Category Theory Foundations

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Abstractions that define the basic objects and constructions of category theory itself — categories, functors and natural transformations, limits and colimits (coequalizer, coproduct, pullback), monoidal and enriched categories, and higher-categorical structures like quasi-categories and 2-groups.

48 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.
• Accessible quasi-category — An infinity-category equivalent to the closure of a small infinity-category under kappa-filtered colimits for some regular cardinal kappa.
• Action groupoid — The groupoid whose objects are points acted on by a group and whose arrows record group elements carrying one point to another.
• 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 theory — A mathematical framework studying objects through composable morphisms, identities, functors, natural transformations, and universal properties.
• 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.
• 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.
• Dominant functor — A functor whose target objects are all retracts of objects in its image.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.