Chain homotopy and the homotopy category $K(\mathcal{A})$

Intuition Beginner

Chain complexes are sequences of objects and arrows. A map of complexes is a compatible map between two such sequences.

Chain homotopy says that two maps can differ by a controlled correction and still carry the same cohomological information. The correction moves one degree in the opposite direction and explains the difference between the two maps as a boundary-type error.

This is the algebraic version of deforming one continuous map into another in topology. The maps may not be equal term by term, but their difference is invisible after passing to cohomology.

The homotopy category keeps complexes but identifies maps that differ by chain homotopy.

Visual Beginner

The diagonal correction data explains why the two maps induce the same cohomology map.

Worked example Beginner

Suppose a complex has a piece that appears and then immediately cancels. A map that changes only this canceling piece should not change the actual cohomology.

A chain homotopy records that cancellation algebraically. It says the difference between two maps is produced by moving into the canceling direction and then applying the differential, plus the same process in the other order.

What this tells us: chain homotopy is the correct notion of sameness before one builds derived categories.

Check your understanding Beginner

Formal definition Intermediate+

Let $C^{\bullet }$ and $D^{\bullet }$ be cochain complexes in an abelian category $\mathcal{A}$, and let $$ f,g^\bullet\to D^\bullet $$ be chain maps. A chain homotopy from $f$ to $g$ is a family of morphisms $$ s^n^n\to D^{n-1} $$ such that $$ f^n-g^n=d_D^{n-1}s^n+s^{n+1}d_C^n $$ for every $n$. One writes $f\simeq g$.

The homotopy category $$ K(\mathcal A) $$ has the same objects as $\mathrm{Ch}(\mathcal{A})$, but morphisms are chain maps modulo chain homotopy. The bounded variants are denoted $$ K^+(\mathcal A),\quad K^-(\mathcal A),\quad K^b(\mathcal A). $$

The shift functor is defined by $$ C[1]^n=C^{n+1},\qquad d_{C[1]}^n=-d_C^{n+1}. $$ The minus sign ensures that the shifted object is again a complex: $$ d_{C[1]}^{n+1}d_{C[1]}^n=0. $$

Counterexamples to common slips

• Chain homotopic maps need not be equal as chain maps.
• A quasi-isomorphism need not be a homotopy equivalence.
• The homotopy category $K(\mathcal{A})$ is not usually abelian, even when $\mathcal{A}$ is abelian.

Key theorem with proof Intermediate+

Theorem (chain-homotopic maps induce the same cohomology map). If $f,g:C^{\bullet }\to D^{\bullet }$ are chain-homotopic maps, then $$ H^n(f)=H^n(g) $$ for every $n$.

Proof. Let $x\in C^n$ be closed, so $d_C^{n}x=0$. The homotopy formula gives $$ f^n(x)-g^n(x)=d_D^{n-1}s^n(x)+s^{n+1}d_C^n(x). $$ The second term vanishes because $x$ is closed. Thus $$ f^n(x)-g^n(x)=d_D^{n-1}s^n(x), $$ which is a boundary in $D^n$.

Therefore $f(x)$ and $g(x)$ define the same cohomology class. Since this holds for every closed representative, the induced maps on $H^n$ agree. $\square$

Bridge. Chain homotopy is the equivalence relation between raw complexes and derived categories. Mapping cones 01.02.32 use it to build triangles, and derived categories 04.03.11 go further by inverting quasi-isomorphisms.

Exercises Intermediate+

Advanced results Master

The homotopy category $K(\mathcal{A})$ is additive, but it is not usually abelian. Kernels and cokernels of maps in $\mathrm{Ch}(\mathcal{A})$ do not descend to an abelian exact structure after quotienting by homotopy. The replacement structure is triangulated, with shift and mapping cones providing the basic triangles.

This is one of Verdier's central insights. The category of complexes modulo homotopy has enough structure to support exact-sequence-like arguments, but the structure is not ordinary abelian exactness. Distinguished triangles replace short exact sequences.

A homotopy equivalence is a chain map $f:C^{\bullet }\to D^{\bullet }$ with a chain map $g:D^{\bullet }\to C^{\bullet }$ such that $gf\simeq {\mathrm{id}}_C$ and $fg\simeq {\mathrm{id}}_D$. Homotopy equivalences become isomorphisms in $K(\mathcal{A})$ and induce isomorphisms on cohomology.

