##  [T-Structure](/t-structure-0) 

 Definition

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

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

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

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

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

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

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

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

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.