In mathematics, a **triangulated category** is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology.

- History
- Definition
- TR 1
- TR 2
- TR 3
- TR 4: The octahedral axiom
- Properties
- Non-functoriality of the cone construction
- Examples
- Are there better axioms?
- Derivators
- Stable ∞-categories
- Cohomology in triangulated categories
- Exact functors and equivalences
- Compactly generated triangulated categories
- t-structures
- Localizing and thick subcategories
- See also
- Notes
- References
- External links

Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space *X* to complexes of sheaves, viewed as objects of the derived category of sheaves on *X*. More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated categories of different origins have been proved or conjectured. For example, the homological mirror symmetry conjecture predicts that the derived category of a Calabi–Yau manifold is equivalent to the Fukaya category of its "mirror" symplectic manifold. Shift operator is a decategorified analogue of 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) from *D* to *D*. It is common to write for integers *n*.

A **triangle** (*X*, *Y*, *Z*, *u*, *v*, *w*) consists of three objects *X*, *Y*, and *Z*, together with morphisms , and . Triangles are generally written in the unravelled form:

or

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] })

- For every object
*X*, the following triangle is exact:

- For every morphism , there is an object
*Z*(called a**cone**or**cofiber**of the morphism*u*) fitting into an exact triangle

- The name "cone" comes from the cone of a map of chain complexes, which in turn was inspired by the mapping cone in topology. It follows from the other axioms that an exact triangle (and in particular the object
*Z*) is determined up to isomorphism by the morphism , although not always up to a unique isomorphism.^{ [4] }

- Every triangle isomorphic to an exact triangle is exact. This means that if

- is an exact triangle, and , , and are isomorphisms, then
- is also an exact triangle.

If

is an exact triangle, then so are the two rotated triangles

and

In view of the last triangle, the object *Z*[−1] is called a **fiber** of the morphism .

The second rotated triangle has a more complex form when and are not isomorphisms but only mutually inverse equivalences of categories, since is a morphism from to , and to obtain a morphism to one must compose with the natural transformation . This leads to complex questions about possible axioms one has to impose on the natural transformations making and into a pair of inverse equivalences. Due to this issue, the assumption that and are mutually inverse isomorphisms is the usual choice in the definition of a triangulated category.

Given two exact triangles and a map between the first morphisms in each triangle, there exists a morphism between the third objects in each of the two triangles that makes everything commute. That is, in the following diagram (where the two rows are exact triangles and *f* and *g* are morphisms such that *gu* = *u′f*), there exists a map *h* (not necessarily unique) making all the squares commute:

Let and be morphisms, and consider the composed morphism . Form exact triangles for each of these three morphisms according to TR 1. The octahedral axiom states (roughly) that the three mapping cones can be made into the vertices of an exact triangle so that "everything commutes".

More formally, given exact triangles

- ,

there exists an exact triangle

such that

This axiom is called the "octahedral axiom" because drawing all the objects and morphisms gives the skeleton of an octahedron, four of whose faces are exact triangles. The presentation here is Verdier's own, and appears, complete with octahedral diagram, in (Hartshorne 1966 ). In the following diagram, *u* and *v* are the given morphisms, and the primed letters are the cones of various maps (chosen so that every exact triangle has an *X*, a *Y*, and a *Z* letter). Various arrows have been marked with [1] to indicate that they are of "degree 1"; e.g. the map from *Z*′ to *X* is in fact from *Z*′ to *X*[1]. The octahedral axiom then asserts the existence of maps *f* and *g* forming an exact triangle, and so that *f* and *g* form commutative triangles in the other faces that contain them:

Two different pictures appear in (Beilinson,Bernstein&Deligne 1982 ) (GelfandandManin ( 2006 ) also present the first one). The first presents the upper and lower pyramids of the above octahedron and asserts that given a lower pyramid, one can fill in an upper pyramid so that the two paths from *Y* to *Y*′, and from *Y*′ to *Y*, are equal (this condition is omitted, perhaps erroneously, from Hartshorne's presentation). The triangles marked + are commutative and those marked "d" are exact:

