Higher stacks

Definition

Higher stacks are a generalization of classical algebraic stacks that incorporate homotopical and derived structures, enabling the study of geometric objects with additional layers of complexity. They arise in derived algebraic geometry and allow for a richer understanding of moduli problems, sheaf theory, and intersection theory by considering not just the points but also higher categorical data associated with them.

5 Must Know Facts For Your Next Test

1. Higher stacks can be thought of as 'spaces' where not only points but also higher-dimensional data play a role, offering insights into moduli spaces and their morphisms.
2. They are crucial in understanding derived moduli problems where one needs to keep track of infinitesimal deformations and obstructions.
3. In higher stacks, morphisms can be understood in terms of homotopies, allowing for a flexible approach to composition and mapping.
4. Higher stacks often employ the language of simplicial sets or $ ext{∞}$-categories to describe their structure, making them compatible with advanced techniques in homotopy theory.
5. They provide tools for extending classical results from algebraic geometry into the realm of derived algebraic geometry, enriching the landscape of geometric intuition.

Review Questions

• How do higher stacks differ from classical algebraic stacks in terms of structure and applications?
  • Higher stacks differ from classical algebraic stacks primarily in their incorporation of homotopical concepts, which allow them to represent more complex geometrical and categorical data. While classical stacks focus on moduli problems related to points and automorphisms, higher stacks extend this by allowing the study of infinitesimal deformations and higher-dimensional morphisms. This results in greater flexibility and power when addressing questions in derived algebraic geometry.
• What role do higher stacks play in derived moduli problems, particularly regarding infinitesimal deformations?
  • Higher stacks play a pivotal role in derived moduli problems by providing a framework for studying infinitesimal deformations and their obstructions. They allow mathematicians to track how small changes in a geometric object affect its moduli space, capturing richer information than traditional methods. The structure of higher stacks helps to understand these deformations systematically, leading to insights about stability conditions and geometric properties.
• Evaluate the implications of using infinity-categories in the study of higher stacks and how this impacts our understanding of geometric objects.
  • The use of infinity-categories in studying higher stacks has significant implications for our understanding of geometric objects. By framing higher stacks within the context of $ ext{∞}$-categories, we can leverage homotopical techniques that reveal deeper relationships between different geometric structures. This approach enhances our ability to perform computations and reason about geometric properties through categorical lenses, thereby opening new pathways for research in both algebraic geometry and topology.

Related terms

Algebraic Stacks: Algebraic stacks are a type of stack that generalizes schemes to allow for the presence of automorphisms, enabling the treatment of objects like moduli spaces more flexibly.

Derived Categories:

Derived categories are a tool used in homological algebra that formalizes the notion of 'complexes' of sheaves, allowing for the extension of various constructions and results from abelian categories.

Infinity-Categories: Infinity-categories are a generalization of categories that allow for 'homotopical' aspects, providing a framework to study higher-dimensional structures and their relationships.