In category theory, a branch of mathematics, a **section** is a right inverse of some morphism. Dually, a **retraction** is a left inverse of some morphism. In other words, if *f* : *X* → *Y* and *g* : *Y* → *X* are morphisms whose composition *f*o*g* : *Y* → *Y* is the identity morphism on *Y*, then *g* is a section of *f*, and *f* is a retraction of *g*.^{ [1] }

Every section is a monomorphism (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative).

In algebra, sections are also called **split monomorphisms** and retractions are also called **split epimorphisms**. In an abelian category, if *f* : *X* → *Y* is a split epimorphism with split monomorphism *g* : *Y* → *X*, then *X* is isomorphic to the direct sum of *Y* and the kernel of *f*. The synonym **coretraction** for section is sometimes seen in the literature, although rarely in recent work.

The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: where is a subspace of is a retraction in the topological sense, if it's a retraction of the inclusion map in the category theory sense. The concept in topology was defined by Karol Borsuk in 1931^{ [2] }.

Borsuk's student, Samuel Eilenberg, was with Saunders Mac Lane the founder of category theory, and since the earliest publications on category theory concerned various topological spaces, one might have expected this term to have initially be used. In fact, their earlier publications, up to, e.g., Mac Lane (1963)'s *Homology*, used the term right inverse. It was not until 1965 when Eilenberg and John Coleman Moore coined the dual term 'coretraction' that Borsuk's term was lifted to category theory in general.^{ [3] } The term coretraction gave way to the term section by the end of the 1960s.

Both use of left/right inverse and section/retraction are commonly seen in the literature: the former use has the advantage that it is familiar from the theory of semigroups and monoids; the latter is considered less confusing by some because one does not have to think about 'which way around' composition goes, an issue that has become greater with the increasing popularity of the synonym *f;g* for *g∘f*.^{ [4] }

In the category of sets, every monomorphism (injective function) with a non-empty domain is a section, and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice.

In the category of vector spaces over a field *K*, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis.

In the category of abelian groups, the epimorphism **Z** → **Z**/2**Z** which sends every integer to its remainder modulo 2 does not split; in fact the only morphism **Z**/2**Z** → **Z** is the zero map. Similarly, the natural monomorphism **Z**/2**Z** → **Z**/4**Z** doesn't split even though there is a non-trivial morphism **Z**/4**Z** → **Z**/2**Z**.

The categorical concept of a section is important in homological algebra, and is also closely related to the notion of a section of a fiber bundle in topology: in the latter case, a section of a fiber bundle is a section of the bundle projection map of the fiber bundle.

Given a quotient space with quotient map , a section of is called a transversal.

- Mac Lane, Saunders (1978).
*Categories for the working mathematician*(2nd ed.). Springer Verlag. - Barry, Mitchell (1965).
*Theory of categories*. Academic Press.

- ↑ Mac Lane (1978, p.19).
- ↑ Borsuk, Karol (1931), "Sur les rétractes",
*Fundamenta Mathematicae*,**17**: 152–170, doi: 10.4064/fm-17-1-152-170 , Zbl 0003.02701 - ↑ Eilenberg, S., & Moore, J. C. (1965).
*Foundations of relative homological algebra*. Memoirs of the American Mathematical Society number 55. American Mathematical Society, Providence: RI, OCLC 1361982. The term was popularised by Barry Mitchell (1965)'s influential*Theory of categories*. - ↑ Cf. e.g., https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-9/

**Category theory** formalizes mathematical structure and its concepts in terms of a labeled directed graph called a *category*, whose nodes are called *objects*, and whose labelled directed edges are called *arrows*. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

In algebra, a **homomorphism** is a structure-preserving map between two algebraic structures of the same type. The word *homomorphism* comes from the ancient Greek language: *ὁμός (homos)* meaning "same" and *μορφή (morphe)* meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German *ähnlich* meaning "similar" to *ὁμός* meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In mathematics, a function *f* from a set *X* to a set *Y* is **surjective**, if for every element *y* in the codomain *Y* of *f*, there is at least one element *x* in the domain *X* of *f* such that *f*(*x*) = *y*. It is not required that *x* be unique; the function *f* may map one or more elements of *X* to the same element of *Y*.

In mathematics, an **abelian category** is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, **Ab**. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very *stable* categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.

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 the context of abstract algebra or universal algebra, a **monomorphism** is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation *X* ↪ *Y*.

In category theory, an **epimorphism** is a morphism *f* : *X* → *Y* that is right-cancellative in the sense that, for all objects *Z* and all morphisms *g*_{1}, *g*_{2}: *Y* → *Z*,

The **snake lemma** is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called *connecting homomorphisms*.

An **exact sequence** is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

**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 branch of mathematics, **duality** is a correspondence between the properties of a category *C* and the dual properties of the opposite category *C*^{op}. Given a statement regarding the category *C*, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category *C*^{op}. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about *C*, then its dual statement is true about *C*^{op}. Also, if a statement is false about *C*, then its dual has to be false about *C*^{op}.

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, an abstract branch of 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 of topological spaces**, often denoted **Top**, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of **Top** and of properties of topological spaces using the techniques of category theory is known as **categorical topology**.

**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, particularly in homotopy theory, a **model category** is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category of topological spaces or of chain complexes, via the acyclic model theorem. The concept was introduced by Daniel G. Quillen (1967).

In mathematics, a **projection** is a mapping of a set into a subset, which is equal to its square for mapping composition. The restriction to a subspace of a projection is also called a *projection*, even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane. The projection of a point is its shadow on the paper sheet. The shadow of a point on the paper sheet is this point itself (idempotency). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the Euclidean space of three dimensions onto a plane in it, like the shadow example. The two main projections of this kind are:

In category theory, a **regular category** is a category with finite limits and coequalizers of a pair of morphisms called **kernel pairs**, satisfying certain *exactness* conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of *images*, without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic.

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 are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

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