In the world of algebra, we often encounter “free” constructions. Think of the free group or the polynomial ring—we take a set of generators and build the most general structure around them, adding only what’s necessary. The process of sheafification, however, feels different. It’s less like building from scratch and more like a sophisticated repair job. It takes a “broken” object called a presheaf, which understands local data but can’t quite piece it together, and masterfully fixes its gluing mechanism to create a well-behaved sheaf.
