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In mathematics, the category **Grp** (or **Gp**^{ [1] }) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

There are two forgetful functors from **Grp**, M: **Grp** → **Mon** from groups to monoids and U: **Grp** → **Set** from groups to sets. M has two adjoints: one right, I: **Mon**→**Grp**, and one left, K: **Mon**→**Grp**. I: **Mon**→**Grp** is the functor sending every monoid to the submonoid of invertible elements and K: **Mon**→**Grp** the functor sending every monoid to the Grothendieck group of that monoid. The forgetful functor U: **Grp** → **Set** has a left adjoint given by the composite KF: **Set**→**Mon**→**Grp**, where F is the free functor; this functor assigns to every set *S* the free group on *S.*

The monomorphisms in **Grp** are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms.

The category **Grp** is both complete and co-complete. The category-theoretical product in **Grp** is just the direct product of groups while the category-theoretical coproduct in **Grp** is the free product of groups. The zero objects in **Grp** are the trivial groups (consisting of just an identity element).

Every morphism *f* : *G* → *H* in **Grp** has a category-theoretic kernel (given by the ordinary kernel of algebra ker f = {*x* in *G* | *f*(*x*) = *e*}), and also a category-theoretic cokernel (given by the factor group of *H* by the normal closure of *f*(*G*) in *H*). Unlike in abelian categories, it is not true that every monomorphism in **Grp** is the kernel of its cokernel.

The category of abelian groups, **Ab**, is a full subcategory of **Grp**. **Ab** is an abelian category, but **Grp** is not. Indeed, **Grp** isn't even an additive category, because there is no natural way to define the "sum" of two group homomorphisms. A proof of this is as follows: The set of morphisms from the symmetric group *S*_{3} of order three to itself, , has ten elements: an element *z* whose product on either side with every element of *E* is *z* (the homomorphism sending every element to the identity), three elements such that their product on one fixed side is always itself (the projections onto the three subgroups of order two), and six automorphisms. If **Grp** were an additive category, then this set *E* of ten elements would be a ring. In any ring, the zero element is singled out by the property that 0*x*=*x*0=0 for all *x* in the ring, and so *z* would have to be the zero of *E*. However, there are no two nonzero elements of *E* whose product is *z*, so this finite ring would have no zero divisors. A finite ring with no zero divisors is a field, but there is no field with ten elements because every finite field has for its order, the power of a prime.

The notion of exact sequence is meaningful in **Grp**, and some results from the theory of abelian categories, such as the nine lemma, the five lemma, and their consequences hold true in **Grp**. The snake lemma however is not true in **Grp**.^{[ dubious – discuss ]}^{[ citation needed ]}

**Grp** is a regular category.

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 meaning "same" and μορφή 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, 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 mathematics, specifically category theory, **adjunction** is a relationship that two functors may have. 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, a branch of mathematics, an **initial object** of a category C is an object I in C such that for every object X in C, there exists precisely one morphism *I* → *X*.

In mathematics, specifically in category theory, a **preadditive category** is another name for an **Ab-category**, i.e., a category that is enriched over the category of abelian groups, **Ab**. That is, an **Ab-category****C** is a category such that every hom-set Hom(*A*,*B*) in **C** has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas:

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*,

In mathematics, specifically in category theory, an **additive category** is a preadditive category **C** admitting all finitary biproducts.

In mathematics, specifically in category theory, a **pre-abelian category** is an additive category that has all kernels and cokernels.

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 sequence of morphisms between objects 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 and its applications to other branches of mathematics, **kernels** are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism *f* : *X* → *Y* is the "most general" morphism *k* : *K* → *X* that yields zero when composed with *f*.

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 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 universal algebra, a **variety of algebras** or **equational class** is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called *finitary algebraic categories*.

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

In mathematics, the **category of rings**, denoted by **Ring**, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

- ↑ Borceux, Francis; Bourn, Dominique (2004).
*Mal'cev, protomodular, homological and semi-abelian categories*. Springer. p. 20. ISBN 1-4020-1961-0.

- Goldblatt, Robert (2006) [1984].
*Topoi, the Categorial Analysis of Logic*(Revised ed.). Dover Publications. ISBN 978-0-486-45026-1 . Retrieved 2009-11-25.

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