This is a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.)
Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also glossary of algebraic topology.
The notations and the conventions used throughout the article are:
subject to the conditions: for any morphisms , and ,
For example, if R is a ring, M a right R-module and N a left R-module, then the tensor product of M and N is
where is the category (called the subdivision category of C) whose objects are symbols for all objects c and all morphisms u in C and whose morphisms are and if and where is induced by F so that would go to and would go to . For example, for functors ,
subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.
For example, a category enriched over sets is an ordinary category.[T]he issue of comparing definitions of weak n-category is a slippery one, as it is hard to say what it even means for two such definitions to be equivalent. [...] It is widely held that the structure formed by weak n-categories and the functors, transformations, ... between them should be a weak (n + 1)-category; and if this is the case then the question is whether your weak (n + 1)-category of weak n-categories is equivalent to mine—but whose definition of weak (n + 1)-category are we using here... ?
Tom Leinster, A survey of definitions of n-category
Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.
where Nat means the set of natural transformations. In particular, the functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property.
In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory. It allows the embedding of any locally small category into a category of functors defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category.
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices.
In category theory, a branch of mathematics, a monad is a monoid in the category of endofunctors of some fixed category. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also useful in the theory of datatypes, the denotational semantics of imperative programming languages, and in functional programming languages, allowing languages without mutable state to do things such as simulate for-loops; see Monad.
In mathematics, the derived categoryD(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences.
In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories without adjoined products may still have an exponential law.
In mathematics, specifically in category theory, hom-sets give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.
In category theory, an end of a functor is a universal dinatural transformation from an object e of X to S.
In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two coherence maps—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors
In category theory, a branch of mathematics, a presheaf on a category is a functor . If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space.
The Grothendieck construction is a construction used in the mathematical field of category theory. It is a fundamental construction in the theory of descent, in the theory of stacks, and in fibred category theory. In categorical logic, the construction is used to model the relationship between a type theory and a logic over that type theory, and allows for the translation of concepts from indexed category theory into fibred category theory, such as Lawvere's concept of hyperdoctrine.
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If is a morphism of geometric or algebraic objects, the corresponding cotangent complex can be thought of as a universal "linearization" of it, which serves to control the deformation theory of . It is constructed as an object in a certain derived category of sheaves on using the methods of homotopical algebra.
This is a glossary of properties and concepts in algebraic topology in mathematics.
In category theory, a branch of mathematics, the density theorem states that every presheaf of sets is a colimit of representable presheaves in a canonical way.
In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition.