Categorical Constructions & Dualities

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Abstractions about categories and their higher-dimensional generalizations — limits, colimits, functors, and dualities such as Stone and Isbell duality — spanning universal constructions like pushouts and the Karoubi envelope, and monoidal, enriched, or n-categorical structures like double categories and triangulated categories.

33 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.

• 2-Yoneda lemma — A bicategorical generalization of Yoneda identifying pseudonatural transformations from a representable pseudofunctor to F with the category F assigns to the representing object.
• Categorical quotient — A universal invariant morphism from an object with group action through which every other invariant morphism factors uniquely.
• Category of sets — The category Set whose objects are sets and morphisms are total functions, with function composition, serving as a foundational categorical universe in which products, coproducts, limits, exponentials, and element-based intuitions are realized.
• Compact closed category — A symmetric monoidal category in which every object has a dual with unit and counit morphisms satisfying the snake identities.
• Complete category — A category possessing a limit for every diagram indexed by a small category, equivalently all small products and equalizers under standard size conventions.
• 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.
• Diagonal functor — The functor sending each object and morphism to a constant tuple or constant diagram, whose adjoints characterize categorical products, coproducts, limits and colimits.
• Diagram (category theory) — A functor from an index category into a target category, encoding a shaped family of objects together with all indexed morphisms and composition relations for limits, colimits, and universal constructions.
• Double category — A two-dimensional categorical structure with objects, horizontal arrows, vertical arrows and squares that compose in both directions subject to an interchange law.
• Five lemma — Infer that the middle vertical morphism in a commutative five-object diagram with exact rows is an isomorphism when the two neighboring outer maps meet the required isomorphism, epimorphism, and monomorphism conditions.
• Giry monad — The probability monad on measurable spaces that sends each space to its measurable space of probability measures.
• Groupoid object — An internal category in which every arrow has an inverse, defined inside a category with suitable pullbacks rather than only inside sets.
• Injective object — A categorical object into which every morphism defined on a subobject extends across the containing monomorphism.
• Interchange law — The coherence equation stating that composing compatible 2-cells horizontally and then vertically gives the same result as composing vertically and then horizontally.
• Isbell duality — An enriched categorical adjunction between presheaves and copresheaves induced by hom-pairing with representable functors.
• Isomorphism of categories — A strict equivalence between categories given by functors whose two composites are exactly the identity functors, producing one-to-one correspondence of objects and morphisms without merely natural isomorphism.
• Karoubi envelope — The universal idempotent completion of a category, adjoining an image object for every idempotent morphism so that all idempotents split.
• Mathematical structure — Endow one or more carrier sets with declared operations, relations, distinguished elements, topology, measure, or other typed data satisfying axioms, so objects are compared by morphisms and isomorphisms that preserve the selected structure rather than incidental presentation.
• Medial magma — A magma whose binary operation satisfies (ab)(cd)=(ac)(bd) for all four elements.
• Monoidal category action — A functor from a monoidal category times another category to that category, equipped with coherent natural isomorphisms expressing associative action and a unit.
• Polyad (mathematics) — A bicategorical generalization of a monad in which a locally punctual indexing bicategory maps into another bicategory, distributing monad-like data across multiple objects.
• Pseudo-abelian category — A preadditive category in which every idempotent splits, equivalently every idempotent has an appropriate kernel and cokernel decomposition.
• Pushout (category theory) — The colimit of a span X←Z→Y, giving the universal object formed by mapping X and Y together while identifying their images of Z.
• 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.
• Rigid category — A monoidal category in which every object has a left and right dual, with evaluation and coevaluation morphisms satisfying triangular identities.
• Simplex category — The category Δ of nonempty finite ordinals [n] and order-preserving maps, whose functors into or out of another category define simplicial and cosimplicial objects.
• 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.
• Tetracategory — A weak four-dimensional categorical structure in which composition and coherence extend tricategorical cells by one dimension rather than holding strictly.
• 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.
• Traced monoidal category — A monoidal category equipped with a trace operation that feeds an output object back into a matching input while satisfying naturality, dinaturality, vanishing, superposing and yanking axioms.
• Triangulated category — An additive category equipped with an autoequivalence and distinguished exact triangles satisfying axioms that abstract exact sequences and homotopy fiber-cofiber sequences.
• Weak n-category — Organize cells through dimension n with composition and units that satisfy associativity and unitality up to coherently related higher cells rather than by strict equality.