procategory

A procategory is a concept in category theory that generalizes the notion of a category. While a standard category consists of objects and morphisms between them, a procategory is a collection of categories that are related by a system of "projection" functors. These projection functors map morphisms from one category to another, preserving some structural properties.

The term "procategory" is derived from the prefix "pro-", which suggests a forward-looking or projected structure. This reflects the idea that procategories can be seen as a way to organize or "project" a family of categories onto a common framework. They are particularly useful when dealing with situations where objects or structures are not static but evolve or change over time, or when comparing different but related mathematical structures.

In essence, a procategory can be thought of as a category where the "objects" are themselves categories, and the "morphisms" are functors between these categories, subject to certain coherence conditions. This allows for the study of relationships and transformations between entire collections of mathematical structures, rather than just individual structures. The formal definition involves inverse systems of categories and their limit. Procategories find applications in various areas of mathematics, including algebraic topology, algebraic geometry, and theoretical computer science.