In mathematics, the category **Ab** has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category:^{ [1] } indeed, every small abelian category can be embedded in **Ab**.^{ [2] }

The zero object of **Ab** is the trivial group {0} which consists only of its neutral element.

The monomorphisms in **Ab** are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.

**Ab** is a full subcategory of **Grp**, the category of *all* groups. The main difference between **Ab** and **Grp** is that the sum of two homomorphisms *f* and *g* between abelian groups is again a group homomorphism:

- (
*f*+*g*)(*x*+*y*) =*f*(*x*+*y*) +*g*(*x*+*y*) =*f*(*x*) +*f*(*y*) +*g*(*x*) +*g*(*y*) - =
*f*(*x*) +*g*(*x*) +*f*(*y*) +*g*(*y*) = (*f*+*g*)(*x*) + (*f*+*g*)(*y*)

The third equality requires the group to be abelian. This addition of morphism turns **Ab** into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category.

In **Ab**, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e. the categorical kernel of the morphism *f* : *A* → *B* is the subgroup *K* of *A* defined by *K* = {*x*∈*A* : *f*(*x*) = 0}, together with the inclusion homomorphism *i* : *K* → *A*. The same is true for cokernels; the cokernel of *f* is the quotient group *C* = *B* / *f*(*A*) together with the natural projection *p* : *B* → *C*. (Note a further crucial difference between **Ab** and **Grp**: in **Grp** it can happen that *f*(*A*) is not a normal subgroup of *B*, and that therefore the quotient group *B* / *f*(*A*) cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that **Ab** is indeed an abelian category.

The product in **Ab** is given by the product of groups, formed by taking the cartesian product of the underlying sets and performing the group operation componentwise. Because **Ab** has kernels, one can then show that **Ab** is a complete category. The coproduct in **Ab** is given by the direct sum; since **Ab** has cokernels, it follows that **Ab** is also cocomplete.

We have a forgetful functor **Ab** → ** Set ** which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function. This functor is faithful, and therefore **Ab** is a concrete category. The forgetful functor has a left adjoint (which associates to a given set the free abelian group with that set as basis) but does not have a right adjoint.

Taking direct limits in **Ab** is an exact functor. Since the group of integers **Z** serves as a generator, the category **Ab** is therefore a Grothendieck category; indeed it is the prototypical example of a Grothendieck category.

An object in **Ab** is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator (**Z**) and an injective cogenerator (**Q**/**Z**).

Given two abelian groups *A* and *B*, their tensor product *A*⊗*B* is defined; it is again an abelian group. With this notion of product, **Ab** is a closed symmetric monoidal category.

**Ab** is not a topos since e.g. it has a zero object.

- Category of modules
- Abelian sheaf - many facts about the category of abelian groups continue to hold for the category of sheaves of abelian groups

In mathematics, specifically category theory, a **functor** is a map 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 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". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

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 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.

**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*.

In mathematics, a **sheaf** is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.

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.

**Mitchell's embedding theorem**, also known as the **Freyd–Mitchell theorem** or the **full embedding theorem**, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.

In mathematics, the category **Grp** 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.

In mathematics, in the area of category theory, a **forgetful 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 signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the **underlying set** of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.

In category theory, a **faithful functor** is a functor that is injective when restricted to each set of morphisms that have a given source and target.

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

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, 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.

- Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics,**211**(Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 - Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics.**5**(2nd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001. - Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004).
*Categorical foundations. Special topics in order, topology, algebra, and sheaf theory*. Encyclopedia of Mathematics and Its Applications.**97**. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.

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