View list of category theory topics for a breakdown of relevant articles.

Background

A survey of categories is an attempt to capture what is commonly observed around various classes of related mathematical structures.

Assume a as punishment lesson. A class Grp of groups consists of all objects getting the "group structure". Supplementary precisely, Grp consists of a lot sets G endowed with the binary operation satisfying a certain placed of axioms. 1 could proceed to prove theorems about groups by making logical deductions from either a placed of axioms. For instance, these are immediately proved from either a axioms that a identity element of a class action is unique.

Instead of focusing but on a person objects (groups) possessing the given structure, when mathematical theories develop traditionally done, category theory emphasizes the morphisms — the structure-preserving processes — between these objects. It turns out that by researching these morphisms, i am entity to study far additional all about a structure of the objects than by only focusing on the structures alone. In a case of groups, a morphisms come the group homomorphisms. The class action homomorphy between ii groups "preserves the group structure" inside the super accurate way — these are the "process" ingesting a single class action to a 2nd, inside how else that carries along data just about a structure of a number 1 class action into the second class action. the learn of class action homomorphisms so will bring a mighty thing for researching general properties of groups & symptoms of the class action axioms.

The similar nature and severity of investigation occurs inside numbers of mathematical theories. The category is an taken for granted formulation of this idea of on mathematical structures to the structure-preserving processes between the children. the orderly learn of categories so allows united states to prove general effects all about any class of mathematical structures & their processes which satisfy the axioms of a category.

The category is itself the nature and severity of mathematical structure, and so i personally might search processes which preserve this structure inside a few feel. Such the run is known as the functor, & it associates to each object of the single category an object of an additional category, and to each morphism in the 1st category a morphism in the 2nd. By researching categories & functors, i am non upright researching a class of mathematical structures & a morphisms between the babies, i am researching the relationships between various classes of mathematical structure.

Several mathematical theories attempt to survey a particular nature and severity of structure by on the structure to an additional simpler, better understood structure. For instance, this is the central underlying theme of algebraic topology — very difficult topologic questions may be translated into algebraical questions which are then good deal more easygoing to solve. Certain "natural constructions", like a fundamental group of a topological space, can be expressed when functors therein way. Different such constructions come typically "naturally related", & this leads to the conception of natural transformation, a way to "map" a single functor to a second. Throughout maths, of these encounters "natural isomorphisms", items that come (essentially) the equivalent inside a "canonical way". Several significant constructions around maths may be exposed in that context.

Historical notes

Categories, functors & natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945. At a start, the notions were applied around topology, especially algebraic topology, as an important a share of the transition from either homology (an intuitive and geometrical conception) to homology theory, an axiomatic approach. It has been claimed, for even instance by or in behalf of Stanislaw Ulam, that comparable ideas were todays in the later on 1930s in the Polish school. These ideas were someways the continuation of the contributions of Emmy Noether in formalizing abstract processes in the first half of the 20th-century. Noether realised that sequentially to know a nature and severity of mathematical structure, of these really needs to realize the processes preserving this structure. Eilenberg & Mac Lane gave an axiomatic formalization of this relation between structures and a processes preserving the two.

Eilenberg/Mac Lane use at times said that their goal was to see natural transformations; sequentially to run that, functors experienced to exist as defined; & to define functors of these requisite categories.

A subsequent development of the theory was powered 1st per computational needs of homological algebra; and then per self-evident needs of algebraic geometry, the field virtually all immune to the Russell-Whitehead view of united foundations. General category theory—an updated universal algebra with many newly features allowing semantic flexibility & higher-order logic—came later; these are nowadays applied throughout math.

Favorite categories known as topoi can even service alternatively to axiomatic set theory as the foundation of maths. These broadly-depending foundational applications of category theory come contentious; however it develop been worked call at quite a few detail, as a comment in or even basis for constructive mathematics. Of these could say, particularly, that taken for granted placed theory however hasn't been replaced per category-theoretic comment on that, in the everyday usage of mathematicians. A idea of bringing category theory into earliest, undergrad teaching (signified per difference between a Birkhoff-Mac Lane & late Mac Lane-Birkhoff abstract algebra texts) has hit noticeable opposition.

