T-structure
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 t-structure on consists of two subcategories of a triangulated category or stable infinity category which abstract the idea of complexes whose cohomology vanishes in positive, respectively negative, degrees. There can be many distinct t-structures on the same category, and the interplay between these structures has implications for algebra and geometry. The notion of a t-structure arose in the work of Beilinson, Bernstein, Deligne, and Gabber on perverse sheaves.
Definition
Fix a triangulated category with translation functor. A t-structure on is a pair of full subcategories, each of which is stable under isomorphism, which satisfy the following three axioms.- If
Suppose that is a t-structure on. In this case, for any integer n, we define to be the full subcategory of whose objects have the form, where is an object of. Similarly, is the full subcategory of objects, where is an object of. More briefly, we define
With this notation, the axioms above may be rewritten as:
- If
The heart of a t-structure is an abelian category, and it is stable under extensions.
A triangulated category with a choice of t-structure is sometimes called a t-category.
Variations
It is clear that, to define a t-structure, it suffices to fix integers m and n and specify and. Some authors define a t-structure to be the pair.The two subcategories and determine each other. An object X is in if and only if for all objects Y in, and vice versa. That is, are left and right orthogonal complements of each other. Consequently, it is enough to specify only one of and. Moreover, because these subcategories are full by definition, it is enough to specify their objects.
The above notation is adapted to the study of cohomology. When the goal is to study homology, slightly different notation is used. A homological t-structure on is a pair such that, if we define
then is a t-structure on. That is, the definition is the same except that upper indices are converted to lower indices and the roles of and are swapped. If we define
then the axioms for a homological t-structure may be written explicitly as
- If X is an object of and Y is an object of, then
- and.
- If A is an object of, then there exists a distinguished triangle such that X is an object of and Y is an object of.
Examples
The natural ''t''-structure
The most fundamental example of a t-structure is the natural t-structure on a derived category. Let be an abelian category, and let be its derived category. Then the natural t-structure is defined by the pair of subcategoriesIt follows immediately that
In this case, the third axiom for a t-structure, the existence of a certain distinguished triangle, can be made explicit as follows. Suppose that is a cochain complex with values in. Define
It is clear that and that there is a short exact sequence of complexes
This exact sequence furnishes the required distinguished triangle.
This example can be generalized to exact categories. There are also similar t-structures for the bounded, bounded above, and bounded below derived categories. If is an abelian subcategory of, then the full subcategory of consisting of those complexes whose cohomology is in has a similar t-structure whose heart is.
Perverse sheaves
The category of perverse sheaves is, by definition, the core of the so-called perverse t-structure on the derived category of the category of sheaves on a complex analytic space X or an algebraic variety over a finite field. As was explained above, the heart of the standard t-structure simply contains ordinary sheaves, regarded as complexes concentrated in degree 0. For example, the category of perverse sheaves on a algebraic curve X is designed so that it contains, in particular, objects of the formwhere is the inclusion of a point, is an ordinary sheaf, is a smooth open subscheme and is a locally constant sheaf on U. Note the presence of the shift according to the dimension of Z and U respectively. This shift causes the category of perverse sheaves to be well-behaved on singular spaces. The simple objects in this category are the intersection cohomology sheaves of subvarieties with coefficients in an irreducible local system.
This t-structure was introduced by Beilinson, Bernstein and Deligne. It was shown by Beilinson that the derived category of the heart is in fact equivalent to the original derived category of sheaves. This is an example of the general fact that a triangulated category may be endowed with several distinct t-structures.
Graded modules
A non-standard example of a t-structure on the derived category of modules over a graded ring has the property that its heart consists of complexeswhere is a module generated by its degree n. This t-structure called geometric t-structure plays a prominent role in Koszul duality.
Spectra
The category of spectra is endowed with a t-structure generated, in the sense above, by a single object, namely the sphere spectrum. The category is the category of connective spectra, i.e., those whose negative homotopy groups vanish.Motives
A conjectural example in the theory of motives is the so-called motivic t-structure. Its existence is closely related to certain standard conjectures on algebraic cycles and vanishing conjectures, such as the Beilinson-Soulé conjecture.Truncation functors
In the above example of the natural t-structure on an abelian category, the distinguished triangle guaranteed by the third axiom was constructed by truncation. As operations on the category of complexes, the truncations and are functorial, and the resulting short exact sequence of complexes is natural in. Using this, it can be shown that there are truncation functors on the derived category and that they induce a natural distinguished triangle.In fact, this is an example of a general phenomenon. While the axioms for a t-structure do not assume the existence of truncation functors, such functors can always be constructed and are essentially unique. Suppose that is a triangulated category and that is a t-structure. The precise statement is that the inclusion functors
admit adjoints. These are functors
such that
Moreover, for any object of, there exists a unique
such that d and the counit and unit of the adjunctions together define a distinguished triangle
Up to unique isomorphism, this is the unique distinguished triangle of the form with and objects of and, respectively. It follows from the existence of this triangle that an object lies in if and only if .
The existence of implies the existence of the other truncation functors by shifting and taking opposite categories. If is an object of, the third axiom for a t-structure asserts the existence of an in and a morphism fitting into a certain distinguished triangle. For each, fix one such triangle and define. The axioms for a t-structure imply that, for any object of, we have
with the isomorphism being induced by the morphism. This exhibits as a solution to a certain universal mapping problem. Standard results on adjoint functors now imply that is unique up to unique isomorphism and that there is a unique way to define on morphisms that makes it a right adjoint. This proves the existence of and hence the existence of all the truncation functors.
