Category-Theoretic Structures¶

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Abstractions about categorical organization through objects, morphisms, functors, limits, extensions, and dualities. They span enriched and higher categories, monoidal structure, universal constructions, factorization, and the completion or localization of categorical systems.

79 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-group — A monoidal groupoid in which every object has a weak inverse, categorifying the notion of a group.
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.
AB5 category — An abelian category with arbitrary coproducts in which filtered colimits of exact sequences remain exact; adding a generator yields a Grothendieck category.
Accessible quasi-category — An infinity-category equivalent to the closure of a small infinity-category under kappa-filtered colimits for some regular cardinal kappa.
Beck's monadicity theorem — A categorical criterion determining when a functor is equivalent to the forgetful functor from algebras for the monad induced by its adjunction.
Cartesian closed category — A category with a terminal object, binary products and exponential objects representing morphisms out of products.
Category of metric spaces — The category whose objects are metric spaces and whose morphisms are nonexpansive maps.
Category of relations — The category Rel whose objects are sets and whose morphisms are binary relations composed by existential relational composition.
Category of representations — A category whose objects are representations of a fixed algebraic structure and whose morphisms are equivariant maps.
Category theory — A mathematical framework studying objects through composable morphisms, identities, functors, natural transformations, and universal properties.
Chu space — A three-part relational structure of points, states, and values whose duality and morphisms generalize topological and linear spaces.
Closed monoidal category — A monoidal category in which tensoring by any object has a right adjoint represented by an internal hom object.
Codensity monad — The monad given by the right Kan extension of a functor along itself when that extension exists.
Coequalizer — A universal quotient-like object that makes two parallel morphisms equal and factors every other morphism that equalizes them uniquely.
Cokernel — The universal quotient of a morphism's codomain that makes the morphism vanish, realized for linear maps as codomain modulo image.
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.
Coproduct — A categorical colimit receiving one morphism from each object and universal among all such cocones.
Cosmos (category theory) — A complete and cocomplete symmetric closed monoidal category chosen as the base of enrichment for categories, functors, natural transformations, limits and tensors.
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.
Dominant functor — A functor whose target objects are all retracts of objects in its image.
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.
Dual (category theory) — The principle that reversing every morphism and composition order converts any categorical statement into a dual statement valid in the opposite category.
Elementary theory of abstract categories — Lawvere's first-order axiomatization of categories and functors, treating objects indirectly through identity arrows and composition rather than through set-theoretic membership.
Envelope (category theory) — A universal embedding of a category or structured object into a larger completed category satisfying a specified closure or completion property.
Essentially surjective functor — A functor whose image contains an object isomorphic to every object in its codomain.
Extensive category — A category with finite coproducts that are disjoint and stable enough that objects over a coproduct decompose equivalently into objects over its summands.
Factorization system — A pair of morphism classes in a category through which every morphism factors, with a unique lifting property characterizing the two classes against one another.
Filtered category — A nonempty category in which every finite diagram admits a compatible cocone, equivalently objects and parallel arrows can be jointly advanced and equalized.
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.
Free category — The category generated by a directed graph whose morphisms are finite composable paths and whose only equations are category axioms.
Generator (category theory) — An object or family of objects whose incoming probes distinguish every unequal pair of parallel morphisms in a category.
Globular set — A sequence of sets of n-cells with source and target maps satisfying globularity equations, forming the presheaf carrier for many higher-category structures.
Grothendieck category — An abelian category with arbitrary coproducts, exact filtered colimits, and a generator.
Groupoid object — An internal category in which every arrow has an inverse, defined inside a category with suitable pullbacks rather than only inside sets.
Image (category theory) — A universal monomorphism through which a morphism factors, generalizing the subset of attained values of a function.
Ind-completion — The free completion of a category under small filtered colimits, whose objects can be represented by filtered diagrams in the original category.
Injective object — A categorical object into which every morphism defined on a subobject extends across the containing monomorphism.
Inserter category — For parallel functors F and G from C to D, the category whose objects are arrows F(X) to G(X) and whose morphisms are C-arrows making the corresponding naturality square commute.