Categorical logic is now the easily-chiseled field according to type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in the setting of the cartesian closed category as non-syntactic description of a lambda calculus. At a super least, a utilise of category theory language allows a single to clarify what exactly these related areas keep close at hand around commons (within an abstract sense).

Categories, objects, and morphisms

Independent articles: category, morphism

The category C consists of the class ob(C) of objects: the class hom(100) of morphisms. Apiece morphism f has the unique source object the & target object b. I write f: the → b, & i say "f is a morphism from a to b". I personally write hom(a, b) [or Hom(a, b), or hom_C(a, b)] to denote a hom-class of tons morphisms from either the to b. (Occasionally authors write Mor(a, b) or even C(the, b).) for each iii objects a, b & c, the binary operation hom(a, b) × hom(b, 100) → hom(the, c) known as composition of morphisms; a composition of f : the → b & g : b → c is written when g ○ f or even gf (A few authors write fg.)

such that a as a result axioms hang on to:

(associativity) whenever f : the → b, g : b → c & h : one hundrefive hundred → d so h ○ (g ○ f) = (h ○ g) ○ f, and (identity) for each object x, there is the morphism Ace_x : 10 → x known as a identity morphism for x, such that for each morphism f : the → b, i have One_b ○ f = f = f ○ Ace_the.

From either these axioms, of these could prove that there exists exactly of these identity morphism for each object. a bit of authors utilize a cold-shoulder variation of the definition where apiece object is identified sustaining the corresponding identity morphism.

Relations among morphisms (like fg = h) might virtually all handily become represented by owning commutative diagrams, where a objects come represented when points & a morphisms when arrows. Indeed, the morphisms of a category come occasionally known as arrows due to the influence of commutative diagrams.

Types of morphisms

The morphism f : the → b is called the monomorphism (or monic) whenever fg_Single = fg_Two implies g_I = g_Two for 100% morphisms g_One, g_Deuce : x → the. an epimorphism (or heroic poem) whenever g_Singlef = g_Iif implies g_One = g_Deuce for tons morphisms g_One, g_Ii : b → x. an isomorphism if there exists the morphism g : b → the by owning fg = Single_b & gf = Single_the.* an endomorphism if a = b. the class of endomorphisms of a is denoted prevent(the). an automorphism if f is both an endomorphism & an isomorphy. the class of automorphisms of a is denoted aut(a).

Note that the morphism which is each heroic & monic is non necessarily an isomorphy! E.g., in The category consisting of ii objects The & B, the identity morphisms, & one morphism f from either A to B, f is each larger-than-life & monic however is non an isomorphy.

Functors

Independent article: functor

Functors come structure-preserving maps between categories. It may be thought of when morphisms in the category of a lot (little) categories.

The (covariant) functor F from either a category C to the category D associates to every object x around C an object F(ten) within D; associates to both morphism f : ten → y the morphism F(f) : F(ten) → F(y)

such that a ensuing ii properties hang on to:

F(One_x) = Single_F(ten) for each object x around C. F(g ○ f) = F(g) ○ F(f) for tons morphisms f : 10 → y & g : y → z.

The contravariant functor F from either C to D occurs as functor that "turns morphisms around" ("reverses all the arrows"). Specifically, F is contravariant if whenever f : 10 → y occurs as morphism inside One hundred, so F(f) : F(y) → F(x). a quickly way to define a contravariant functor is as a covariant functor from either the paired category C^op to D.

Natural transformations and isomorphisms

Independent article: natural transformation

The natural transformation occurs as relation between ii functors. Functors typically describe "natural constructions" & natural transformations so describe "natural homomorphisms" between deuce such constructions. Another time deuce quite different constructions yield "the same" symptom; this is expressed by the natural isomorphy between them functors.

