Lectures related to Sheaves: Hartshorne I.1

Covers concrete categories with sets and structures, including Ens, Gr, Ab, and Vectk.

Introduces limits and colimits in a category, covering their properties and uniqueness.

Explores free abelian groups, homomorphisms, and exact sequences in the context of different categories.

Introduces two fundamental examples of simplicial sets: the nerve of a small category and the singular simplicial set of a topological space.

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Covers exact sequences, torsion, divisibility, and operations on abelian groups.

Explores group actions on spaces, including fixed points and non-trivial actions.

Provides an overview of categories, functors, and natural transformations in mathematics.

Explores limits and colimits in functor categories, focusing on equalizers, pullbacks, and their significance in category theory.