---
tag: 2GD9V7SA0E
kind: concept
name: t-structure
msc: [18G80, 14F08]
sources:
  - url: https://en.wikipedia.org/wiki/T-structure
same_as:
  wikidata: Q21071553
  wikipedia: T-structure
---

## Summary

A t-structure on a [[triangulated category|T2V7GJJ2EN]] is a pair of full subcategories, of "non-positive" and "non-negative" objects.
They behave like the complexes with cohomology only in degrees $\le 0$ and $\ge 0$ in a [[derived category|5ST7NVJFSM]].
It gives truncation functors and cohomology functors with values in its heart, which is always an abelian category.
t-structures were introduced by Beilinson, Bernstein and Deligne to construct perverse sheaves as the heart of a non-standard t-structure.

## Explanations

### L1 human

In the derived category of an abelian category, a complex can be cut into its low degree part and its high degree part.
The original abelian category is recovered as the complexes with cohomology only in degree $0$.
A t-structure is an abstract way of making such a cut in any triangulated category.
Different t-structures on the same triangulated category can have quite different hearts.
This is how one category can hide several abelian categories, such as sheaves and perverse sheaves.