definitegluing

Definite gluing is a technique used in the context of algebraic geometry and category theory, particularly in the study of derived categories and their enhancements. It is a method for constructing new objects from given ones in a way that respects certain properties and structures.

The term "definite gluing" was introduced by Bondal and Kapranov in their work on derived categories. The idea is to glue together two categories along a common subcategory in a way that is "definite," meaning that the gluing process is well-defined and respects the categorical structures involved. This process can be used to construct new categories from old ones, often with desirable properties.

In more concrete terms, definite gluing involves taking two categories, A and B, and a functor F from a third category C to both A and B. The goal is to construct a new category D that contains both A and B, with C identified in both, and such that the inclusion functors from A and B to D respect the functors from C. The category D is said to be the definite gluing of A and B along C.

Definite gluing has applications in various areas of mathematics, including the study of derived categories of coherent sheaves on algebraic varieties, the construction of enhanced triangulated categories, and the study of t-structures. It is a powerful tool for constructing and understanding complex mathematical objects in a structured and systematic way.