# Category theory

Last updated

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, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

## Contents

A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the source and the target of the morphism. One often says that a morphism is an arrow that maps its source to its target. Morphisms can be composed if the target of the first morphism equals the source of the second one, and morphism composition has similar properties as function composition (associativity and existence of identity morphisms). Morphisms are often some sort of function, but this is not always the case. For example, a monoid may be viewed as a category with a single object, whose morphisms are the elements of the monoid.

The second fundamental concept of category is the concept of a functor, which plays the role of a morphism between two categories ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}:}$ it maps objects of ${\displaystyle C_{1}}$ to objects of ${\displaystyle C_{2}}$ and morphisms of ${\displaystyle C_{1}}$ to morphisms of ${\displaystyle C_{2}}$ in such a way that sources are mapped to sources and targets are mapped to targets (or, in the case of a contravariant functor, sources are mapped to targets and vice-versa). A third fundamental concept is a natural transformation that may be viewed as a morphism of functors.

## Categories, objects, and morphisms

### Categories

A categoryC consists of the following three mathematical entities:

• A class ob(C), whose elements are called objects;
• A class hom(C), whose elements are called morphisms or maps or arrows.
Each morphism f has a source object a and target object b.
The expression f : ab, would be verbally stated as "f is a morphism from a to b".
The expression hom(a, b) – alternatively expressed as homC(a, b), mor(a, b), or C(a, b) – denotes the hom-class of all morphisms from a to b.
• A binary operation ∘, called composition of morphisms, such that
for any three objects a, b, and c, we have
: hom(b, c) × hom(a, b) → hom(a, c).
The composition of f : ab and g : bc is written as gf or gf, [lower-alpha 1] governed by two axioms:
1. Associativity: If f : ab, g : bc, and h : cd then
h ∘ (gf) = (hg) ∘ f
2. Identity: For every object x, there exists a morphism 1x : xx called the identity morphism for x,
such that
for every morphism f : ab, we have
1bf = f = f ∘ ida. [lower-alpha 2]
From the axioms, it can be proved that there is exactly one identity morphism for every object.
Some authors[ who? ] deviate from the definition just given, by identifying each object with its identity morphism.

### Morphisms

Relations among morphisms (such as fg = h) are often depicted using commutative diagrams, with "points" (corners) representing objects and "arrows" representing morphisms.

Morphisms can have any of the following properties. A morphism f : ab is a:

• monomorphism (or monic) if fg1 = fg2 implies g1 = g2 for all morphisms g1, g2 : xa.
• epimorphism (or epic) if g1f = g2f implies g1 = g2 for all morphisms g1, g2 : bx.
• bimorphism if f is both epic and monic.
• isomorphism if there exists a morphism g : ba such that fg = 1b and gf = 1a. [lower-alpha 3]
• endomorphism if a = b. end(a) denotes the class of endomorphisms of a.
• automorphism if f is both an endomorphism and an isomorphism. aut(a) denotes the class of automorphisms of a.
• retraction if a right inverse of f exists, i.e. if there exists a morphism g : ba with fg = 1b.
• section if a left inverse of f exists, i.e. if there exists a morphism g : ba with gf = 1a.

Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent:

• f is a monomorphism and a retraction;
• f is an epimorphism and a section;
• f is an isomorphism.

## Functors

Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories.

A (covariant) functor F from a category C to a category D, written F : CD, consists of:

• for each object x in C, an object F(x) in D; and
• for each morphism f : xy in C, a morphism F(f) : F(x) → F(y) in D,

such that the following two properties hold:

• For every object x in C, F(1x) = 1F(x);
• For all morphisms f : xy and g : yz, F(gf) = F(g) ∘ F(f).

A contravariant functor F: CD is like a covariant functor, except that it "turns morphisms around" ("reverses all the arrows"). More specifically, every morphism f : xy in C must be assigned to a morphism F(f) : F(y) → F(x) in D. In other words, a contravariant functor acts as a covariant functor from the opposite category Cop to D.

## Natural transformations

A natural transformation is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors.

If F and G are (covariant) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism ηX : F(X) → G(X) in D such that for every morphism f : XY in C, we have ηYF(f) = G(f) ∘ ηX; this means that the following diagram is commutative:

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that ηX is an isomorphism for every object X in C.

## Other concepts

### Universal constructions, limits, and colimits

Using the language of category theory, many areas of mathematical study can be categorized. Categories include sets, groups and topologies.

Each category is distinguished by properties that all its objects have in common, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered atomic, i.e., we do not know whether an object A is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find universal properties that uniquely determine the objects of interest.

Numerous important constructions can be described in a purely categorical way if the category limit can be developed and dualized to yield the notion of a colimit.

### Equivalent categories

It is a natural question to ask: under which conditions can two categories be considered essentially the same, in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called equivalence of categories, which is given by appropriate functors between two categories. Categorical equivalence has found numerous applications in mathematics.

### Further concepts and results

The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading.

