Chain Homotopy, Quasi-Isomorphisms, and Derived Vocabulary

Chain homotopy and quasi-isomorphism compare complexes at two different resolutions. A chain homotopy is explicit degree-shifting data showing that two maps differ by a boundary. A chain-homotopy equivalence has an inverse up to such data. A quasi-isomorphism asks only that a specified chain map induce isomorphisms on every homology group. Consequently,

$$\text{chain isomorphism} \Longrightarrow \text{chain-homotopy equivalence} \Longrightarrow \text{quasi-isomorphism},$$

but neither converse holds for general complexes of modules. The mapping cone makes the distinction testable: a map is a quasi-isomorphism exactly when its cone is acyclic, whereas it is a chain-homotopy equivalence exactly when its cone is contractible. Derived language then makes quasi-isomorphisms formally invertible without pretending that they possess degree-by-degree inverses.

Required background. Categories, Functors, Natural Transformations, and Universal Properties and Chains, Homology, Cohomology, and Exact Sequences. They supply the categories, kernels, images, homology, and long exact sequences used below.

Chain complexes and grading conventions

Let $R$ be a ring. Unless stated otherwise, all complexes are chain complexes of left $R$-modules,

$$\cdots\longrightarrow C_{n+1} \xrightarrow{\partial_{C,n+1}}C_n \xrightarrow{\partial_{C,n}}C_{n-1} \longrightarrow\cdots, \qquad \partial_{C,n}\partial_{C,n+1}=0.$$

The complexes may be unbounded, although the worked examples are finite. The same definitions make sense in an additive category when only sums and homotopies are needed. Kernels, images, homology, acyclicity, and the long exact homology sequence require an abelian setting.

A chain map $f:C\to D$ is a degree-zero family $f_n:C_n\to D_n$ satisfying

$$\partial_{D,n}f_n=f_{n-1}\partial_{C,n} \qquad\text{for every }n.$$

The equation is the typed reason that $f_n$ sends cycles to cycles and boundaries to boundaries. It therefore induces $H_n(f):H_n(C)\to H_n(D)$.

Chain homotopy compares maps

Let $f,g:C\to D$ be chain maps. A chain homotopy from $g$ to $f$ is a degree-$+1$ family

$$h_n:C_n\longrightarrow D_{n+1}$$

such that

$$(f-g)_n = \partial_{D,n+1}h_n+h_{n-1}\partial_{C,n}. \tag{1}$$

Both terms on the right have type $C_n\to D_n$. We write $f\simeq g$. Equation (1) does not say that $f_n=g_n$; it says that their difference is controlled by the differentials.

Why homotopic maps agree on homology

Take a cycle $z\in C_n$, so $\partial_Cz=0$. Equation (1) gives

$$f_n(z)-g_n(z) = \partial_Dh_n(z)+h_{n-1}(0) = \partial_Dh_n(z).$$

Thus $f_n(z)$ and $g_n(z)$ differ by a boundary in $D_n$, and hence

$$H_n(f)[z]=H_n(g)[z].$$

Therefore chain-homotopic maps induce the same map on every homology group. The converse is false in general: equality of induced homology maps need not produce the degree-$+1$ maps in (1).

Homotopy is compatible with composition. If $a:B\to C$ and $b:D\to E$ are chain maps, then $f\simeq g$ implies $f\circ a\simeq g\circ a$ and $b\circ f\simeq b\circ g$. This is why maps modulo chain homotopy form a category rather than merely a set of ad hoc equivalence classes. These chain-complex, homotopy, and induced-homology conventions agree with Stacks Project Authors 2026, §12.13, tags 010V, 010X, 010Z, 0110, 0113, 0115, and 0116.

Homotopy equivalence keeps chain-level control

A chain map $f:C\to D$ is a chain-homotopy equivalence if there is a chain map $g:D\to C$ with

$$g f\simeq\operatorname{id}_C, \qquad f g\simeq\operatorname{id}_D.$$

The map $g$ is a homotopy inverse, not necessarily an actual inverse in the category of chain complexes. Applying homology and using the preceding proof shows that $H_n(g)$ is the inverse of $H_n(f)$ for every $n$. Every chain-homotopy equivalence is therefore a quasi-isomorphism.

