Scope and manuscript
The attached full manuscript adapts Jan R. Strooker's Introduction to Categories, Homological Algebra and Sheaf Cohomology to the language of infinity-categories. It follows the four parts of the source through representability, exactness, derived functors, and sheaf cohomology. The appendix-level concordance in Chapter 30 locates the classical constructions. The proofs below give the core comparisons. The PDF contains their full development, the classical recovery statements, and computations on finite sites, spheres, cyclic actions, and the relative projective line.
We use the homotopy convention $\pi_i X=H^{-i}(X)$, so suspension is the cochain shift $[1]$. Write $W_{a,b}X=\tau_{\le b}\tau_{\ge a}X$. We call a $t$-structure left complete if $X\to\lim_n\tau_{\le n}X$ is an equivalence for all $X$, and right complete if $\operatorname{colim}_m\tau_{\ge-m}X\to X$ is an equivalence for all $X$. Size conventions and foundational references appear in the PDF.
Representability and extensions
In a stable infinity-category $\mathcal C$, the space of cofiber sequences $A\to E\to B$ with specified outer terms is $\operatorname{Map}_{\mathcal C}(B,A[1])$. The boundary sends a sequence to a point of this mapping space, and taking the fiber of that boundary gives the inverse. These constructions preserve the coherences because both are functorial on the specified diagrams. Pushout in $A$ acts by postcomposition on the boundary, and pullback in $B$ acts by precomposition. Thus Baer addition is induced by the diagonal of $B$ and the codiagonal of $A[1]$.
The spaces $\operatorname{Map}(B,A[n])$, based at zero, are successive infinite loop spaces of the mapping spectrum. Their homotopy in degree $i\ge0$ is $\operatorname{Hom}_{h\mathcal C}(B,A[n-i])$. For an abelian category and its derived enhancement, the components recover the usual Yoneda extension classes. The comparison with higher extension categories in the exact setting uses the results of Retakh, Klemenc, and Børve–Trygsland.
Finite-window mapping comparison
Let $X$ admit connective reconstruction and $Y$ admit coconnective reconstruction in a stable $t$-category $\mathcal C$. There is a natural equivalence $$ \operatorname{Map}_{\mathcal C}(X,Y)\simeq \lim_{m,n\ge0}\operatorname{Map}_{\mathcal C} (W_{-m,n}X,W_{-m,n}Y). $$ The two indices are inverse in the mapping-space diagram, whose transitions apply the appropriate truncation to both arguments.
Indeed, the hypotheses and the mapping-space tests for limits and colimits first give the limit of $\operatorname{Map}(\tau_{\ge-m}X,\tau_{\le n}Y)$. The left adjunction of $\tau_{\le n}$ replaces the source by $W_{-m,n}X$. The right adjunction of $\tau_{\ge-m}$ then replaces the target by $W_{-m,n}Y$. Naturality gives the transitions.
Consequently, an exact $t$-exact functor between left and right complete stable $t$-categories is fully faithful if it induces isomorphisms on $\operatorname{Hom}(A,B[q])$ for all heart objects $A,B$ and $q\ge0$. These tests give equivalences of the mapping spectra on heart objects: the remaining groups vanish by orthogonality. Finite Postnikov induction in each variable gives the equivalences for bounded objects. The displayed formula then gives full faithfulness. An equivalence of hearts alone omits the positive extension degrees. For example, the forgetful functor from $H\mathbb Z$-modules to spectra agrees on hearts, but the stable operation $\mathrm{Sq}^2$ gives a degree-two map of mod-two Eilenberg–Mac Lane spectra that does not come from $\mathrm D(\mathbb Z)$.
Finite cohomological amplitude
Let $F:\mathcal C\to\mathcal D$ be exact with $F(\mathcal C_{\ge0})\subseteq\mathcal D_{\ge-d}$ and $F(\mathcal C_{\le0})\subseteq\mathcal D_{\le0}$. Then for $a\le b$ there is a natural equivalence $$ W_{a,b}F(X)\simeq W_{a,b}F(W_{a,b+d}X). $$ To prove this, apply $F$ to $\tau_{\ge a}X\to X\to\tau_{\le a-1}X$. The last term lands in degrees at most $a-1$, so the first map becomes an equivalence after retaining the output window. Apply $F$ to the comparison from $\tau_{\ge a}X$ to $W_{a,b+d}X$. Its fiber is $F(\tau_{\ge b+d+1}X)$ and lies in degrees at least $b+1$, so this comparison also becomes an equivalence. This proves the equivalence by the resulting span.
If $\pi_{-d}F(A)\ne0$ for a heart object $A$, the input layer in degree $b+d$ is necessary, as witnessed by $X=A[b+d]$. For a nonempty finite poset with maximum chain length $d+1$, derived limit has the stated amplitude bound: the constant diagram has the projective resolution whose $n$th term is the sum of the free evaluation diagrams at the last vertices of strict $n$-chains. At each vertex the augmented chain complex contracts by appending that vertex. The face poset of the boundary of a $(d+1)$-simplex realizes the upper endpoint of the bound.
Satellites as a Kan extension
For an abelian category $\mathcal A$ with enough injectives and a left exact additive functor $T:\mathcal A\to\mathrm{Ab}$, the degree-$n$ enriched left Kan extension along the degree-zero inclusion into the graded Yoneda extension category is $$ \int^{A\in\mathcal A}\operatorname{Ext}^n(A,B)\otimes_{\mathbb Z}T(A) \simeq R^nT(B). $$ The enriched hom group from $(A,r)$ to $(B,s)$ is $\operatorname{Ext}^{s-r}(A,B)$ for $s\ge r$ and zero otherwise. The enriched coend balances the Yoneda action against the action of $T$ on maps. Its comparison with $R^nT(B)$ applies the derived functor to the extension arrow $A\to B[n]$ and takes $H^0$.