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.
Join (simplicial sets) — A monoidal operation combining two simplicial sets so simplices consist of an ordered simplex from the first followed by one from the second, corresponding under realization to topological join.
Kan extension — A universal way to extend a functor along another functor, with left and right Kan extensions respectively initial and terminal among compatible factorizations.
Karoubi envelope — The universal idempotent completion of a category, adjoining an image object for every idempotent morphism so that all idempotents split.
Kernel (category theory) — The universal morphism into an object's domain that is annihilated by a given morphism, equivalently the equalizer of that morphism and zero in a category with zero morphisms.
Krull–Schmidt category — An additive category in which every object decomposes into finitely many indecomposables uniquely up to permutation and isomorphism.
Localization of a category — A universal construction that formally makes a chosen class of morphisms invertible in a category.
Monoid (category theory) — An object in a monoidal category equipped with associative multiplication and a two-sided unit expressed by coherent morphism diagrams.
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.
Nodal decomposition — A category-theoretic factorization of a morphism as a strong epimorphism, followed by a bimorphism, followed by a strong monomorphism.
Opposite category — The category obtained by retaining every object and reversing the direction of every morphism and composition order.
Opposite simplicial set — The simplicial set obtained by precomposing with the order-reversing automorphism of the simplex category, extending categorical arrow reversal to higher categorical models.
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.
Presheaf (category theory) — A contravariant set-valued functor on a category, assigning data to each object and restriction maps to each morphism.
Pseudo-abelian category — A preadditive category in which every idempotent splits, equivalently every idempotent has an appropriate kernel and cokernel decomposition.
Pullback (category theory) — The categorical limit of two morphisms sharing a codomain.
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.
Quasi-category — A simplicial set satisfying every inner horn-filling condition, modeling an infinity-category with composition coherent up to higher homotopy.
Refinement (category theory) — A categorical construction that replaces an object's structure through a universal morphism from a chosen class, dual to an envelope construction.
Ribbon category — A rigid braided monoidal category equipped with a twist compatible with braiding and duality.
Rigid category — A monoidal category in which every object has a left and right dual, with evaluation and coevaluation morphisms satisfying triangular identities.
Semigroupoid — A category-like partial algebra with objects, composable morphisms and associative composition but without requiring an identity morphism at every object.
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.
Simplicially enriched category — A category whose hom-objects are simplicial sets and whose composition and identities are simplicial maps, encoding higher homotopies between morphisms.
Skeletonization of fusion categories — Reduction of a fusion category to skeletal simple-object labels, fusion rules, and coherence data.
Small set (category theory) — A set belonging to a fixed foundational universe used to bound categorical size.
Span (category theory) — A diagram of two morphisms with common domain, used as a generalized relation or correspondence between their codomains.
Subcategory — A category whose objects and morphisms are selected from a parent category while retaining the same sources, targets, identity morphisms, and composition.
Subobject — An equivalence class of monomorphisms into an object, abstracting the notion of a subset, subgroup, or subspace inside an arbitrary category.
Tetracategory — A weak four-dimensional categorical structure in which composition and coherence extend tricategorical cells by one dimension rather than holding strictly.
Theory of categories — The ontological project of identifying the highest and most general kinds of being and the fundamental distinctions among entities.
Topological category (enriched category theory) — A category whose hom-sets carry topological-space structure and whose identity and composition maps are continuous, usually formalized as enrichment over compactly generated Hausdorff spaces.
Tower of objects — An inverse sequence in a category: objects indexed by nonnegative integers with compatible maps from every later stage to each earlier stage.
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.
Twisted diagonal (category theory) — A category whose objects are arrows of a category and whose morphisms are oppositely directed domain-codomain squares.
Unitary modular tensor category — A modular tensor category equipped with compatible Hilbert-space and dagger structure making braiding, duality and fusion unitary.
Waldhausen category — A category equipped with designated cofibrations and weak equivalences satisfying gluing axioms so its algebraic K-theory spectrum can be constructed by the S-construction.