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.
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.
of full subcategories, each of which is stable under isomorphism, which satisfy the following three axioms.
More briefly, we define With this notation, the axioms above may be rewritten as: The heart or core of the t-structure is the full subcategory
A triangulated category with a choice of t-structure is sometimes called a t-category.
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 (possibly singular) algebraic curve X (or analogously a possibly singular surface) is designed so that it contains, in particular, objects of the form where
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 is an example of the general fact that a triangulated category may be endowed with several distinct t-structures.
[6] The category of spectra is endowed with a t-structure generated, in the sense above, by a single object, namely the sphere spectrum.
is the category of connective spectra, i.e., those whose negative homotopy groups vanish.
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.
implies the existence of the other truncation functors by shifting and taking opposite categories.
is defined as As the name suggests, this is a cohomological functor in the usual sense for a triangulated category.
, we obtain a long exact sequence In applications to algebraic topology, the cohomology functors may be denoted
is an exact functor (in the usual sense for triangulated categories, that is, up to a natural equivalence it commutes with translation and preserves distinguished triangles).
are bounded below derived categories of sheaves on a space X, an open subset U, and the closed complement F of U.
This works, in particular, when the sheaves in question are left modules over a sheaf of rings
Many t-structures arise by means of the following fact: in a triangulated category with arbitrary direct sums, and a set
A t-structure on an ∞-category can be notated either homologically or cohomologically, just as in the case of a triangulated category.
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.
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.
is a localization functor, then the set S of all morphisms f for which Lf is an equivalence is quasisaturated.
It is also possible to form a left or right completion with respect to a t-structure.
The left and right completions are themselves stable ∞-categories which inherit a canonical t-structure.
is replaced by the opposite inclusion and the other two axioms kept the same, the resulting notion is called a co-t-structure or weight structure.