A finite interval contracts onto either endpoint

Let $I$ be the chain complex

$$0\longrightarrow \mathbb Z e \xrightarrow{\partial} \mathbb Z v_0\oplus\mathbb Z v_1 \longrightarrow0, \qquad \partial e=v_1-v_0,$$

in degrees $1$ and $0$. Let $S=\mathbb Z[0]$ be concentrated in degree zero. Define

$$p(v_0)=p(v_1)=1, \qquad i(1)=v_0.$$

Then $pi=\operatorname{id}_S$. The degree-$+1$ map

$$H(v_0)=0, \qquad H(v_1)=e, \qquad H(e)=0$$

satisfies

$$\operatorname{id}_I-ip=\partial H+H\partial.$$

For example, the right side sends $v_1$ to $\partial e=v_1-v_0$ and sends $e$ to $H(v_1-v_0)=e$. Thus $I$ and $S$ are homotopy equivalent even though their terms are not degreewise isomorphic. The homotopy records exactly how the edge and the difference of its endpoints cancel.

A complex $K$ is contractible when $\operatorname{id}_K\simeq0$. A contractible complex is acyclic because its identity induces both the identity and the zero map on homology. The reverse implication depends on the coefficient category and fails for general $R$-modules.

Quasi-isomorphism asks only about homology

A chain map $f:C\to D$ is a quasi-isomorphism if

$$H_n(f):H_n(C)\xrightarrow{\ \cong\ }H_n(D)$$

for every $n$. This is a property of a specified map. Merely listing abstract isomorphisms $H_n(C)\cong H_n(D)$ does not construct a chain map and therefore does not establish a quasi-isomorphism.

Four common comparison levels are now distinct:

• equality means the complexes have the same graded modules and the same differential, not merely corresponding data;
• a chain isomorphism has an actual inverse commuting with the differentials;
• a chain-homotopy equivalence has an inverse only up to explicit homotopies; and
• a quasi-isomorphism need only become invertible after applying every homology functor.

The next construction detects exactly how far a chain map is from being a quasi-isomorphism or a homotopy equivalence.

The mapping cone tests the difference

For a chain map $f:C\to D$, define its mapping cone by

$$\operatorname{Cone}(f)_n=D_n\oplus C_{n-1}$$

and

$$\partial_{\operatorname{Cone}}(y,x) = \bigl(\partial_Dy+f(x),-\partial_Cx\bigr). \tag{2}$$

The indices in (2) are determined by $y\in D_n$ and $x\in C_{n-1}$. Applying the differential twice gives

$$\begin{aligned} \partial_{\operatorname{Cone}}^2(y,x) &= \bigl( \partial_D^2y+\partial_Df(x)-f(\partial_Cx), \partial_C^2x \bigr)\\ &=(0,0), \end{aligned}$$

where the middle terms cancel because $f$ is a chain map. The minus sign in (2) is essential.

Define the suspension by

$$(\Sigma C)_n=C_{n-1}, \qquad \partial_{\Sigma C}=-\partial_C.$$

There is a short exact sequence of complexes

$$0\longrightarrow D \longrightarrow\operatorname{Cone}(f) \longrightarrow\Sigma C \longrightarrow0,$$

whose long exact homology sequence contains

$$\cdots\longrightarrow H_n(C) \xrightarrow{H_n(f)}H_n(D) \longrightarrow H_n(\operatorname{Cone}(f)) \longrightarrow H_{n-1}(C) \longrightarrow\cdots.$$

Exactness immediately gives the first cone criterion:

$$f\text{ is a quasi-isomorphism} \quad\Longleftrightarrow\quad \operatorname{Cone}(f)\text{ is acyclic}. \tag{3}$$

There is a stronger chain-level criterion:

$$f\text{ is a chain-homotopy equivalence} \quad\Longleftrightarrow\quad \operatorname{Cone}(f)\text{ is contractible}. \tag{4}$$

This is Hughes and Ranicki 1996, Lemma 3.6(iii), p. 32, PDF; their cone uses an equivalent sign convention.