Repeated truncation for a t-structure behaves similarly to repeated truncation for complexes. If, then there are natural transformations
which yield natural equivalences
Cohomology functors
The nth cohomology functor is defined asAs the name suggests, this is a cohomological functor in the usual sense for a triangulated category. That is, for any distinguished triangle, we obtain a long exact sequence
In applications to algebraic topology, the cohomology functors may be denoted instead of. The cohomology functors take values in the heart. By one of the repeated truncation identities above, up to natural equivalence it is equivalent to define
For the natural t-structure on a derived category, the cohomology functor is, up to quasi-isomorphism, the usual nth cohomology group of a complex. However, considered as functors on complexes, this is not true. Consider, for example, as defined in terms of the natural t-structure. By definition, this is
This complex is non-zero in degrees and, so it is clearly not the same as the zeroth cohomology group of the complex. However, the non-trivial differential is an injection, so the only non-trivial cohomology is in degree, where it is, the zeroth cohomology group of the complex. It follows that the two possible definitions of are quasi-isomorphic.
A t-structure is non-degenerate if the intersection of all, as well as the intersection of all, consists only of zero objects. For a non-degenerate t-structure, the collection of functors is conservative. Moreover, in this case, may be identified with the full subcategory of those objects for which for .
Exact functors
For, let be a triangulated category with a fixed t-structure Suppose that is an exact functor. Then is:- Left t-exact if,
- Right t-exact if, and
- t-exact if it is both left and right t-exact.
The motivation for the study of one-sided t-exactness properties is that they lead to one-sided exactness properties on hearts. Let be the inclusion. Then there is a composite functor
It can be shown that if is left exact, then is also left exact, and that if is also left exact, then.
If is left t-exact, and if is in , then there is a natural isomorphism .
If are exact functors with left adjoint to, then is right t-exact if and only if is left t-exact, and in this case, are a pair of adjoint functors.
Constructions of ''t''-structures
Let be a t-structure on. If n is an integer, then the translation by n t-structure is. The dual t-structure is the t-structure on the opposite category defined by.Let be a triangulated subcategory of a triangulated category. If is a t-structure on, then
is a t-structure on if and only if is stable under the truncation functor. When this condition holds, the t-structure is called the induced t-structure. The truncation and cohomology functors for the induced t-structure are the restriction to of those on. Consequently, the inclusion of in is t-exact, and.
To construct the category of perverse sheaves, it is important to be able to define a t-structure on a category of sheaves over a space by working locally in that space. The precise conditions necessary for this to be possible can be abstracted somewhat to the following setup. Suppose that there are three triangulated categories and two morphisms
satisfying the following properties.
- There are two sequences of triples of adjoint functors and.
- The functors,, and are full and faithful, and they satisfy.
- There are unique differentials making, for every K in, exact triangles
This t-structure is said to be the gluing of the t-structures on U and F. The intended use cases are when,, and are bounded below derived categories of sheaves on a space X, an open subset U, and the closed complement F of U. The functors and are the usual pullback and pushforward functors. This works, in particular, when the sheaves in question are left modules over a sheaf of rings on X and when the sheaves are ℓ-adic sheaves.
Many t-structures arise by means of the following fact: in a triangulated category with arbitrary direct sums, and a set of compact objects in, the subcategories
can be shown to be a t-structure. The resulting t-structure is said to be generated by .
Given an abelian subcategory of a triangulated category, it is possible to construct a subcategory of and a t-structure on that subcategory whose heart is.
On stable ∞-categories
The elementary theory of t-structures carries over to the case of ∞-categories with few changes. Let be a stable ∞-category. A t-structure on is defined to be a t-structure on its homotopy category . A t-structure on an ∞-category can be notated either homologically or cohomologically, just as in the case of a triangulated category.Suppose that is an ∞-category with homotopy category and that is a t-structure on. Then, for each integer n, we define and to be the full subcategories of spanned by the objects in and, respectively. Define
to be the inclusion functors. Just as in the case of a triangulated category, these admit a right and a left adjoint, respectively, the truncation functors
These functors satisfy the same repeated truncation identities as in the triangulated category case.
The heart of a t-structure on is defined to be the ∞-subcategory. The category is equivalent to the nerve of its homotopy category. The cohomology functor is defined to be, or equivalently.
The existence of means that is, by definition, a localization functor. In fact, there is a bijection between t-structures on and certain kinds of localization functors called t-localizations. These are localization functors L whose essential image is closed under extension, meaning that if is a fiber sequence with X and Z in the essential image of L, then Y is also in the essential image of L. Given such a localization functor L, the corresponding t-structure is defined by
t-localization functors can also be characterized in terms of the morphisms f for which Lf is an equivalence. A set of morphisms S in an ∞-category is quasisaturated if it contains all equivalences, if any 2-simplex in with two of its non-degenerate edges in S has its third non-degenerate edge in S, and if it is stable under pushouts. If is a localization functor, then the set S of all morphisms f for which Lf is an equivalence is quasisaturated. Then L is a t-localization functor if and only if S is the smallest quasisaturated set of morphisms containing all morphisms.
The derived category of an abelian category has several subcategories corresponding to different boundedness conditions. A t-structure on a stable ∞-category can be used to construct similar subcategories. Specifically,
These are stable subcategories of. One says that is left bounded if, right bounded if, and bounded if.
It is also possible to form a left or right completion with respect to a t-structure. This is analogous to formally adjoining directed limits or directed colimits. The left completion of is the homotopy limit of the diagram
The right completion is defined dually. The left and right completions are themselves stable ∞-categories which inherit a canonical t-structure. There is a canonical map from to either of its completions, and this map is t-exact. We say that is left complete or right complete if the canonical map to its left or right completion, respectively, is an equivalence.
Related concepts
If the requirement, is replaced by the opposite inclusionand the other two axioms kept the same, the resulting notion is called a co-t-structure or weight structure.