A pair (D^{≤0}, D^{≥0}) of full subcategories of a triangulated category D, closed under shifts and extensions, whose intersection (the heart) is an abelian category; t-structures organize cohomological truncations and produce cohomology functors H^i.

Principle

Objects are filtered by cohomological degrees: every object fits into a distinguished triangle relating its truncations, and the subcategories satisfy orthogonality and shift axioms that define coherent truncation functors.

Demonstration

On the derived category D^b(A) of an abelian category A, the standard t-structure has D^{≤0} the complexes with vanishing cohomology in positive degrees and D^{≥0} those with vanishing cohomology in negative degrees; its heart is equivalent to A.

Misapplication

Treating the heart as equal to the whole triangulated category or assuming a given triangulated category admits a unique t-structure; using truncation functors without verifying the required orthogonality and closure under extensions.

Consequence

A t-structure yields canonical cohomology functors, abelian hearts suitable for doing homological algebra, spectral sequences from filtrations, and a bridge between triangulated and abelian categories.

Reversal

A weight structure (co-t-structure) reverses the role of truncation axes: its heart behaves like the category of pure 'weights' rather than cohomological degrees, producing different filtrations and orthogonality conditions.

Boundary

Applies only to triangulated categories with the required closure and orthogonality properties; not every triangulated category admits a t-structure, and t-structures are not intrinsic invariants of objects but additional structure choices.

Semantic Tension

T-structure versus weight structure: both give hearts and filtrations but encode different grading philosophies (cohomological degree vs weight); confusion also arises between the abstract heart and concrete abelian categories of sheaves or modules.

Synthesis

A t-structure is an extra triangulated-category structure partitioning objects into cohomological halves so that their intersection is an abelian heart, enabling truncation, cohomology functors, and the transfer of homological techniques from abelian to triangulated settings.