[Paper Review] Introduction to the language of stacks and gerbes

Ieke Moerdijk|ArXiv.org|Dec 19, 2002

Homotopy and Cohomology in Algebraic Topology3 references54 citations

TL;DR

This paper provides a concise, accessible introduction to stacks and gerbes using non-abelian Čech cohomology, focusing on their role in classifying geometric structures like principal bundles and bundle gerbes. It establishes connections between gerbes, sheaves of groupoids, and differential geometry, showing how curvature classes in De Rham cohomology arise from groupoid connections on bundle gerbes.

ABSTRACT

This is an introduction to gerbes for topologists, with emphasis on non-abelian cohomology.

Motivation & Objective

• To provide a self-contained, pedagogical introduction to stacks and gerbes for topology students without prior algebraic geometry background.
• To explain how gerbes with a given band are classified by non-abelian Čech cocycles of degree 2 using the language of fibered categories and stacks.
• To connect abstract gerbe theory with differential geometry by introducing bundle gerbes and their curvature classes in De Rham cohomology.
• To demonstrate the existence of groupoid connections on bundle gerbes and their relation to curvature forms via Mayer-Vietoris arguments.

Proposed method

• Uses the framework of presheaves, sheaves, and étale spaces to build intuition for stacks via categorical adjunctions.
• Introduces fibered categories, prestacks, and stacks as analogues of presheaves, separated presheaves, and sheaves.
• Defines gerbes via bands (liens) and constructs their classification using non-abelian Čech cocycles in degree 2.
• Shows that every equivalence class of gerbes with a fixed band arises from a sheaf of groupoids.
• Applies Mayer-Vietoris techniques to pull back curvature forms from $M \times_X M$ to $U \times_X U$ and then to $X$.
• Constructs a groupoid connection on a bundle gerbe using pullbacks of ordinary connections on associated $S^1$-bundles and their tensor product.

Experimental results

Research questions

• RQ1 How can gerbes be classified using non-abelian Čech cohomology in degree 2?
• RQ2 What is the categorical relationship between fibered categories, prestacks, and stacks, and how does it mirror the sheaf-theoretic hierarchy?
• RQ3 How do groupoid connections on bundle gerbes relate to curvature forms in De Rham cohomology?
• RQ4 What conditions ensure the existence of a groupoid connection on a bundle gerbe, and how can it be explicitly constructed?
• RQ5 How does the curvature class of a bundle gerbe coincide with the cohomology class derived from its Čech cocycle?

Key findings

• Every equivalence class of gerbes with a fixed band is represented by a sheaf of groupoids, establishing a categorical realization of non-abelian degree 2 cohomology.
• The curvature 2-form $\kappa$ of a groupoid connection on a bundle gerbe satisfies the cocycle condition $\pi_{12}^*(\kappa) + \pi_{23}^*(\kappa) = \pi_{13}^*(\kappa)$ on $M \times_X M \times_X M$.
• By Mayer-Vietoris, the curvature $\kappa$ on $U \times_X U$ lifts to a 2-form $\lambda$ on $U$ such that $\kappa = \pi_2^*(\lambda) - \pi_1^*(\lambda)$.
• The 3-form $\xi$ on $X$ defined by $d\lambda = \pi^*(\xi)$ is closed and integral, and its cohomology class $[\xi] \in H^3_{\text{DR}}(X)$ matches the class from the Čech cocycle.
• A groupoid connection on a bundle gerbe always exists, constructed via pullbacks of ordinary connections on $S^1$-bundles over $M$.
• The induced connection on $G \to M \times_X M$ is compatible with the groupoid structure, and its curvature satisfies the required cocycle identity.