Cohomologia de Cech e gerbes

Abstract

The investigation for global properties of mathematical objects often relies on an understanding of their local behavior. However, when one attempts to assemble local data from different regions of a space, obstructions may rise, revealing intrinsic topological information about the space itself. Sheaf Theory organizes mathematical information defined on local regions, while Cech Cohomology measures when this local information cannot be consistently combined to form a global object. This work develop these theories, starting from categorical foundations, including monoidal categories and 2-categories, and advancing toward sheaf theory and Cech cohomology. Subsequently, these tools are applied to the study of vector bundles, presenting a geometric model for the first Chern class, which arises within the context of Cech cohomology as the class that classifies complex line bundles up to isomorphism, thus providing a geometric interpretation of the cohomology group $H^2(X, \mathbb{Z})$. Finally, we introduce gerbes as structures that generalize line bundles within a 2-categorical framework. Using Cech cohomology and the language of 2-categories, it is described how gerbes relate to third-degree cohomology classes, offering a geometric model of $H^3(X, \mathbb{Z})$.