The second diagram is a more innovative presentation. Exact triangles are presented linearly, and the diagram emphasizes the fact that the four triangles in the "octahedron" are connected by a series of maps of triangles, where three triangles (namely, those completing the morphisms from *X* to *Y*, from *Y* to *Z*, and from *X* to *Z*) are given and the existence of the fourth is claimed. One passes between the first two by "pivoting" about *X*, to the third by pivoting about *Z*, and to the fourth by pivoting about *X*′. All enclosures in this diagram are commutative (both trigons and the square) but the other commutative square, expressing the equality of the two paths from *Y*′ to *Y*, is not evident. All the arrows pointing "off the edge" are degree 1:

This last diagram also illustrates a useful intuitive interpretation of the octahedral axiom. In triangulated categories, triangles play the role of exact sequences, and so it is suggestive to think of these objects as "quotients", and . In those terms, the existence of the last triangle expresses on the one hand

- (looking at the triangle ), and
- (looking at the triangle ).

Putting these together, the octahedral axiom asserts the "third isomorphism theorem":

If the triangulated category is the derived category *D*(*A*) of an abelian category *A*, and *X*, *Y*, *Z* are objects of *A* viewed as complexes concentrated in degree 0, and the maps and are monomorphisms in *A*, then the cones of these morphisms in *D*(*A*) are actually isomorphic to the quotients above in *A*.

Finally, Neeman ( 2001 ) formulates the octahedral axiom using a two-dimensional commutative diagram with 4 rows and 4 columns. Beilinson,Bernstein,andDeligne ( 1982 ) also give generalizations of the octahedral axiom.

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

- Given an exact triangle

- 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 , 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, , and every epimorphism is a projection .^{ [6] }A related point is that one should not talk about "injectivity" or "surjectivity" for morphisms in a triangulated category. Every morphism that is not an isomorphism has a nonzero "cokernel"*Z*(meaning that there is an exact triangle ) and also a nonzero "kernel", namely*Z*[−1].

One of the technical complications with triangulated categories is the fact the cone construction is not functorial. For example, given a ring and the partial map of distinguished triangles

in , there are two maps which complete this diagram. This could be the identity map, or the zero map

both of which are commutative. The fact there exist two maps is a shadow of the fact that a triangulated category is a tool which encodes homotopy limits and colimit. One solution for this problem was proposed by Grothendieck where not only the derived category is considered, but the derived category of diagrams on this category. Such an object is called a Derivator.

- Vector spaces over a field
*k*form an elementary triangulated category in which*X*[1] =*X*for all*X*. An exact triangle is a sequence of*k*-linear maps (writing the same map twice) which is exact at*X*,*Y*and*Z*. - If
*A*is an additive category (for example, an abelian category), define the homotopy category to have as objects the chain complexes in*A*, and as morphisms the homotopy classes of morphisms of complexes. Then is a triangulated category.^{ [7] }The shift*X*[1] is the complex*X*moved one step to the left (and with differentials multiplied by −1). An exact triangle in is a triangle which is isomorphic in to the triangle associated to some map of chain complexes. (Here denotes the mapping cone of a chain map.) - The derived category
*D*(*A*) of an abelian category*A*is a triangulated category.^{ [8] }It is constructed from the category of complexes*C*(*A*) by localizing with respect to all quasi-isomorphisms. That is, formally adjoin an inverse morphism for every quasi-isomorphism. The objects of*D*(*A*) are unchanged; that is, they are chain complexes. An exact triangle in*D*(*A*) is a triangle which is isomorphic in*D*(*A*) to the triangle associated to some map of chain complexes.

A key motivation for the derived category is that derived functors on*A*can be viewed as functors on the derived category.^{ [9] }Some natural subcategories of*D*(*A*) are also triangulated categories, for example the subcategory of complexes*X*whose cohomology objects in*A*vanish for*i*sufficiently negative, sufficiently positive, or both, called , respectively. - In topology, the stable homotopy category is a triangulated category.
^{ [10] }The objects are spectra, the shift*X*[1] is the suspension (or equivalently the delooping ), and the exact triangles are the cofiber sequences. A distinctive feature of the stable homotopy category (compared to the unstable homotopy category) is that fiber sequences are the same as cofiber sequences. In fact, in any triangulated category, exact triangles can be viewed as fiber sequences and also as cofiber sequences. - In modular representation theory of a finite group
*G*, the stable module category StMod(*kG*) is a triangulated category. Its objects are the representations of*G*over a field*k*, and the morphisms are the usual ones modulo those that factor via projective (or equivalently injective)*kG*-modules. More generally, the stable module category is defined for any Frobenius algebra in place of*kG*.

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.

