In the mathematical field of category theory, the **category of sets**, denoted as **Set**, is the category whose objects are sets. The arrows or morphisms between sets *A* and *B* are the total functions from *A* to *B*, and the composition of morphisms is the composition of functions.

Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind.

The axioms of a category are satisfied by **Set** because composition of functions is associative, and because every set *X* has an identity function id_{X} : *X* → *X* which serves as identity element for function composition.

The epimorphisms in **Set** are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.

The empty set serves as the initial object in **Set** with empty functions as morphisms. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects in **Set**.

The category **Set** is complete and co-complete. The product in this category is given by the cartesian product of sets. The coproduct is given by the disjoint union: given sets *A*_{i} where *i* ranges over some index set *I*, we construct the coproduct as the union of *A*_{i}×{*i*} (the cartesian product with *i* serves to ensure that all the components stay disjoint).

**Set** is the prototype of a concrete category; other categories are concrete if they are "built on" **Set** in some well-defined way.

Every two-element set serves as a subobject classifier in **Set**. The power object of a set *A* is given by its power set, and the exponential object of the sets *A* and *B* is given by the set of all functions from *A* to *B*. **Set** is thus a topos (and in particular cartesian closed and exact in the sense of Barr).

**Set** is not abelian, additive nor preadditive.

Every non-empty set is an injective object in **Set**. Every set is a projective object in **Set** (assuming the axiom of choice).

The finitely presentable objects in **Set** are the finite sets. Since every set is a direct limit of its finite subsets, the category **Set** is a locally finitely presentable category.

If *C* is an arbitrary category, the contravariant functors from *C* to **Set** are often an important object of study. If *A* is an object of *C*, then the functor from *C* to **Set** that sends *X* to Hom_{C}(*X*,*A*) (the set of morphisms in *C* from *X* to *A*) is an example of such a functor. If *C* is a small category (i.e. the collection of its objects forms a set), then the contravariant functors from *C* to **Set**, together with natural transformations as morphisms, form a new category, a functor category known as the category of presheaves on *C*.

In Zermelo–Fraenkel set theory the collection of all sets is not a set; this follows from the axiom of foundation. One refers to collections that are not sets as proper classes. One cannot handle proper classes as one handles sets; in particular, one cannot write that those proper classes belong to a collection (either a set or a proper class). This is a problem because it means that the category of sets cannot be formalized straightforwardly in this setting. Categories like **Set** whose collection of objects forms a proper class are known as large categories, to distinguish them from the small categories whose objects form a set.

One way to resolve the problem is to work in a system that gives formal status to proper classes, such as NBG set theory. In this setting, categories formed from sets are said to be *small* and those (like **Set**) that are formed from proper classes are said to be *large*.

Another solution is to assume the existence of Grothendieck universes. Roughly speaking, a Grothendieck universe is a set which is itself a model of ZF(C) (for instance if a set belongs to a universe, its elements and its powerset will belong to the universe). The existence of Grothendieck universes (other than the empty set and the set of all hereditarily finite sets) is not implied by the usual ZF axioms; it is an additional, independent axiom, roughly equivalent to the existence of strongly inaccessible cardinals. Assuming this extra axiom, one can limit the objects of **Set** to the elements of a particular universe. (There is no "set of all sets" within the model, but one can still reason about the class *U* of all inner sets, i.e., elements of *U*.)

In one variation of this scheme, the class of sets is the union of the entire tower of Grothendieck universes. (This is necessarily a proper class, but each Grothendieck universe is a set because it is an element of some larger Grothendieck universe.) However, one does not work directly with the "category of all sets". Instead, theorems are expressed in terms of the category **Set**_{U} whose objects are the elements of a sufficiently large Grothendieck universe *U*, and are then shown not to depend on the particular choice of *U*. As a foundation for category theory, this approach is well matched to a system like Tarski–Grothendieck set theory in which one cannot reason directly about proper classes; its principal disadvantage is that a theorem can be true of all **Set**_{U} but not of **Set**.

Various other solutions, and variations on the above, have been proposed.^{ [1] }^{ [2] }^{ [3] }

The same issues arise with other concrete categories, such as the category of groups or the category of topological spaces.

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 category theory, a branch of mathematics, a **Grothendieck topology** is a structure on a category *C* that makes the objects of *C* act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a **site**.

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 category theory, a branch of mathematics, **group objects** are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous.

In category theory, a branch of mathematics, an **enriched category** generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms associated with every pair of objects is replaced by an object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint.

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 category theory, the **product** of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a **universe** is a collection that contains all the entities one wishes to consider in a given situation.

In category theory, the **coproduct**, or **categorical sum**, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

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 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, 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.

**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.

In mathematics, specifically in category theory, *F*-**algebras** generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor *F*, the *signature*.

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

In category theory, a branch of mathematics, a **diagram** is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a *function* from a fixed index *set* to the class of *sets*. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a *functor* from a fixed index *category* to some *category*.

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.

The theory of **accessible categories** is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" of the operations needed to generate their objects.

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.

In mathematics, a **Grothendieck category** is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.

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