Triangulated category — Verdier axioms TR1-TR4 and the octahedral axiom

Intuition Beginner

In an abelian category, short exact sequences are the basic way to say that one object is built from two others.

In homotopy and derived categories, short exact sequences no longer behave well enough. The replacement is a triangle: three objects connected in a cycle, with a shift at the end.

The guiding example is the mapping cone. A map between complexes produces a cone, and the source, target, and cone form a triangle. This triangle carries the same kind of information that a short exact sequence carries in ordinary homological algebra.

A triangulated category is a category where these triangle patterns are taken as part of the structure.

Visual Beginner

The triangle is the derived-category replacement for a short exact sequence.

Worked example Beginner

A map of complexes has a mapping cone. The cone measures the difference between the source and target.

The source, target, and cone form a standard triangle. When we apply cohomology, that triangle produces a long exact sequence.

What this tells us: triangles are not decoration. They are the exactness language that survives after passing to homotopy and derived categories.

Check your understanding Beginner

Formal definition Intermediate+

A triangulated category is an additive category $T$ equipped with an autoequivalence $$ [1]\to T $$ called the shift, and a class $\Delta$ of diagrams $$ X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1], $$ called distinguished triangles, satisfying Verdier's axioms.

The axioms are:

• TR1 (existence and normalization). Triangles isomorphic to distinguished triangles are distinguished; every object has the zero triangle; every morphism $u:X\to Y$ extends to some distinguished triangle.
• TR2 (rotation). A triangle is distinguished if and only if its rotated triangle $Y\to Z\to X[1]\to Y[1]$ is distinguished, with the standard sign on the rotated final map.
• TR3 (morphisms of triangles). Given two distinguished triangles and maps on the first two objects that commute with the first arrows, there exists a map on the third objects completing a morphism of triangles.
• TR4 (octahedral axiom). For composable maps $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$, the triangles attached to $f$, $g$, and $gf$ can be chosen so that their cones fit into one compatible octahedral diagram.

In the homotopy category $K(\mathcal{A})$, the model distinguished triangles are the cone triangles $$ C^\bullet\xrightarrow{f}D^\bullet\to \operatorname{Cone}(f)\to C^\bullet[1]. $$ In the derived category $D(\mathcal{A})$, distinguished triangles are the images of such triangles after localization at quasi-isomorphisms.

Counterexamples to common slips

• A triangulated category is not usually abelian.
• A distinguished triangle is not just any three composable maps.
• TR4 is not optional bookkeeping; it controls how cones behave under composition.

Key theorem with proof Intermediate+

Theorem (two-of-three for morphisms of triangles). Let $$ \begin{matrix} X&\to&Y&\to&Z&\to&X[1]\ \downarrow&&\downarrow&&\downarrow&&\downarrow\ X'&\to&Y'&\to&Z'&\to&X'[1] \end{matrix} $$ be a morphism of distinguished triangles. If two of the three vertical maps $X\to X^{\prime }$, $Y\to Y^{\prime }$, and $Z\to Z^{\prime }$ are isomorphisms, then the third is an isomorphism.

Proof. Apply the cohomological functor represented by any object $A$: $$ \operatorname{Hom}_T(A,-). $$ One of the fundamental consequences of the triangulated axioms is that applying ${\mathrm{Hom}}_T(A,-)$ to a distinguished triangle gives a long exact sequence of abelian groups.

The morphism of triangles therefore produces a morphism between two long exact sequences. If two of the three vertical maps in the triangle are isomorphisms, then the corresponding maps in the long exact Hom sequences are isomorphisms in the matching degrees.

The ordinary five lemma applied to these long exact sequences forces the remaining Hom map to be an isomorphism for every test object $A$. By the Yoneda principle, the remaining vertical map in $T$ is an isomorphism. $\square$

Bridge. This theorem explains why triangles behave like exact sequences: exactness can be tested by applying Hom functors. The next unit 04.03.11 localizes the homotopy category at quasi-isomorphisms and inherits this triangular structure.

Exercises Intermediate+

Advanced results Master

The triangulated-category axioms abstract the behavior of exact triangles in $K(\mathcal{A})$, $D(\mathcal{A})$, and the stable homotopy category. The shift functor represents suspension or degree shift, depending on context. Distinguished triangles represent cofiber sequences, cone sequences, or derived short exact sequences.

TR1 says that enough triangles exist and that the class of distinguished triangles is invariant under isomorphism. TR2 says the triangle can be rotated without leaving the distinguished class. TR3 says compatible maps between two sides of triangles can be extended to the third side. TR4 says composable maps create a coherent octahedron of cones.

The octahedral axiom is the least intuitive but most structurally important axiom. For $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$, choose cone triangles for $f$, $g$, and $gf$. The axiom asserts that there is a triangle $$ \operatorname{Cone}(f)\to \operatorname{Cone}(gf)\to \operatorname{Cone}(g)\to \operatorname{Cone}(f)[1] $$ compatible with the original three cone triangles. This is the cone-of-a-composite bookkeeping law.