One alternative proposal is the theory of derivators proposed in Pursuing stacks by Grothendieck in the 80s^{ [11] }^{pg 191}, and later developed in the 90s in his manuscript on the topic. Essentially, these are a system of homotopy categories given by the diagram categories for a category with a class of weak equivalences . These categories are then related by the morphisms of diagrams . This formalism has the advantage of being able to recover the homotopy limits and colimits, which replaces the cone construction.

Another alternative built is the theory of stable ∞-categories. The homotopy category of a stable ∞-category is canonically triangulated, and moreover mapping cones become essentially unique (in a precise homotopical sense). Moreover, a stable ∞-category naturally encodes a whole hierarchy of compatibilities for its homotopy category, at the bottom of which sits the octahedral axiom. Thus, it is strictly stronger to give the data of a stable ∞-category than to give the data of a triangulation of its homotopy category. Nearly all triangulated categories that arise in practice come from stable ∞-categories. A similar (but more special) enrichment of triangulated categories is the notion of a dg-category.

In some ways, stable ∞-categories or dg-categories work better than triangulated categories. One example is the notion of an exact functor between triangulated categories, discussed below. For a smooth projective variety *X* over a field *k*, the bounded derived category of coherent sheaves comes from a dg-category in a natural way. For varieties *X* and *Y*, every functor from the dg-category of *X* to that of *Y* comes from a complex of sheaves on by the Fourier–Mukai transform.^{ [12] } By contrast, there is an example of an exact functor from to that does not come from a complex of sheaves on .^{ [13] } In view of this example, the "right" notion of a morphism between triangulated categories seems to be one that comes from a morphism of underlying dg-categories (or stable ∞-categories).

Another advantage of stable ∞-categories or dg-categories over triangulated categories appears in algebraic K-theory. One can define the algebraic K-theory of a stable ∞-category or dg-category *C*, giving a sequence of abelian groups for integers *i*. The group has a simple description in terms of the triangulated category associated to *C*. But an example shows that the higher K-groups of a dg-category are not always determined by the associated triangulated category.^{ [14] } Thus a triangulated category has a well-defined group, but in general not higher K-groups.

On the other hand, the theory of triangulated categories is simpler than the theory of stable ∞-categories or dg-categories, and in many applications the triangulated structure is sufficient. An example is the proof of the Bloch–Kato conjecture, where many computations were done at the level of triangulated categories, and the additional structure of ∞-categories or dg-categories was not required.

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

the sequence in *A* is exact. Since an exact triangle determines an infinite sequence of exact triangles in both directions,

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

A key example is: for each object *B* in a triangulated category *D*, the functors and 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 determines two long exact sequences of abelian groups:

and

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

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

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 in *A*. That is, an exact triangle in *D*(*A*) determines a long exact sequence in *A*:

using that .

An **exact functor** (also called **triangulated functor**) from a triangulated category *D* to a triangulated category *E* is an additive functor 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 (where the first denotes the translation functor of *D* and the second denotes the translation functor of *E*), such that whenever

is an exact triangle in *D*,

is an exact triangle in *E*.

An **equivalence** of triangulated categories is an exact functor that is also an equivalence of categories. In this case, there is an exact functor 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 commutes with direct sums. Explicitly, this means that for every family of objects in *D* indexed by a set *S*, the natural homomorphism of abelian groups 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 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, .

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 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 .
^{ [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, a cohomological functor which takes coproducts to products. Then *H* is representable. (That is, there is an object *W* of *D* such that for all *X*.) For another version, let *D* be a compactly generated triangulated category, *T* any triangulated category. If an exact functor 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 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 with kernel *S*.^{ [24] } (That is, for every object *X* in *D* there is an exact triangle with *Y* in *S* and *LX* in the right orthogonal .) 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 of compact objects is a thick subcategory of .

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.

