Triangulated category

Triangulated categories were introduced independently by Dieter Puppe (1962) and Jean-Louis Verdier (1963), although Puppe's axioms were less complete (lacking the octahedral axiom (TR 4)).[1] Puppe was motivated by the stable homotopy category. Verdier's key example was the derived category of an abelian category, which he also defined, developing ideas of Alexander Grothendieck. The early applications of derived categories included coherent duality and Verdier duality, which extends Poincaré duality to singular spaces.

A shift or translation functor on a category D is an additive automorphism (or for some authors, an auto-equivalence) ${\displaystyle \Sigma }$ from D to D. It is common to write ${\displaystyle X[n]=\Sigma ^{n}X}$ for integers n.

A triangle (X, Y, Z, u, v, w) consists of three objects X, Y, and Z, together with morphisms ${\displaystyle u\colon X\to Y}$, ${\displaystyle v\colon Y\to Z}$ and ${\displaystyle w\colon Z\to X[1]}$. Triangles are generally written in the unravelled form:

${\displaystyle X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1],}$

or

${\displaystyle X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}}$

for short.

A triangulated category is an additive category D with a translation functor and a class of triangles, called exact triangles[2] (or distinguished triangles), satisfying the following properties (TR 1), (TR 2), (TR 3) and (TR 4). (These axioms are not entirely independent, since (TR 3) can be derived from the others.[3])

Here are some simple consequences of the axioms for a triangulated category D.

• Given an exact triangle

${\displaystyle X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1]}$

in D, the composition of any two successive morphisms is zero. That is, vu = 0, wv = 0, u[1]w = 0, and so on.[5]

• Given a morphism ${\displaystyle u\colon X\to Y}$, TR 1 guarantees the existence of a cone Z completing an exact triangle. Any two cones of u are isomorphic, but the isomorphism is not always uniquely determined.[4]

• Every monomorphism in D is the inclusion of a direct summand, ${\displaystyle X\to X\oplus Y}$, and every epimorphism is a projection ${\displaystyle X\oplus Y\to X}$.[6] A related point is that one should not talk about "injectivity" or "surjectivity" for morphisms in a triangulated category. Every morphism ${\displaystyle X\to Y}$ that is not an isomorphism has a nonzero "cokernel" Z (meaning that there is an exact triangle ${\displaystyle X\to Y\to Z\to X[1]}$) and also a nonzero "kernel", namely Z[−1].

Some experts suspect[11]^{pg 190} (see, for example, (Gelfand & Manin 2006, Introduction, Chapter IV)) that triangulated categories are not really the "correct" concept. The essential reason is that the cone of a morphism is unique only up to a non-unique isomorphism. In particular, the cone of a morphism does not in general depend functorially on the morphism (note the non-uniqueness in axiom (TR 3), for example). This non-uniqueness is a potential source of errors. The axioms work adequately in practice, however, and there is a great deal of literature devoted to their study.

Triangulated categories admit a notion of cohomology, and every triangulated category has a large supply of cohomological functors. A cohomological functor F from a triangulated category D to an abelian category A is a functor such that for every exact triangle

${\displaystyle X\to Y\to Z\to X[1],\ }$

the sequence ${\displaystyle F(X)\to F(Y)\to F(Z)}$ in A is exact. Since an exact triangle determines an infinite sequence of exact triangles in both directions,

${\displaystyle \cdots \to Z[-1]\to X\to Y\to Z\to X[1]\to \cdots ,\ }$

a cohomological functor F actually gives a long exact sequence in the abelian category A:

${\displaystyle \cdots \to F(Z[-1])\to F(X)\to F(Y)\to F(Z)\to F(X[1])\to \cdots .\ }$

A key example is: for each object B in a triangulated category D, the functors ${\displaystyle \operatorname {Hom} (B,{\text{-}})}$ and ${\displaystyle \operatorname {Hom} ({\text{-}},B)}$ are cohomological, with values in the category of abelian groups.[15] (To be precise, the latter is a contravariant functor, which can be considered as a functor on the opposite category of D.) That is, an exact triangle ${\displaystyle X\to Y\to Z\to X[1]}$ determines two long exact sequences of abelian groups:

${\displaystyle \cdots \to \operatorname {Hom} (B,X[i])\to \operatorname {Hom} (B,Y[i])\to \operatorname {Hom} (B,Z[i])\to \operatorname {Hom} (B,X[i+1])\to \cdots }$

and

${\displaystyle \cdots \to \operatorname {Hom} (Z,B[i])\to \operatorname {Hom} (Y,B[i])\to \operatorname {Hom} (X,B[i])\to \operatorname {Hom} (Z,B[i+1])\to \cdots .}$

For particular triangulated categories, these exact sequences yield many of the important exact sequences in sheaf cohomology, group cohomology, and other areas of mathematics.

One may also use the notation

${\displaystyle \operatorname {Ext} ^{i}(B,X)=\operatorname {Hom} (B,X[i])}$

for integers i, generalizing the Ext functor in an abelian category. In this notation, the first exact sequence above would be written:

${\displaystyle \cdots \to \operatorname {Ext} ^{i}(B,X)\to \operatorname {Ext} ^{i}(B,Y)\to \operatorname {Ext} ^{i}(B,Z)\to \operatorname {Ext} ^{i+1}(B,X)\to \cdots .\ }$

For an abelian category A, another basic example of a cohomological functor on the derived category D(A) sends a complex X to the object ${\displaystyle H^{0}(X)}$ in A. That is, an exact triangle ${\displaystyle X\to Y\to Z\to X[1]}$ in D(A) determines a long exact sequence in A:

${\displaystyle \cdots \to H^{i}(X)\to H^{i}(Y)\to H^{i}(Z)\to H^{i+1}(X)\to \cdots ,}$

using that ${\displaystyle H^{0}(X[i])\cong H^{i}(X)}$.

An exact functor (also called triangulated functor) from a triangulated category D to a triangulated category E is an additive functor ${\displaystyle F\colon D\to E}$ which, loosely speaking, commutes with translation and sends exact triangles to exact triangles.[16]

In more detail, an exact functor comes with a natural isomorphism ${\displaystyle \eta \colon F\Sigma \to \Sigma F}$ (where the first ${\displaystyle \Sigma }$ denotes the translation functor of D and the second ${\displaystyle \Sigma }$ denotes the translation functor of E), such that whenever

${\displaystyle X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1]}$

is an exact triangle in D,

${\displaystyle F(X){\xrightarrow {F(u)}}F(Y){\xrightarrow {F(v)}}F(Z){\xrightarrow {\eta _{X}F(w)}}F(X)[1]}$

is an exact triangle in E.

An equivalence of triangulated categories is an exact functor ${\displaystyle F\colon D\to E}$ that is also an equivalence of categories. In this case, there is an exact functor ${\displaystyle G\colon E\to D}$ such that FG and GF are naturally isomorphic to the respective identity functors.

Let D be a triangulated category such that direct sums indexed by an arbitrary set (not necessarily finite) exist in D. An object X in D is called compact if the functor ${\displaystyle {\text{Hom}}_{D}(X,{\text{-}})}$ commutes with direct sums. Explicitly, this means that for every family of objects ${\displaystyle Y_{i}}$ in D indexed by a set S, the natural homomorphism of abelian groups ${\displaystyle \oplus _{i\in S}\mathrm {Hom} _{D}(X,Y_{i})\to \mathrm {Hom} _{D}(X,\oplus _{i\in S}Y_{i})}$ is an isomorphism. This is different from the general notion of a compact object in category theory, which involves all colimits rather than only coproducts.

For example, a compact object in the stable homotopy category ${\displaystyle h{\cal {S}}}$ is a finite spectrum.[17] A compact object in the derived category of a ring, or in the quasi-coherent derived category of a scheme, is a perfect complex. In the case of a smooth projective variety X over a field, the category Perf(X) of perfect complexes can also be viewed as the bounded derived category of coherent sheaves, ${\displaystyle D_{\text{coh}}^{\text{b}}(X)}$.