• The functor category DC has as objects the functors from C to D and as morphisms the natural transformations of such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes representable functors in functor categories.
• Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by "reversing all the arrows". If one statement is true in a category C then its dual is true in the dual category Cop. This duality, which is transparent at the level of category theory, is often obscured in applications and can lead to surprising relationships.
• Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties.

### Higher-dimensional categories

Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes".

For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e., processes which allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism.

This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of ω-category corresponding to the ordinal number ω.

Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept introduced by Ronald Brown. For a conversational introduction to these ideas, see John Baez, 'A Tale of n-categories' (1996).

## Historical notes

It should be observed first that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation [...]

Eilenberg and Mac Lane (1945) [1]

Whilst specific examples of functors and natural transformations had been given by Samuel Eilenberg and Saunders Mac Lane in a 1942 paper on group theory, [2] these concepts were introduced in a more general sense, together with the additional notion of categories, in a 1945 paper by the same authors [1] (who discussed applications of category theory to the field of algebraic topology). [3] Their work was an important part of the transition from intuitive and geometric homology to homological algebra, Eilenberg and Mac Lane later writing that their goal was to understand natural transformations, which first required the definition of functors, then categories.

Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland. Eilenberg was Polish, and studied mathematics in Poland in the 1930s. Category theory is also, in some sense, a continuation of the work of Emmy Noether (one of Mac Lane's teachers) in formalizing abstract processes; [4] Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure (homomorphisms).[ citation needed ] Eilenberg and Mac Lane introduced categories for understanding and formalizing the processes (functors) that relate topological structures to algebraic structures (topological invariants) that characterize them.

Category theory was originally introduced for the need of homological algebra, and widely extended for the need of modern algebraic geometry (scheme theory). Category theory may be viewed as an extension of universal algebra, as the latter studies algebraic structures, and the former applies to any kind of mathematical structure and studies also the relationships between structures of different nature. For this reason, it is used throughout mathematics. Applications to mathematical logic and semantics (categorical abstract machine) came later.

Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics. A topos can also be considered as a specific type of category with two additional topos axioms. These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics. Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.

Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in functional programming and domain theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus. At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense).

Category theory has been applied in other fields as well. For example, John Baez has shown a link between Feynman diagrams in physics and monoidal categories. [5] Another application of category theory, more specifically: topos theory, has been made in mathematical music theory, see for example the book The Topos of Music, Geometric Logic of Concepts, Theory, and Performance by Guerino Mazzola.

More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those of William Lawvere and Rosebrugh (2003) and Lawvere and Stephen Schanuel (1997) and Mirroslav Yotov (2012).

## Notes

1. Some authors compose in the opposite order, writing fg or fg for gf. Computer scientists using category theory very commonly write f ; g for gf
2. Instead of the notation 1x, the identity morphism for x may be denoted as idx.
3. Note that a morphism that is both epic and monic is not necessarily an isomorphism! An elementary counterexample: in the category consisting of two objects A and B, the identity morphisms, and a single morphism f from A to B, f is both epic and monic but is not an isomorphism.

## Related Research Articles

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 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, a category is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

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, an epimorphism is a morphism f : XY that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: YZ,

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.

In mathematics, a monoidal category is a category equipped with a bifunctor

The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows, where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

In category theory, a branch of mathematics, a monad is a monoid in the category of endofunctors. 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 and in functional programming languages, allowing languages with non-mutable states to do things such as simulate for-loops; see Monad.

In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of X. Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.

In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

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.

In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.

In category theory, a branch of mathematics, a closed category is a special kind of category.

Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space X to another topological space Y is associated the pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.

## References

### Citations

1. Eilenberg, Samuel; Mac Lane, Saunders (1945). "General theory of natural equivalences" (PDF). Transactions of the American Mathematical Society. 58: 247. doi:10.1090/S0002-9947-1945-0013131-6. ISSN   0002-9947. Archived (PDF) from the original on 2022-10-10.
2. Eilenberg, S.; Mac Lane, S. (1942). . Annals of Mathematics. 43 (4): 757–831. doi:10.2307/1968966. ISSN   0003-486X. JSTOR   1968966 via JSTOR.
3. Marquis, Jean-Pierre (2019). "Category Theory". Stanford Encyclopedia of Philosophy . Department of Philosophy, Stanford University . Retrieved 26 September 2022.
4. Reck, Erich (2020). The Prehistory of Mathematical Structuralism (1st ed.). Oxford University Press. pp. 215–219. ISBN   9780190641221.
5. Baez, J.C.; Stay, M. (2009). "Physics, topology, logic and computation: A Rosetta stone". New Structures for Physics. Lecture Notes in Physics. Vol. 813. pp. 95–172. arXiv:. doi:10.1007/978-3-642-12821-9_2. ISBN   978-3-642-12820-2. S2CID   115169297.

### Sources

• Marquis, Jean-Pierre (2008). From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory. Springer. ISBN   978-1-4020-9384-5.