- ↑ Puppe (1962, 1967); Verdier (1963, 1967).
- ↑ Weibel (1994), Definition 10.2.1.
- ↑ J. Peter May,
*The axioms for triangulated categories*. - 1 2 Weibel (1994), Remark 10.2.2.
- ↑ Weibel (1994), Exercise 10.2.1.
- ↑ Gelfand & Manin (2006), Exercise IV.1.1.
- ↑ Kashiwara & Schapira (2006), Theorem 11.2.6.
- ↑ Weibel (1994), Corollary 10.4.3.
- ↑ Weibel (1994), section 10.5.
- ↑ Weibel (1994), Theorem 10.9.18.
- 1 2 Grothendieck. "Pursuing Stacks".
*thescrivener.github.io*. Archived (PDF) from the original on 30 Jul 2020. Retrieved 2020-09-17. - ↑ Toën (2007), Theorem 8.15.
- ↑ Rizzardo et al. (2019), Theorem 1.4.
- ↑ Dugger & Shipley (2009), Remark 4.9.
- ↑ Weibel (1994), Example 10.2.8.
- ↑ Weibel (1994), Definition 10.2.6.
- ↑ Neeman (2001), Remark D.1.5.
- ↑
*Stacks Project, Tag 09IS*,*Stacks Project, Tag 09M1*. - ↑ Neeman (2001), Lemma D.1.3.
- ↑ Neeman (1996), Theorems 3.1 and 4.1.
- ↑ Neeman (1996), Example 4.2.
- ↑ Beilinson et al. (1982), Definition 1.3.1.
- ↑ Neeman (2001), Introduction, after Remark 1.4.
- ↑ Krause (2010), Theorem, Introduction.
- ↑ Neeman (2001), Remark 3.2.7.
- ↑ Ravenel (1992), Theorem 3.4.3.

In mathematics, specifically category theory, a **functor** is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

In mathematics, especially in category theory and homotopy theory, a **groupoid** generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In category theory, a branch of mathematics, a **natural transformation** provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category.

In mathematics, specifically category theory, **adjunction** is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as **adjoint functors**, one being the **left adjoint** and the other the **right adjoint**. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

In mathematics, specifically in category theory, a **pre-abelian category** is an additive category that has all kernels and cokernels.

**Homological algebra** is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, certain functors may be *derived* to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

In mathematics, the **Ext functors** are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext^{1} classifies extensions of one module by another.

In category theory, a **faithful functor** is a functor that is injective on hom-sets, and a **full functor** is surjective on hom-sets. A functor that has both properties is called a **fully faithful functor**.

In mathematics, the **derived category***D*(*A*) of an abelian category *A* is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on *A*. The construction proceeds on the basis that the objects of *D*(*A*) should be chain complexes in *A*, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences.

In mathematics, **Brown's representability theorem** in homotopy theory gives necessary and sufficient conditions for a contravariant functor *F* on the homotopy category *Hotc* of pointed connected CW complexes, to the category of sets **Set**, to be a representable functor.

In the branch of mathematics called homological algebra, a ** t-structure** is a way to axiomatize the properties of an abelian subcategory of a derived category. A

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, especially homological algebra, a **differential graded category**, often shortened to **dg-category** or **DG category**, is a category whose morphism sets are endowed with the additional structure of a differential graded -module.

In homological algebra in mathematics, the **homotopy category***K(A)* of chain complexes in an additive category *A* is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes *Kom(A)* of *A* and the derived category *D(A)* of *A* when *A* is abelian; unlike the former it is a triangulated category, and unlike the latter its formation does not require that *A* is abelian. Philosophically, while *D(A)* turns into isomorphisms any maps of complexes that are quasi-isomorphisms in *Kom(A)*, *K(A)* does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, *K(A)* is more understandable than *D(A)*.

In homological algebra, the **mapping cone** is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map *f* being acyclic means that the map is a quasi-isomorphism; if we pass to the derived category of complexes, this means that *f* is an isomorphism there, which recalls the familiar property of maps of groups, modules over a ring, or elements of an arbitrary abelian category that if the kernel and cokernel both vanish, then the map is an isomorphism. If we are working in a t-category, then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core.

In mathematics, **Grothendieck's six operations**, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the **six-functor formalism**. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes *f* : *X* → *Y*. The basic insight was that many of the elementary facts relating cohomology on *X* and *Y* were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold. The six operations formalism has since been shown to apply to contexts such as *D*-modules on algebraic varieties, sheaves on locally compact topological spaces, and motives.

In algebraic geometry, a **presheaf with transfers** is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite correspondences to the category of abelian groups.

In mathematics, **derived noncommutative algebraic geometry**, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Some basic examples include the bounded derived category of coherent sheaves on a smooth variety, , called its derived category, or the derived category of perfect complexes on an algebraic variety, denoted . For instance, the derived category of coherent sheaves on a smooth projective variety can be used as an invariant of the underlying variety for many cases. Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name.

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*

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.