A triangulated category D is compactly generated if

• D has arbitrary (not necessarily finite) direct sums;
• There is a set S of compact objects in D such that for every nonzero object X in D, there is an object Y in S with a nonzero map ${\displaystyle Y[n]\to X}$ for some integer n.

Many naturally occurring "large" triangulated categories are compactly generated:

• The derived category of modules over a ring R is compactly generated by one object, the R-module R.
• The quasi-coherent derived category of a quasi-compact quasi-separated scheme is compactly generated by one object.[18]
• The stable homotopy category is compactly generated by one object, the sphere spectrum ${\displaystyle S^{0}}$.[19]

Amnon Neeman generalized the Brown representability theorem to any compactly generated triangulated category, as follows.[20] Let D be a compactly generated triangulated category, ${\displaystyle H\colon D^{\text{op}}\to {\text{Ab}}}$ a cohomological functor which takes coproducts to products. Then H is representable. (That is, there is an object W of D such that ${\displaystyle H(X)\cong {\text{Hom}}(X,W)}$ for all X.) For another version, let D be a compactly generated triangulated category, T any triangulated category. If an exact functor ${\displaystyle F\colon D\to T}$ sends coproducts to coproducts, then F has a right adjoint.

The Brown representability theorem can be used to define various functors between triangulated categories. In particular, Neeman used it to simplify and generalize the construction of the exceptional inverse image functor ${\displaystyle f^{!}}$ for a morphism f of schemes, the central feature of coherent duality theory.[21]

For every abelian category A, the derived category D(A) is a triangulated category, containing A as a full subcategory (the complexes concentrated in degree zero). Different abelian categories can have equivalent derived categories, so that it is not always possible to reconstruct A from D(A) as a triangulated category.

Alexander Beilinson, Joseph Bernstein and Pierre Deligne described this situation by the notion of a t-structure on a triangulated category D.[22] A t-structure on D determines an abelian category inside D, and different t-structures on D may yield different abelian categories.

Let D be a triangulated category with arbitrary direct sums. A localizing subcategory of D is a strictly full triangulated subcategory that is closed under arbitrary direct sums.[23] To explain the name: if a localizing subcategory S of a compactly generated triangulated category D is generated by a set of objects, then there is a Bousfield localization functor ${\displaystyle L\colon D\to D}$ with kernel S.[24] (That is, for every object X in D there is an exact triangle ${\displaystyle Y\to X\to LX\to Y[1]}$ with Y in S and LX in the right orthogonal ${\displaystyle S^{\perp }}$.) For example, this construction includes the localization of a spectrum at a prime number, or the restriction from a complex of sheaves on a space to an open subset.

A parallel notion is more relevant for "small" triangulated categories: a thick subcategory of a triangulated category C is a strictly full triangulated subcategory that is closed under direct summands. (If C is idempotent-complete, a subcategory is thick if and only if it is also idempotent-complete.) A localizing subcategory is thick.[25] So if S is a localizing subcategory of a triangulated category D, then the intersection of S with the subcategory ${\displaystyle D^{\text{c}}}$ of compact objects is a thick subcategory of ${\displaystyle D^{\text{c}}}$.

For example, Devinatz–Hopkins–Smith described all thick subcategories of the triangulated category of finite spectra in terms of Morava K-theory.[26] The localizing subcategories of the whole stable homotopy category have not been classified.

• Fourier–Mukai transform
• Six operations
• Perverse sheaf
• D-module
• Beilinson–Bernstein localization
• Module spectrum
• Semiorthogonal decomposition
• Bridgeland stability condition

Some textbook introductions to triangulated categories are:

• Gelfand, Sergei; Manin, Yuri (2006), "IV. Triangulated Categories", Methods of homological algebra, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, doi:10.1007/978-3-662-12492-5, ISBN 978-3540435839, MR 1950475
• Kashiwara, Masaki; Schapira, Pierre (2006), Categories and sheaves, Grundlehren der mathematischen Wissenschaften, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-27950-4, ISBN 978-3-540-27949-5, MR 2182076
• Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.