One direction constructs a contracting homotopy on the cone from a homotopy inverse of $f$. Conversely, write a contracting homotopy of the cone in block form relative to $D_n\oplus C_{n-1}$. Its component $D_n\to C_n$ is a chain map $g:D\to C$, while the diagonal components of $\partial H+H\partial=\operatorname{id}$ supply homotopies $fg\simeq\operatorname{id}_D$ and $gf\simeq\operatorname{id}_C$. Criteria (3) and (4) expose the precise gap: acyclic need not mean contractible. For the homotopy, acyclic-complex, and cone comparison—including the acyclic-cone criterion—see Weibel 1994, Chapter 1, §§1.4–1.5, especially Corollary 1.5.4, p. 19.

A quasi-isomorphism without a homotopy inverse

Over $\mathbb Z$, consider

$$P: 0\longrightarrow\mathbb Z \xrightarrow{\,2\,}\mathbb Z \longrightarrow0$$

in degrees $1$ and $0$, and let $M=(\mathbb Z/2\mathbb Z)[0]$. Reduction modulo $2$ defines a chain map $q:P\to M$: its degree-zero component is $q_0:\mathbb Z\to\mathbb Z/2$, and its degree-one component is zero. Since

$$H_1(P)=0, \qquad H_0(P)=\mathbb Z/2,$$

$q$ is a quasi-isomorphism.

It is not a chain-homotopy equivalence. Any homomorphism $s_0:\mathbb Z/2\to\mathbb Z$ is zero, so no chain map $s:M\to P$ can make $qs$ the identity. Moreover, $M$ has no nonzero degree-$+1$ homotopy, so $qs\simeq\operatorname{id}_M$ would force literal equality.

The cone displays the same obstruction. With convention (2), it is

$$0\longrightarrow\mathbb Z \xrightarrow{\,-2\,}\mathbb Z \xrightarrow{\ \bmod 2\ }\mathbb Z/2 \longrightarrow0$$

in degrees $2,1,0$. This complex is exact, hence acyclic. It is not contractible: a contracting homotopy in degree zero would give a homomorphism $\mathbb Z/2\to\mathbb Z$ splitting reduction modulo $2$, which does not exist. Thus this one finite example simultaneously disproves

$$\text{quasi-isomorphism} \Longrightarrow \text{chain-homotopy equivalence}$$

and

$$\text{acyclic} \Longrightarrow \text{contractible}$$

for complexes of abelian groups.

Why coefficients matter

For complexes of algebraic vector spaces over a field, every short exact sequence splits. Choosing complements to boundaries and cycles shows that every acyclic complex is noncanonically contractible. By the cone criteria, every quasi-isomorphism of such bare vector-space complexes has a homotopy inverse.

That conclusion does not automatically survive in categories of modules, sheaves, topological or completed vector spaces, differential operators, local observables, or differential graded algebras. Even when an underlying vector-space splitting exists, it may fail to be continuous, local, covariant, multiplicative, or compatible with a pairing. The ambient category is part of the assertion.

What derived language changes

Write $\operatorname{Ch}(R)$ for the category of chain complexes of $R$-modules and chain maps. The homotopy category $K(R)$ has the same objects, but its morphisms are chain maps modulo chain homotopy. Thus homotopic maps become equal in $K(R)$, and chain-homotopy equivalences become isomorphisms there.

The derived category is obtained by formally inverting all quasi-isomorphisms:

$$D(R)=K(R)[\mathrm{Qis}^{-1}].$$

A morphism in $D(R)$ can be represented by a roof

$$C\xleftarrow[\sim]{\ s\ }C'\xrightarrow{\ a\ }D, \qquad s\text{ a quasi-isomorphism},$$

which is read formally as $a\circ s^{-1}$. The symbol $s^{-1}$ here does not assert that $s$ has a degreewise chain inverse. Different roofs can represent the same derived morphism, and their equivalence is governed by the localization construction.

Derived language therefore does more than discard a complex and retain its homology groups. Over a general ring, a complex need not split as a direct sum of its homology objects, and extension data can remain visible in $D(R)$. What changes is the class of maps declared invertible: quasi-isomorphisms are treated as equivalences while chain-level representatives and their relations still matter. The cone-and-shift construction of the homotopy category and its localization is developed in Stacks Project Authors 2026, §13.9, tag 014E and Stacks Project Authors 2026, §13.11, tag 05RR.