For $0\to B\to I\to C\to0$ with $I$ injective, the weight $\operatorname{Ext}^1(-,B)$ is the cokernel of $\operatorname{Hom}(-,I)\to\operatorname{Hom}(-,C)$. The coend preserves cokernels and evaluates the two represented weights by Yoneda, giving $\operatorname{coker}(T(I)\to T(C))=R^1T(B)$. In higher degrees the two sides are identified by dimension shifting with degree $n-1$ for $C$. The comparison is compatible with these boundaries because they are composition with the extension class. Degree zero is the additive Yoneda formula. The manuscript gives the dual right Kan construction for left derived functors and compares it with spectral enrichment.
Derived sheaves and hyperdescent
For a small ordinary site and a fixed ordinary ring $R$, write $a$ for exact sheafification of presheaves of $R$-modules, with fully faithful right adjoint $i$. Passing to derived infinity-categories gives an exact localization $\mathrm D(a)$ with fully faithful right adjoint $\mathrm Ri$. If $I$ is $K$-injective as a complex of sheaves, $iI$ is $K$-injective as a complex of presheaves: for every acyclic presheaf complex $P$, the mapping complex into $iI$ is the mapping complex from $aP$ into $I$, which is acyclic. The counit is represented by $aiI=I$. Exactness of $a$ identifies the inverted maps as the local quasi-isomorphisms, tested by sheafification of all cohomology presheaves. The right adjoint takes $K$ to the diagram of derived section objects $U\mapsto \mathrm R\Gamma(U,K)$.
A hypercover of $U$ gives a local quasi-isomorphism from the realization of its free module diagrams to the free module on $U$. Mapping into a local object therefore gives the descent totalization. Conversely the cofiber of the unit for a hyperdescent diagram satisfies hyperdescent and has zero homotopy sheaves. Taking each infinite loop space level gives a hypercomplete space-valued sheaf with trivial homotopy sheaves, hence a terminal object. This shows that the local derived diagrams are exactly the hyperdescent diagrams in these coefficients.
The following example separates this descent condition from left completeness. Let $X$ be the one-point compactification of $\coprod_{n\ge1}S^n$, and let $A_n$ be the extension by zero of the constant field $k$-sheaf on the clopen $S^n$. The coconnective truncations of $K=\bigoplus_{n\ge1}A_n[n]$ have limit $\prod_{n\ge1}A_n[n]$. At the added point the stalk of $K$ is zero. For the product, derived sections on the neighborhood containing the added point and the spheres $S^n$ for $n\ge N$ have degree-zero homotopy $\prod_{n\ge N}k$. Each factor comes from the top cohomology of its sphere. The stalk of the homotopy sheaf at the added point is the colimit under deleting finitely many coordinates, namely $\prod_n k/\bigoplus_n k$, which is nonzero. Thus the Postnikov comparison fails in the derived category even though its coefficient diagrams satisfy hyperdescent.
A prescribed differential on a finite site
Let $d\ge2$, let $P_d$ be the face poset of the boundary of a $(d+1)$-simplex, and use constant diagrams of vector spaces $A,B$ over a field $k$. The choice of an orientation gives $$ \operatorname{Hom}(\underline A,\underline B[d]) \cong \operatorname{Hom}_k(A,B) $$ in the stable diagram category. For $u:A\to B$ take the corresponding map $\widehat u$ and its fiber $K_u$. The homotopy sheaves in degrees $0,d-1$ are $\underline A,\underline B$, independent of $u$. At every vertex the coefficient splits. With $F=\lim_{P_d}$, the map $F(\widehat u)$ has only one possible nonzero matrix entry between equal-degree summands, and that entry is $u$, as can be seen by composing $\widehat u$ with constant sections of $\underline A$. The section object is therefore $$ F(K_u)\simeq A[-d]\oplus B[d-1]\oplus\operatorname{fib}(u:A\to B). $$ The hypercohomology spectral sequence has two rows, in cohomological degrees $0,1-d$, and only columns $0,d$. The first possible differential is $d_d$ from $(0,0)$ to $(d,1-d)$. The two-layer triangle identifies it with $u$ up to the sign of the orientation. This gives a finite-site computation in which stalkwise complexes and homotopy sheaves agree, while the section objects depend on the higher extension.
Sources and preparation
- J. R. Strooker, Introduction to Categories, Homological Algebra and Sheaf Cohomology, Cambridge Tracts in Mathematics 73 (1978). Publisher's front matter
- J. Lurie, Higher Topos Theory and Higher Algebra.
- E. D. Børve and P. Trygsland, A theorem of Retakh for exact infinity-categories and higher extension functors.
- J. Klemenc, The stable hull of an exact infinity-category.
- N. Spaltenstein, Resolutions of unbounded complexes.
- D. Dugger, S. Hollander, and D. C. Isaksen, Hypercovers and simplicial presheaves.
- D. Clausen and A. Mathew, Hyperdescent and étale K-theory.
The PDF has the complete bibliography and identifies the inputs from prior work. The manuscript was prepared with AI assistance and typeset with LaTeX. The byline is anonymous for editorial consideration.