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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, a functor is a mapping between categories. Functors were first considered in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, where algebraic objects (such as the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
) are associated to
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s, and maps between these algebraic objects are associated to
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
and
Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. ...
, respectively. The latter used ''functor'' in a
linguistic Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
context; see
function word In linguistics, function words (also called functors) are words that have little lexical meaning or have ambiguous meaning and express grammatical relationships among other words within a sentence, or specify the attitude or mood of the speake ...
.


Definition

Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y) in ''D'' such that the following two conditions hold: ** F(\mathrm_) = \mathrm_\,\! for every object X in ''C'', ** F(g \circ f) = F(g) \circ F(f) for all morphisms f \colon X \to Y\,\! and g \colon Y\to Z in ''C''. That is, functors must preserve identity morphisms and composition of morphisms.


Covariance and contravariance

There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor ''F'' from ''C'' to ''D'' as a mapping that *associates each object X in ''C'' with an object F(X) in ''D'', *associates each morphism f \colon X\to Y in ''C'' with a morphism F(f) \colon F(Y) \to F(X) in ''D'' such that the following two conditions hold: **F(\mathrm_X) = \mathrm_\,\! for every object X in ''C'', **F(g \circ f) = F(f) \circ F(g) for all morphisms f \colon X\to Y and g \colon Y\to Z in ''C''. Note that contravariant functors reverse the direction of composition. Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a ''covariant'' functor on the opposite category C^\mathrm. Some authors prefer to write all expressions covariantly. That is, instead of saying F \colon C\to D is a contravariant functor, they simply write F \colon C^ \to D (or sometimes F \colon C \to D^) and call it a functor. Contravariant functors are also occasionally called ''cofunctors''. There is a convention which refers to "vectors"—i.e., vector fields, elements of the space of sections \Gamma(TM) of a tangent bundle TM—as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections \Gamma\mathord\left(T^*M\right) of a
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...
T^*M—as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as ^ = \Lambda^i_j x^j for \mathbf' = \boldsymbol\mathbf or \omega'_i = \Lambda^j_i \omega_j for \boldsymbol' = \boldsymbol\boldsymbol^\textsf. In this formalism it is observed that the coordinate transformation symbol \Lambda^j_i (representing the matrix \boldsymbol^\textsf) acts on the basis vectors "in the same way" as on the "covector coordinates": \mathbf_i = \Lambda^j_i\mathbf_j—whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: \mathbf^i = \Lambda^i_j \mathbf^j). This terminology is contrary to the one used in category theory because it is the covectors that have ''pullbacks'' in general and are thus ''contravariant'', whereas vectors in general are ''covariant'' since they can be ''pushed forward''. See also
Covariance and contravariance of vectors In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notat ...
.


Opposite functor

Every functor F \colon C\to D induces the opposite functor F^\mathrm \colon C^\mathrm\to D^\mathrm, where C^\mathrm and D^\mathrm are the opposite categories to C and D. By definition, F^\mathrm maps objects and morphisms in the identical way as does F. Since C^\mathrm does not coincide with C as a category, and similarly for D, F^\mathrm is distinguished from F. For example, when composing F \colon C_0\to C_1 with G \colon C_1^\mathrm\to C_2, one should use either G\circ F^\mathrm or G^\mathrm\circ F. Note that, following the property of opposite category, \left(F^\mathrm\right)^\mathrm = F.


Bifunctors and multifunctors

A bifunctor (also known as a binary functor) is a functor whose domain is a product category. For example, the Hom functor is of the type . It can be seen as a functor in ''two'' arguments. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other. A multifunctor is a generalization of the functor concept to ''n'' variables. So, for example, a bifunctor is a multifunctor with .


Properties

Two important consequences of the functor
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s are: * ''F'' transforms each
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
in ''C'' into a commutative diagram in ''D''; * if ''f'' is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
in ''C'', then ''F''(''f'') is an isomorphism in ''D''. One can compose functors, i.e. if ''F'' is a functor from ''A'' to ''B'' and ''G'' is a functor from ''B'' to ''C'' then one can form the composite functor from ''A'' to ''C''. Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the category of small categories. A small category with a single object is the same thing as a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.


