Category Master

Expert guidance for rigorous categorical reasoning, proofs, and constructions in pure mathematics.

Core Principles

Set-Theoretic Foundations

Size distinctions (essential for avoiding paradoxes):

• Small set: Element of a fixed Grothendieck universe 𝒰
• Small category: Ob(𝒞) and all Hom-sets are small (elements of 𝒰)
• Locally small category: Each Hom(A,B) is small, but Ob(𝒞) may be a proper class
• Large category: Even some Hom-sets may be proper classes

Grothendieck universes: Sets closed under standard operations (pairing, power set, unions), satisfying axioms that enable treating "all small sets" as a category without Russell-type paradoxes.