Category Theory & Higher Structures

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Abstractions about categories, functors, and their higher-dimensional generalizations, covering foundational categorical constructions (mathematical categories, subterminal objects, joins of categories, Yoneda extensions), higher-categorical and homotopical structures (3-categories, higher stacks, pseudomonads, simplicial localization), and invariants like K-theory.

18 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.

• 3-Category — A higher category with objects and composable morphisms in dimensions one through three, under specified coherence laws.
• Amnestic Functor — A functor for which any source isomorphism mapped to a target identity is necessarily already an identity morphism.
• Bijective proof — A proof of equal cardinality by an explicit invertible map pairing every object in one combinatorial class with exactly one object in another.
• Category of Manifolds — The category whose objects are manifolds of a declared C^p class and whose morphisms are C^p maps, with variants fixing model spaces, dimension, boundary, or smoothness conventions.
• Higher Stack — A higher-categorical stack: a presheaf valued in spaces, infinity-groupoids, or higher categories that satisfies descent by gluing compatible local objects together with all levels of equivalence and coherence data.
• Injective and Projective Model Structure — Dual model structures on a diagram category Fun(I,C) with objectwise weak equivalences; injective cofibrations or projective fibrations are objectwise, and the complementary class is determined by lifting when existence hypotheses hold.
• Join of Categories — The small-category construction that preserves two input categories, adds one morphism from every left object to every right object and none backward, and forms an associative monoidal product.
• K-theory — A family of functorial invariants that forms groups or spectra from stable classes of vector bundles, projective modules, automorphisms, or related structures, connecting topology, algebraic geometry, operator algebras, and index theory.
• Mathematical Category — A mathematical category is a structure consisting of objects, morphisms between objects, identity morphisms, and an associative partial composition law with matching domains and codomains, optionally enriched or equipped with additional categorical structure.
• Monoidal Category — A category with a bifunctorial product, unit object, and coherent natural associativity and unit isomorphisms.
• Monoidal Monad — A monad on a monoidal category equipped with coherent lax-monoidal comparison maps compatible with the monad unit and multiplication.
• Monoidal Natural Transformation — A natural transformation between monoidal functors whose components also commute with their tensor comparison and unit maps.
• Motive (algebraic geometry) — An object in a specified motivic category that packages algebraic-geometric correspondences and their cohomological realizations.
• Pseudomonad (category theory) — A higher-categorical monad whose unit and multiplication laws hold up to coherent invertible cells.
• Quasi-Isomorphism — A morphism of complexes inducing degreewise isomorphisms on homology or cohomology.
• Simplicial Localization — A localization of a category at chosen weak equivalences into a simplicial category whose mapping-space components recover ordinary localized morphisms while retaining higher homotopy data.
• Subterminal Object — An object of a category into which every object has at most one morphism; when a terminal object exists, subterminal objects are precisely its subobjects.
• Yoneda Extension — For a functor from a small category into a cocomplete category, the essentially unique colimit-preserving functor on the presheaf category whose restriction along the Yoneda embedding recovers the original functor.