Examples

;
Diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...
: For categories ''C'' and ''J'', a diagram of type ''J'' in ''C'' is a covariant functor D \colon J\to C. ; (Category theoretical) presheaf:For categories ''C'' and ''J'', a ''J''-presheaf on ''C'' is a contravariant functor D \colon C\to J.In the special case when J is Set, the category of sets and functions, ''D'' is called a presheaf on ''C''. ; Presheaves (over a topological space): If ''X'' is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, then the
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s in ''X'' form a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
Open(''X'') under inclusion. Like every partially ordered set, Open(''X'') forms a small category by adding a single arrow if and only if U \subseteq V. Contravariant functors on Open(''X'') are called ''
presheaves In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
'' on ''X''. For instance, by assigning to every open set ''U'' the
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
of real-valued continuous functions on ''U'', one obtains a presheaf of algebras on ''X''. ; Constant functor: The functor which maps every object of ''C'' to a fixed object ''X'' in ''D'' and every morphism in ''C'' to the identity morphism on ''X''. Such a functor is called a ''constant'' or ''selection'' functor. ; : A functor that maps a category to that same category; e.g., polynomial functor. ; : In category ''C'', written 1''C'' or id''C'', maps an object to itself and a morphism to itself. The identity functor is an endofunctor. ; Diagonal functor: The diagonal functor is defined as the functor from ''D'' to the functor category ''D''''C'' which sends each object in ''D'' to the constant functor at that object. ; Limit functor: For a fixed index category ''J'', if every functor has a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
(for instance if ''C'' is complete), then the limit functor assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem. This requires a suitable version of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant). ; Power sets functor: The power set functor maps each set to its
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
and each function f \colon X \to Y to the map which sends U \in \mathcal(X) to its image f(U) \in \mathcal(Y). One can also consider the contravariant power set functor which sends f \colon X \to Y to the map which sends V \subseteq Y to its inverse image f^(V) \subseteq X. For example, if X = \ then F(X) = \mathcal(X) = \. Suppose f(0) = \ and f(1) = X. Then F(f) is the function which sends any subset U of X to its image f(U), which in this case means \ \mapsto f(\) = \, where \mapsto denotes the mapping under F(f), so this could also be written as (F(f))(\)= \. For the other values, \ \mapsto f(\) = \ = \,\ \ \mapsto f(\) = \ = \,\ \ \mapsto f(\) = \ = \. Note that f(\) consequently generates the trivial topology on X. Also note that although the function f in this example mapped to the power set of X, that need not be the case in general. ; : The map which assigns to every
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
its
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
and to every
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself. ; Fundamental group: Consider the category of pointed topological spaces, i.e. topological spaces with distinguished points. The objects are pairs , where ''X'' is a topological space and ''x''0 is a point in ''X''. A morphism from to is given by a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
map with . To every topological space ''X'' with distinguished point ''x''0, one can define the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
based at ''x''0, denoted . This is the group of
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
classes of loops based at ''x''0, with the group operation of concatenation. If is a morphism of pointed spaces, then every loop in ''X'' with base point ''x''0 can be composed with ''f'' to yield a loop in ''Y'' with base point ''y''0. This operation is compatible with the homotopy
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
and the composition of loops, and we get a group homomorphism from to . We thus obtain a functor from the category of pointed topological spaces to the category of groups. In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the fundamental
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *'' Group'' with a partial func ...
instead of the fundamental group, and this construction is functorial. ; Algebra of continuous functions: A contravariant functor from the category of
topological spaces In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
(with continuous maps as morphisms) to the category of real
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
s is given by assigning to every topological space ''X'' the algebra C(''X'') of all real-valued continuous functions on that space. Every continuous map induces an
algebra homomorphism In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF( ...
by the rule for every ''φ'' in C(''Y''). ; Tangent and cotangent bundles: The map which sends every
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
to its tangent bundle and every smooth map to its
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is a covariant functor from the category of differentiable manifolds to the category of
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s. Doing this constructions pointwise gives the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
above. ; Group actions/representations: Every group ''G'' can be considered as a category with a single object whose morphisms are the elements of ''G''. A functor from ''G'' to Set is then nothing but a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of ''G'' on a particular set, i.e. a ''G''-set. Likewise, a functor from ''G'' to the category of vector spaces, Vect''K'', is a linear representation of ''G''. In general, a functor can be considered as an "action" of ''G'' on an object in the category ''C''. If ''C'' is a group, then this action is a group homomorphism. ; Lie algebras: Assigning to every real (complex)
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
its real (complex)
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
defines a functor. ; Tensor products: If ''C'' denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
V \otimes W defines a functor which is covariant in both arguments. ; Forgetful functors: The functor which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor. Functors like these, which "forget" some structure, are termed ''
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
s''. Another example is the functor which maps a ring to its underlying additive
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
. Morphisms in Rng ( ring homomorphisms) become morphisms in Ab (abelian group homomorphisms). ; Free functors: Going in the opposite direction of forgetful functors are free functors. The free functor sends every set ''X'' to the free group generated by ''X''. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See
free object 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 eleme ...
. ; Homomorphism groups: To every pair ''A'', ''B'' of abelian groups one can assign the abelian group Hom(''A'', ''B'') consisting of all group homomorphisms from ''A'' to ''B''. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor (where Ab denotes the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of ...
with group homomorphisms). If and are morphisms in Ab, then the group homomorphism : is given by . See Hom functor. ; Representable functors: We can generalize the previous example to any category ''C''. To every pair ''X'', ''Y'' of objects in ''C'' one can assign the set of morphisms from ''X'' to ''Y''. This defines a functor to Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor . If and are morphisms in ''C'', then the map is given by . Functors like these are called representable functors. An important goal in many settings is to determine whether a given functor is representable.