In case F & G come (covariant) functors between a categories C & D, so the natural transformation from either F to G associates to each object x around C the morphism η_x : F(ten) → G(10) around D such that for each morphism f : x → y around C, i have η_y ○ F(f) = G(f) ○ η_x; this means that a as a consequence diagram is commutative:

Them functors F & G come known as naturally isomorphous in case there is the natural transformation from either F to G such that η_x is an isomorphy for each object x around C.

Universal constructions, limits, and colimits

Independent articles: universal property, limit (category theory)

Using a language of category theory, several areas of mathematical learn may be cast into appropriate categories, like the categories of 100% sets, groups, topologies, so in. These categories sure enough develop a bit of objects that come "special" around a certain way, like the empty set or the product of two topologies. Eventually, in the definition of the category, objects come considered to exist as atomlike; i personally.e. i personally don't understand, whether an object The occurs as placed, the topology, or even any more abstract construct. Hence, a challenge is to define favorite objects while forgoing on to the internal structure one objects. However how can you define a empty placed while forgoing on to elements, or even a product topology forswearing on to open sets?

A guide is to characterize these objects inside terms of their relations to more objects, equally from a morphisms of the various categories. So a project is to locate universal properties that uniquely determine a objects of interest. Indeed, it turns out that many crucial constructions may be described inside the strictly categoric way. A central construct which is required for this purpose is known as flat set boundaries, & may be dualized to yield the notion of a colimit.

Equivalent categories

Independent articles: equivalence of categories, isomorphism of categories

These are the natural wonder to ask, under which conditions deuce categories may exist as considered to be "essentially the same", in a feel that theorems astir of these category could readily become transformed into theorems all about the more category. the major convienence a single employs to describe such a situation is known as equivalence of categories. These are from appropriate functors between 2 categories. Flat equivalence has obtained many applications inside math.

Further concepts and results

A definitions of categories & functors provide just a super fundamentals of flat algebra. Extra significant topics come employed following. Although there are hard interrelatedness between everthing one topics, a given choose may be considered as a guideline for farther reading.

A functor category D^C has as objects a functors from either C to D & when morphisms a natural transformations of such functors. A Yoneda lemma is one of the best known basic final result of category theory; it describes representable functors inside functor categories. Duality: Every statement, theorem, or even definition inside category theory has the dual which is fundamentally found by "reversing all the arrows". Whenever of these statement is admittedly within the category C so its dual is admittedly in the dual category C^op. This duality, which is transparent at a level of category theory, is typically obscured within applications & can lead to surprising relationships. Adjoint functors: A functor even may be left (or perfect) adjoint to an additional functor that maps in the paired counsel. Such the pair of adjoint functors occasionally arises from either the constructiin defined by the universal property; it may be seen as a other abstract & mighty look at on universal properties.

Higher-dimensional categories

Several of a above conception, especially equivalence of categories, adjoint functor pairs, & functor categories, may be placed into the context of higher-dimensional categories. Briefly, in case i personally assume the morphism between 2 objects as a "process taking us from one object to another", so higher-dimensional categories allow the states to fruitfully generalise this by looking for "higher-dimensional processes".

E.g., (nonindulgent) 2-category is a category together with "morphisms between morphisms", we.e. processes which allow u.s. to transform 1 morphism into an additional. I might so "compose" these "bimorphisms" each horizontally & vertically, & i postulate the Two-two-dimensional "exchange law" to hang on to, on them composition laws. In that context, a standard lesson is Cat, a Two-category of completely (little) categories, & in that case, bimorphisms of morphisms come just natural transformations of morphisms in the usual feel. An additional basic case is to assume the Two-category using one object—which are actually basically monoidal categories.

This run may be extended for a lot natural numbers n, & which are actually known as n-categories. There exists potentially the notion of ω-category corresponding to the ordinal number ω. For the colloquial introduction to these ideas, view [http://math.ucr.edu/home/baez/week73.html John Baez: The Tale of n-categories].