The converse fails in general: a quasi-isomorphism may induce isomorphisms on all cohomology objects without having a homotopy inverse. Derived categories are designed to force all quasi-isomorphisms to become invertible. Thus the path is $$ \mathrm{Ch}(\mathcal A)\to K(\mathcal A)\to D(\mathcal A). $$

Bounded homotopy categories are essential in algebraic geometry. For coherent sheaves, one often works with $K^b$ or $D^b$ because finite cohomological amplitude is part of the geometry. For injective or projective resolutions, $K^{+}$ and $K^{-}$ control one-sided resolutions.

Synthesis. Chain homotopy is the first categorical compression of complexes. It identifies maps that act the same on cohomology for a structural reason, creates the homotopy category, and prepares the mapping-cone triangles that make triangulated categories possible.

Full proof set Master

Proposition 1 (chain homotopy is compatible with composition). If $f\simeq g:C^{\bullet }\to D^{\bullet }$ and $u:D^{\bullet }\to E^{\bullet }$, then $uf\simeq ug$. If $v:B^{\bullet }\to C^{\bullet }$, then $fv\simeq gv$.

Proof. If $f-g=d_{D}s+s{d}_C$, then $$ u(f-g)=u d_Ds+u s d_C=d_E(us)+(us)d_C, $$ using that $u$ is a chain map. Thus $us$ is a homotopy from $uf$ to $ug$. The precomposition case is similar: $$ (f-g)v=d_D(sv)+(sv)d_B. $$ Therefore homotopy classes compose well. $\square$

Proposition 2 (homotopy equivalences are cohomology isomorphisms). If $f:C^{\bullet }\to D^{\bullet }$ is a homotopy equivalence, then $H^n(f)$ is an isomorphism for all $n$.

Proof. Let $g$ be a homotopy inverse. Since $gf\simeq {\mathrm{id}}_C$ and $fg\simeq {\mathrm{id}}_D$, the key theorem gives $$ H(g)H(f)=H(gf)=H(\mathrm{id}_C) $$ and $$ H(f)H(g)=H(fg)=H(\mathrm{id}_D). $$ Thus $H(g)$ is the inverse of $H(f)$ in every degree. $\square$

Proposition 3 (the shifted differential squares to zero). With $C[1{]}^n=C^{n+1}$ and $d_{C[1]}^n=-d_C^{n+1}$, the shifted object is a complex.

Proof. The composite in degree $n$ is $$ d_{C[1]}^{n+1}d_{C[1]}^n=(-d_C^{n+2})(-d_C^{n+1}) =d_C^{n+2}d_C^{n+1}=0. $$ Therefore $C[1]$ is again a cochain complex. $\square$

Connections Master

• Chain complexes 01.02.30. Chain homotopy is a relation between morphisms of complexes.

• Mapping cones 01.02.32. Mapping cones use the shift convention and become the source of standard distinguished triangles.

• Triangulated categories 04.03.10. The homotopy category is the motivating example for triangulated structure.

• Derived category 04.03.11. The derived category is obtained from $K(\mathcal{A})$ by inverting quasi-isomorphisms.

• Derived functors 04.03.12. Resolutions are used up to homotopy and then interpreted in the derived category.

Historical & philosophical context Master

Chain homotopy entered algebra from topology, where homotopic maps induce the same maps on homology. Algebraic topology needed a chain-level relation that could prove this without returning to point-set geometry.

Cartan-Eilenberg and Mac Lane made chain homotopy part of the standard language of homological algebra ^{[Cartan-Eilenberg]}. Gelfand-Manin present it as the bridge from complexes to homotopy categories ^{[Gelfand-Manin]}.

Verdier's derived-category framework reframed the homotopy category as the staging ground for triangulated categories and localization ^[Verdier]. The philosophical point is that exactness survives in a new form after quotienting by homotopy: not as short exact sequences, but as triangles.

Bibliography Master

@book{GelfandManinChainHomotopy,
  author = {Gelfand, Sergei I. and Manin, Yuri I.},
  title = {Methods of Homological Algebra},
  publisher = {Springer},
  year = {1996}
}

@book{Weibel1994ChainHomotopy,
  author = {Weibel, Charles A.},
  title = {An Introduction to Homological Algebra},
  publisher = {Cambridge University Press},
  year = {1994}
}

@book{CartanEilenberg1956ChainHomotopy,
  author = {Cartan, Henri and Eilenberg, Samuel},
  title = {Homological Algebra},
  publisher = {Princeton University Press},
  year = {1956}
}

@book{Verdier1996DerivedCategories,
  author = {Verdier, Jean-Louis},
  title = {Des categories derivees des categories abeliennes},
  series = {Asterisque},
  volume = {239},
  publisher = {Societe Mathematique de France},
  year = {1996}
}