Translating to cochain language

BRST and BV complexes are usually cohomologically graded. For

$$\cdots\longrightarrow C^{n-1} \xrightarrow{d_C^{n-1}}C^n \xrightarrow{d_C^n}C^{n+1} \longrightarrow\cdots,$$

a cochain homotopy has degree $-1$:

$$h^n:C^n\longrightarrow D^{n-1},$$

and the homotopy equation is

$$f^n-g^n = d_D^{n-1}h^n+h^{n+1}d_C^n. \tag{5}$$

The cochain cone convention corresponding to (2) is

$$\operatorname{Cone}(f)^n=D^n\oplus C^{n+1},$$ $$d_{\operatorname{Cone}}(y,x) = \bigl(d_Dy+f(x),-d_Cx\bigr).$$

With homology replaced by cohomology, the definitions and both cone criteria are unchanged. Writing the grading and cone convention explicitly avoids the most common sign error.

A controlled BV-facing example

Let $k$ be a field and let $B$ be the two-term cochain complex

$$B^0=ku, \qquad B^1=kv, \qquad d u=v.$$

For any cochain complex $C$, set $A=C\oplus B$. Let $p:A\to C$ be projection and $i:C\to A$ inclusion. Then $pi=\operatorname{id}_C$. Define a degree-$-1$ map $H:A\to A$ by

$$H(v)=u, \qquad H(u)=0, \qquad H|_C=0.$$

On both $u$ and $v$ one checks that

$$\operatorname{id}_A-ip=dH+Hd.$$

Thus $B$ is contractible and $A\simeq C$. This is the linear-algebra pattern behind recognizing a contractible BRST or BV doublet: the pair contributes no cohomology, and the displayed $H$ supplies stronger chain-level cancellation.

Actual free BV theories contain more data than a bare cochain complex. For example, the formulation of Benini, Musante, and Schenkel uses a complex of linear differential operators together with a compatible shifted pairing and a Green witness; its time-slice statements ask appropriate maps of cochain-complex-valued theories to be quasi-isomorphisms. Eliminating a pair in that setting requires compatibility with the relevant locality, differential, pairing, support, and functorial structures. The calculation above alone does not establish equivalence of QFTs, preservation of brackets, locality, unitarity, or quantization. For the free-BV objects, Green witnesses, and quasi-isomorphism-valued time-slice comparison, see Benini, Musante, and Schenkel 2024, §§2.1 and 2.4, Definitions 2.9 and 2.11, and Theorem 3.13.

The later page Derived Critical Loci and Derived Gauge Quotients will use this vocabulary in a geometric setting. This page does not construct derived critical loci or gauge quotients; it supplies the comparison language needed to state what their weak equivalences mean.

Common pitfalls

Comparing homology groups without a map. Abstract isomorphisms $H_n(C)\cong H_n(D)$ do not constitute a quasi-isomorphism. One needs a chain map inducing isomorphisms, or an explicitly controlled zigzag of such maps.

Reversing the implication chain. A homotopy equivalence is always a quasi-isomorphism, but the finite $\mathbb Z\to\mathbb Z/2$ example shows that a quasi-isomorphism need not have a homotopy inverse.

Equating acyclic with contractible. Acyclicity is a homology statement; contractibility requires a degree-shifting witness for the identity. They coincide for split complexes, not in every coefficient category.

Dropping the cone sign. For the conventions on this page, the second cone component has differential $-\partial_C$ in chain grading and $-d_C$ in cochain grading. Without the minus sign, the cross terms in the square of the differential add instead of canceling.

Treating a formal inverse as a chain inverse. Localization makes a quasi-isomorphism invertible in $D(R)$. It does not manufacture a degreewise inverse in $\operatorname{Ch}(R)$ or a homotopy inverse in $K(R)$.

Inferring physical equivalence from cohomology alone. A quasi-isomorphism of underlying complexes says nothing by itself about locality, products, brackets, pairings, support conditions, positivity, unitarity, or quantization. Those structures require their own compatible maps and theorems.

Check your understanding

Type check. If $h_n:C_n\to D_{n+1}$, verify the types of both terms in $\partial_Dh+h\partial_C$ on $C_n$.

Answer check. The first composite is $C_n\xrightarrow{h_n}D_{n+1}\xrightarrow{\partial_{D,n+1}}D_n$. The second is $C_n\xrightarrow{\partial_{C,n}}C_{n-1}\xrightarrow{h_{n-1}}D_n$.

Proof checkpoint. Prove directly that a contractible complex is acyclic.

Answer check. If $\operatorname{id}_C=\partial H+H\partial$ and $z$ is a cycle, then $z=\partial H(z)$ is a boundary. Hence every homology class vanishes.

Cone calculation. Compute the cone of $q:(0\to\mathbb Z\xrightarrow{2}\mathbb Z\to0)\to\mathbb Z/2[0]$ and check exactness at its middle term.

Answer check. The cone is $0\to\mathbb Z\xrightarrow{-2}\mathbb Z\xrightarrow{\bmod 2}\mathbb Z/2 \to0$. The image of $-2$ is $2\mathbb Z$, exactly the kernel of reduction modulo $2$.

Counterexample check. Why can no contracting homotopy exist for that acyclic cone?

Answer check. Its degree-zero equation would require a homomorphism $s:\mathbb Z/2\to\mathbb Z$ with $(\bmod 2)\circ s=\operatorname{id}$. Every such homomorphism is zero, so the required splitting is impossible.

Cochain transfer. For the pair $du=v$, verify the homotopy equation first on $u$ and then on $v$.

Answer check. On $u$, $dH(u)+Hd(u)=H(v)=u$. On $v$, $dH(v)+Hd(v)=d(u)=v$. Thus $dH+Hd$ is the identity on the pair.

What has been established

Chain homotopy supplies explicit higher-degree data relating maps; chain-homotopy equivalence preserves a complex up to that data; and quasi-isomorphism preserves homology through a specified map. Mapping cones separate acyclicity from contractibility and provide practical criteria for both notions. The categories $\operatorname{Ch}(R)$, $K(R)$, and $D(R)$ then record, respectively, strict chain maps, maps modulo homotopy, and maps after quasi-isomorphisms have been formally inverted.

For the chapter-wide comparison of these notions with higher and local-to-global structures, continue eventually to Derived, Higher, and Factorization Frameworks: a Boundary Map.

References

• Marco Benini, Giorgio Musante, and Alexander Schenkel, “Quantization of Lorentzian Free BV Theories: Factorization Algebra vs Algebraic Quantum Field Theory,” Letters in Mathematical Physics 114 (2024), article 36, DOI:10.1007/s11005-024-01784-1, arXiv:2212.02546v2, §§ 1 and 2.1; § 2.4, Definitions 2.9 and 2.11; Definitions 3.1, 3.3, and 3.5; Theorem 3.13; and Propositions 4.3 and 4.6. Free BV cochain complexes, quasi-isomorphism-valued time-slice statements, shifted pairings, and Green witnesses.

• Bruce Hughes and Andrew Ranicki, Ends of Complexes, Cambridge Tracts in Mathematics 123, Cambridge University Press (1996), Chapter 3, Lemma 3.6(iii), p. 32, author-hosted text, PDF. A chain map is a chain-homotopy equivalence exactly when its mapping cone is contractible; their cone convention differs by standard signs.

• The Stacks Project Authors, Derived Categories, The Stacks Project (continuously updated; accessed August 11, 2026), Stacks Project Authors, § 13.9, tag 014E and Stacks Project Authors, § 13.11, tag 05RR. Cones, shifts, homotopy categories, and construction of the derived category by inverting quasi-isomorphisms.

• The Stacks Project Authors, Homological Algebra, The Stacks Project (continuously updated; accessed August 11, 2026), Stacks Project Authors, § 12.13, tags 010V, 010X, 010Z, 0110, 0113, 0115, and 0116. Chain and cochain complexes, homotopies, homotopy equivalences, quasi-isomorphisms, acyclic complexes, and induced maps on homology. The current text was checked together with the project’s correction history.

• Charles A. Weibel, An Introduction to Homological Algebra, Cambridge University Press (1994), Chapter 1, §§ 1.4–1.5, DOI:10.1017/CBO9781139644136.002. Chain homotopies, homotopy equivalences, split and nonsplit acyclic complexes, and mapping cones; see especially Corollary 1.5.4, p. 19, for the acyclic-cone criterion. The author’s correction list changes “split complexes” to “exact complexes” on p. 19, line $-7$.