Category Theory¶

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70 domain-specific abstractions whose origin domain is Category Theory.

AB5 category — An abelian category with arbitrary coproducts in which filtered colimits of exact sequences remain exact; adding a generator yields a Grothendieck category.
Action groupoid — The groupoid whose objects are points acted on by a group and whose arrows record group elements carrying one point to another.
Associativity Isomorphism — A natural family of isomorphisms rebracketing a categorical tensor product, constrained by the pentagon coherence identity.
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.
Categorical Lift — A morphism through an object over a target that restores a prescribed commutative triangle or fills a commutative-square lifting problem.
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 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.
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.
Coherent category — Equip a regular category with finite unions of subobjects that remain stable under pullback, providing categorical semantics for finite-limit, existential, and finite-disjunctive reasoning.
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.
Coproduct — A categorical colimit receiving one morphism from each object and universal among all such cocones.
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.
Free category — The category generated by a directed graph whose morphisms are finite composable paths and whose only equations are category axioms.
Functor Category — For fixed categories C and D, the category whose objects are functors C→D and whose morphisms are natural transformations, with identities and composition defined componentwise.
Generator (category theory) — An object or family of objects whose incoming probes distinguish every unequal pair of parallel morphisms in a category.
Grothendieck category — An abelian category with arbitrary coproducts, exact filtered colimits, and a generator.
Grothendieck topology — A categorical covering structure that designates compatible families of morphisms as covers, enabling sheaves and cohomology on categories whose objects need not be open subsets of a space.
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.
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.
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.
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.
Localizing Subcategory — A Serre subcategory whose exact quotient functor admits a right-adjoint section, so the objects it annihilates define a recoverable Gabriel localization of an abelian category.
Monoid (category theory) — An object in a monoidal category equipped with associative multiplication and a two-sided unit expressed by coherent morphism diagrams.
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.
Pointed set — A set equipped with one distinguished basepoint, with morphisms required to preserve that point, forming a category that adds a canonical zero-like reference to otherwise unstructured sets.
Presheaf (category theory) — A contravariant set-valued functor on a category, assigning data to each object and restriction maps to each morphism.
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.
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.
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.
Regular Category — A finitely complete category in which every morphism has a pullback-stable regular-epimorphism–monomorphism image factorization—equivalently, kernel-pair quotients exist and regular epimorphisms remain regular under pullback.
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.
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.
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.
Triangulated category — An additive category equipped with an autoequivalence and distinguished exact triangles satisfying axioms that abstract exact sequences and homotopy fiber-cofiber sequences.
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.