Relation to other categorical concepts

Let ''C'' and ''D'' be categories. The collection of all functors from ''C'' to ''D'' forms the objects of a category: the functor category. Morphisms in this category are
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s between functors. Functors are often defined by universal properties; examples are the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
, the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
and
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of groups or vector spaces, construction of free groups and modules,
direct Direct may refer to: Mathematics * Directed set, in order theory * Direct limit of (pre), sheaves * Direct sum of modules, a construction in abstract algebra which combines several vector spaces Computing * Direct access (disambiguation), ...
and
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
limits. The concepts of limit and colimit generalize several of the above. Universal constructions often give rise to pairs of adjoint functors.


Computer implementations

Functors sometimes appear in
functional programming In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions tha ...
. For instance, the programming language Haskell has a
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently ...
Functor where fmap is a
polytypic function In programming language theory and type theory, polymorphism is the provision of a single interface to entities of different types or the use of a single symbol to represent multiple different types.: "Polymorphic types are types whose operati ...
used to map functions (''morphisms'' on ''Hask'', the category of Haskell types) between existing types to functions between some new types.See https://wiki.haskell.org/Category_theory/Functor#Functors_in_Haskell for more information.


See also

* Functor category * Kan extension * Pseudofunctor


Notes


References

* .


External links

* * see and the variations discussed and linked to there. * André Joyal
CatLab
a wiki project dedicated to the exposition of categorical mathematics * formal introduction to category theory. * J. Adamek, H. Herrlich, G. Stecker
Abstract and Concrete Categories-The Joy of Cats
*
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...
:
Category Theory
— by Jean-Pierre Marquis. Extensive bibliography.
List of academic conferences on category theory
* Baez, John, 1996

An informal introduction to higher order categories.
WildCats
is a
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
package for
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimiza ...
. Manipulation and visualization of objects,
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s, categories, functors,
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s, universal properties.
The catsters
a YouTube channel about category theory.
Video archive
of recorded talks relevant to categories, logic and the foundations of physics.
Interactive Web page
which generates examples of categorical constructions in the category of finite sets. {{Functors