A concise summary with applications is:

• Kashiwara, Masaki; Schapira, Pierre (2002), "Chapter I. Homological Algebra", Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, doi:10.1007/978-3-662-02661-8, ISBN 978-3540518617, MR 1074006

Some more advanced references are:

• Beilinson, A.A.; Bernstein, J.; Deligne, P. (2018) [1982], "Faisceaux pervers", Astérisque, Société Mathématique de France, Paris, 100, ISBN 978-2-85629-878-7, MR 0751966
• Dugger, Daniel; Shipley, Brooke (2009), "A curious example of triangulated-equivalent model categories which are not Quillen equivalent", Algebraic and Geometric Topology, 9: 135–166, arXiv:0710.3070, doi:10.2140/agt.2009.9.135, MR 2482071
• Hartshorne, Robin (1966), "Chapter I. The Derived Category", Residues and duality, Lecture Notes in Mathematics 20, vol. 20, Springer-Verlag, pp. 20–48, doi:10.1007/BFb0080482, ISBN 978-3-540-03603-6, MR 0222093
• Krause, Henning (2010), "Localization theory for triangulated categories", Triangulated categories, London Mathematical Society Lecture Note Series, vol. 375, Cambridge University Press, pp. 161–235, arXiv:0806.1324, doi:10.1017/CBO9781139107075.005, ISBN 9780521744317, MR 2681709, S2CID 50160822
• Neeman, Amnon (1996), "The Grothendieck duality theorem via Bousfield's techniques and Brown representability", Journal of the American Mathematical Society, 9: 205–236, doi:10.1090/S0894-0347-96-00174-9, MR 1308405
• Neeman, Amnon (2001), Triangulated categories, Annals of Mathematics Studies, Princeton University Press, doi:10.1515/9781400837212, ISBN 978-0691086866, MR 1812507, S2CID 242258794
• Puppe, Dieter (1962), "On the formal structure of stable homotopy theory", Colloquium on algebraic topology, Aarhus Universitet Matematisk Institute, pp. 65–71, Zbl 0139.41106
• Puppe, Dieter (1967), "Stabile Homotopietheorie. I.", Mathematische Annalen, 169 (2): 243–274, doi:10.1007/BF01362348, MR 0211400, S2CID 122663283
• Ravenel, Douglas (1992), Nilpotence and periodicity in stable homotopy theory, Princeton University Press, ISBN 9780691025728, MR 1192553
• Rizzardo, Alice; Van den Bergh, Michel; Neeman, Amnon (2019), "An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves", Inventiones Mathematicae, 216 (3): 927–1004, arXiv:1410.4039, Bibcode:2019InMat.216..927R, doi:10.1007/s00222-019-00862-9, MR 3955712, S2CID 253743362
• Toën, Bertrand (2007), "The homotopy theory of dg-categories and derived Morita theory", Inventiones Mathematicae, 167 (3): 615–667, arXiv:math/0408337, doi:10.1007/s00222-006-0025-y, MR 2276263, S2CID 9445008
• Verdier, Jean-Louis (1977) [1963], "Catégories dérivées: quelques résultats (état 0)", Cohomologie étale (SGA 4 1/2) (PDF), Lecture Notes in Mathematics, vol. 569, Springer, pp. 262–311, doi:10.1007/BFb0091525, ISBN 978-3-540-08066-4, MR 3727440
• Verdier, Jean-Louis (1996) [1967], Des catégories dérivées des catégories abéliennes, Astérisque, vol. 239, Société Mathématique de France, MR 1453167

• J. Peter May, The axioms for triangulated categories
• The Stacks Project Authors, The Stacks Project