Categories, Quotients & Dualities

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Abstractions about concrete and quotient categories, subquotients, categorical duality, and tilting relationships between algebraic structures.

5 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Concrete category — A category equipped with a faithful functor to Set, allowing its objects to be regarded as sets with structure and its morphisms as distinguishable structure-preserving functions.
• Quotient category — Keep a category's objects while replacing each hom-set by equivalence classes of morphisms under a composition-compatible congruence, so composition descends and the projection is universal for identifying equivalent arrows.
• Stone duality — Relate Boolean and related ordered algebraic structures contravariantly to compact topological or ordered spaces so algebraic elements become distinguished subsets and homomorphisms reverse into continuous maps.
• Subquotient — Obtain an algebraic object by first selecting a subobject and then quotienting it by a compatible normal subobject or congruence.
• Tilting theory — Use a self-orthogonal finite-projective-dimension module or analogous tilting object to construct an endomorphism algebra and transport controlled homological information between representation categories.