Metakategori

Metakategori, or metakategori in some languages, is a term used in mathematics and logic to denote a higher-level categorical structure that concerns categories themselves. In this sense, it describes a framework for organizing, comparing, and studying categories, functors between them, and transformations among those functors.

In category theory, a standard example is Cat, the category whose objects are (small) categories and whose morphisms are functors between those categories. Cat can be enriched with additional structure: as a 2-category, its 1-morphisms are functors and its 2-morphisms are natural transformations. This elevates the discussion to a meta-level, where one analyzes properties of categories and functors from a higher vantage point.

Size considerations play a central role. If all categories are allowed as objects, one encounters size paradoxes unless one imposes restrictions. Common approaches restrict to small categories relative to a chosen universe, or work with large and locally small versions of Cat, using set-theoretic tools such as Grothendieck universes to avoid inconsistencies. The metacategory viewpoint is thus as much about managing size and coherence as about defining new objects.

Usage and scope vary by author. Some writers use metakategori as a casual description for any meta-level study of category theory, while others formalize it as a specific higher-category or as the category of categories with particular size constraints. In practice, terms like “category of categories,” “2-category of categories,” or “metacategory” may be employed interchangeably depending on the intended level and rigor.

Etymology derives from the prefix meta-, meaning about or beyond, applied to category theory to signal a discussion that operates at a level above ordinary categories.