-heart realization theorem or left completeness of the entire ordinary sheaf category. The classical floor consists of bounded-below injective resolutions, their mapping complexes, enough injectives, and exact coproducts and filtered colimits of module sheaves. The required sequential derived colimit is constructed below by a mapping telescope, rather than assumed from a general unbounded derived-category existence theorem. *Written by GPT-6.1 Sol (OpenAI). Self-checked by the writing AI. CC0 1.0.* The injective mapping complexes give full faithfulness, and a mapping telescope realizes every sheaf with the stated lower section bound. ## SH02-BSR-1 — The actual functor {#SH02-BSR-1} Choose a bounded-below injective resolution \(K\to I\) and set \[ T_XK(U)=\Gamma(U;I). \tag{BSR.1} \] The [injective spectral Čech calculation](../modern-proper-support-construction.html#SH02-SXM-8) proves descent for this coefficient complex. It also proves that an injective sheaf \(J\), viewed as a heart object, has section values \(J(U)[0]\). The differential graded section functors respect composition, differentials and mapping complexes, so (BSR.1) is an enhanced functor rather than a selection of objects and cohomology groups. Bounded-below injective complexes model the classical derived infinity-category; replacing a resolution by a chain-homotopy equivalent one gives the same functor. Cones, shifts and zero objects are computed in this model, so \(T_X\) is exact. Its stalk is the original stalk complex, since filtered colimits of modules are exact. It is t-exact for the stated t-structures: a classical lower bound has an injective resolution with that same lower term bound, whereas a classical upper cohomology-sheaf bound is detected on its stalks. In particular it identifies the ordinary hearts. ## SH02-BSR-2 — A bounded stalk comparison without recognition {#SH02-BSR-2} **Lemma.** If \(E\) is an ordinary module-valued infinity-sheaf with a single lower section bound \(a\), and all its stalk complexes are zero, then \(E=0\). **Proof.** Shift to \(a=0\). The presheaf \(U\mapsto H^0(E(U))\) is already an ordinary sheaf. To see this, use covering descent for \(E\). Its Čech totalization has section degree and Čech degree both nonnegative; total degree zero is exactly the equalizer of the degree-zero groups in Čech degrees zero and one. Products commute with degree-zero cohomology for bounded-below complexes of modules. Thus the ordinary sheaf gluing equation holds. The same argument is valid for any open cover, not just a finite one. Its stalk is \(H^0(E_x)\), because filtered coefficient colimits are exact. Hence it is the zero ordinary sheaf, and \(H^0(E(U))=0\) on every open. The section lower bound has improved to one. Shift by one and repeat. After the \(n\)-th repetition all section cohomology through degree \(n-1\) is zero. There were no negative groups to begin with, so every section complex is zero. This is an objectwise equivalence to the zero sheaf. \(\square\) Consequently a stalk equivalence between two objects with a common lower section bound is an equivalence: its fibre retains that lower bound, has zero stalks, and the lemma applies. This lemma supplies the bounded separation needed below without invoking bounded recognition circularly. ## SH02-BSR-3 — Spectral Ext into a heart injective {#SH02-BSR-3} Let \(J\) be an injective ordinary module sheaf, regarded as a heart object of the infinity-sheaf category. For every ordinary module sheaf \(M\), \[ \operatorname{Hom}(M,J[q])= \begin{cases} \operatorname{Hom}_{k_X}(M,J),&q=0,\\ 0,&q\ne0. \end{cases} \tag{BSR.2} \] Negative degrees vanish by the t-structure, and degree zero is the heart identification. Here is a proof for all positive degrees, including the local-to-global step. For an open \(U\), the free local module \(k_U\), extended by zero, represents sections in the spectral module-sheaf category. This follows directly by sheafifying the free coefficient presheaf on the open representable. It is a heart object: its stalk is \(k\) in \(U\) and zero outside, and spectral sheafification retains its section lower bound zero. Mapping it to \(J[q]\) gives \(H^q(J(U)[0])\), hence zero for \(q>0\). There is an epimorphism \[ P=\bigoplus_{(U,s),\ s\in M(U)}k_U\longrightarrow M. \tag{BSR.3} \] It is stalkwise surjective: every stalk element has a section representative. The coproduct is the coproduct of these heart objects, also in the spectral category. For clarity, its section lower bound zero is preserved by the coproduct and sheafification; its stalks are the direct sums of the degree-zero stalks and have no other cohomology. It is therefore the same ordinary heart coproduct. Maps out of that coproduct are products of the maps from its summands. Thus \(\operatorname{Hom}(P,J[q])=0\) for every \(q>0\). Let \(N\) be the kernel in (BSR.3). The short exact heart sequence gives the fibre sequence \(N\to P\to M\). For \(q=1\), its long exact mapping sequence identifies \(\operatorname{Hom}(M,J[1])\) with the cokernel of \(\operatorname{Hom}(P,J)\to\operatorname{Hom}(N,J)\), which is zero by ordinary injectivity of \(J\). For \(q\geq2\), it identifies \(\operatorname{Hom}(M,J[q])\) with \(\operatorname{Hom}(N,J[q-1])\). Induction on \(q\), applied to all ordinary sheaves at once, proves (BSR.2). ## SH02-BSR-POSTNIKOV — The bounded sheaf colimit {#SH02-BSR-POSTNIKOV} If an ordinary infinity-sheaf \(E\) has one actual section lower bound \(a\), its finite cohomological truncations \(E_n=\tau_{\leq n}E\), \(n\geq a\), retain that bound. Their colimit \(E'=\operatorname{colim}_n E_n\) in the ordinary infinity-sheaf category retains it too. Indeed this is the sheafification of the pointwise filtered colimit; the filtered-colimit and sheafification stability of this t-structure part is the stated categorical floor, DAG VIII, Proposition 2.1.3(7), with grading reversed. One may also use the plus construction: its covering limits, filtered colimits and transfinite iterations preserve the same lower section bound. The canonical map \(E'\to E\) is a stalk equivalence: stalks preserve sheaf colimits, coefficient filtered colimits are exact, and every fixed stalk cohomology group has stabilized in the finite truncations. BSR-2 therefore makes this actual map an equivalence. This proves bounded Postnikov-colimit convergence before full faithfulness or recognition is used. It makes no assertion that derived sections on an arbitrary open commute with a filtered colimit. For \(E=T_XK\), exactness and t-exactness in BSR-1 give the canonical comparisons \(E_n=T_X(\tau_{\leq n}K)\), including their truncation maps. Thus the preceding sheaf-category colimit calculation applies to precisely these classical finite truncations. ## SH02-BSR-4 — Full faithfulness on mapping spectra {#SH02-BSR-4} First take a target consisting of one injective \(J\) in degree zero. For any \(K\in D^+(k_X)\) the natural comparison gives \[ \begin{gathered} \operatorname{Hom}(T_XK,J[q])\\ \simeq\operatorname{Hom}_{k_X}(H^{-q}K,J)\\ \simeq H^q\operatorname{Hom}^{\bullet}(K,J). \end{gathered} \tag{BSR.4} \] First take \(K_n=\tau_{\leq n}K\), which has a finite cohomological filtration. BSR-3 applies to every heart quotient and gives \[ \operatorname{Hom}(T_XK_n,J[q]) =\operatorname{Hom}_{k_X}(H^{-q}K_n,J). \tag{BSR.4a} \] The filtration's long exact mapping sequences, or its finite spectral sequence with only the injective row, give this equality with the actual comparison induced by \(T_X\). It agrees on the heart, and its finite extension calculation is natural in the truncation maps. We now justify passage to the possibly unbounded-above source. BSR-POSTNIKOV proves \(T_XK=\operatorname{colim}_n T_XK_n\) in the sheaf category, without recognition. Mapping that colimit into \(J\) gives the inverse limit of the spectra \(M_n=\operatorname{Map}(T_XK_n,J)\). For every integer \(t\), the finite calculation gives \[ \pi_t M_n=\operatorname{Hom}_{k_X}(H^tK_n,J). \tag{BSR.4b} \] This group tower and its maps are eventually constant, with value \(\operatorname{Hom}_{k_X}(H^tK,J)\). The homotopy inverse limit is the fibre of \(1-\mathrm{shift}\) on the product of the spectra \(M_n\). On each homotopy-group tower the same map is surjective: in the isomorphism tail choose one starting coordinate and solve recursively forwards by the inverse transition maps, then solve backwards through the finite initial segment. Every coordinate uses only finitely many operations. The resulting long exact homotopy sequence has zero cokernel term from the next homotopy-group tower, and hence \[ \begin{gathered} \pi_t\operatorname{Map}(T_XK,J)=\lim_n\pi_t M_n\\ =\operatorname{Hom}_{k_X}(H^tK,J). \end{gathered} \tag{BSR.4c} \] The compatible finite comparisons induce this identification, so it is the original natural comparison on every homotopy group. This is a limit calculation for mapping spectra, with no exactness of products or inverse limits of arbitrary sheaves. It supplies the first equality of (BSR.4) for the full source range. No orthogonality from a high-degree source to a low-degree target is presumed. The second equality is the ordinary exactness of \(\operatorname{Hom}_{k_X}(-,J)\). Running this calculation for every shift \(q\), including after shifting the source, proves equivalence of the whole mapping spectra into \(J\). Their homotopy groups are these shifted morphism groups; a map of spectra inducing all these isomorphisms is an equivalence. Now let \(L\) have a bounded-below injective resolution \(I\), with \(I^m=0\) for \(m