In derived categories, a short exact sequence of complexes $$ 0\to A^\bullet\to B^\bullet\to C^\bullet\to 0 $$ gives a distinguished triangle $$ A^\bullet\to B^\bullet\to C^\bullet\to A^\bullet[1]. $$ Thus the derived category remembers exact sequences, but it remembers them through triangles rather than through kernels and cokernels.

Cohomological functors convert distinguished triangles into long exact sequences. This is how ordinary cohomology, Ext, sheaf cohomology, and stable homotopy groups extract computable algebra from triangulated structure.

Synthesis. Triangulated categories are the exactness language of derived mathematics. They preserve the long-exact-sequence power of abelian categories while working in settings where kernels and cokernels are no longer the right primitive objects.

Full proof set Master

Proposition 1 (cone triangles in $K(\mathcal{A})$ are distinguished). In the homotopy category of complexes, every cochain map $f:C^{\bullet }\to D^{\bullet }$ determines a distinguished triangle $$ C^\bullet\xrightarrow{f}D^\bullet\to \operatorname{Cone}(f)\to C^\bullet[1]. $$

Proof. The homotopy category $K(\mathcal{A})$ is constructed from complexes modulo chain homotopy. The mapping cone from 01.02.32 is a complex, and the natural inclusion and projection maps are chain maps. Verdier's distinguished triangles in $K(\mathcal{A})$ are defined to be those isomorphic to such cone triangles. Therefore every map determines a distinguished triangle by construction. $\square$

Proposition 2 (cohomological functors produce long exact sequences). If $H:T\to \mathcal{B}$ is a cohomological functor from a triangulated category to an abelian category, then every distinguished triangle $$ X\to Y\to Z\to X[1] $$ produces a long exact sequence $$ \cdots\to H(X[n])\to H(Y[n])\to H(Z[n])\to H(X[n+1])\to\cdots. $$

Proof. This is the definition of cohomological functor: it sends distinguished triangles to exact sequences, and compatibility with shift extends the exact sequence through all degrees. Representable functors ${\mathrm{Hom}}_T(A,-)$ and ${\mathrm{Hom}}_T(-,A)$ are the fundamental examples, with one covariant and one contravariant convention. $\square$

Proposition 3 (octahedral axiom as cone-composition law). In $K(\mathcal{A})$, the octahedral axiom for $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$ is realized by the short exact comparison between the three mapping cones.

Proof. Write the cone complexes for $f$, $g$, and $gf$ using the block differential convention of 01.02.32. There are natural chain maps $$ \operatorname{Cone}(f)\to \operatorname{Cone}(gf)\to \operatorname{Cone}(g) $$ induced by the identity maps on the shifted source and target components together with $g$. Direct computation with the cone differentials shows that these maps commute with differentials. Their cone triangle supplies $$ \operatorname{Cone}(f)\to \operatorname{Cone}(gf)\to \operatorname{Cone}(g)\to \operatorname{Cone}(f)[1], $$ and the compatibility squares with the three original cone triangles are precisely the octahedral diagram. $\square$

Connections Master

• Chain homotopy and $K(\mathcal{A})$ 01.02.31. The homotopy category is the first major example of a triangulated category.

• Mapping cone 01.02.32. Cone triangles are the model distinguished triangles.

• Abelian categories 01.02.33. Triangulated exactness replaces abelian exactness after passing to homotopy and derived categories.

• Derived category 04.03.11. Derived categories inherit their triangulated structure from homotopy categories.

• Derived functors 04.03.12. Total derived functors are naturally triangulated functors.

• $t$-structures 04.03.18. A $t$-structure recovers an abelian heart inside a triangulated category.

Historical & philosophical context Master

Verdier introduced triangulated categories in his thesis as the formal framework for derived categories ^[Verdier]. The axioms captured the behavior of mapping cones while avoiding dependence on a chosen cone construction.

Gelfand-Manin and Weibel present triangulated categories as the grammar of derived homological algebra ^{[Gelfand-Manin] [Weibel]}. Kashiwara-Schapira made the same grammar central for sheaf theory and microlocal analysis ^{[Kashiwara-Schapira]}.

The philosophical move is substantial: exactness is no longer a property of kernels and cokernels, but a property of triangles. This is why derived categories can be useful even though they are not abelian. They retain the long exact sequences that matter for cohomology while discarding exact structures that are too rigid for homotopy.

Bibliography Master

@book{Verdier1996Triangulated,
  author = {Verdier, Jean-Louis},
  title = {Des categories derivees des categories abeliennes},
  series = {Asterisque},
  volume = {239},
  publisher = {Societe Mathematique de France},
  year = {1996}
}

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

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

@book{KashiwaraSchapiraTriangulated,
  author = {Kashiwara, Masaki and Schapira, Pierre},
  title = {Categories and Sheaves},
  publisher = {Springer},
  year = {2006}
}

@book{NeemanTriangulated,
  author = {Neeman, Amnon},
  title = {Triangulated Categories},
  series = {Annals of Mathematics Studies},
  volume = {148},
  publisher = {Princeton University Press},
  year = {2001}
}