Theory of the categories

The theory of the categories studies the mathematical structures and the relations which they maintain, made abstraction of the objects which have these structures.

The categories are used in the majority of the mathematical branches and certain sectors of the theoretical Informatique and in the mathematical physics. They form a unifying concept. This theory was installation by Samuel Eilenberg and Saunders Mac Lane in 1942-1945, in bond with the algebraic Topologie.

Basic elements

The study of the categories, very abstract, was justified by the abundance of characteristics common to various classes related to mathematical structures.

Here an example. The class Grp of the groups includes/understands all the objects having a “structure of group”. More precisely, Grp includes/understands all the Ensemble S G provided with a binary Relation which satisfies a certain whole of Axiome S. Of the Théorème S can thus be proven by carrying out logical deductions starting from this set of axioms. For example, they bring the direct proof that the element identity of a group is single.

Instead of studying simply the object alone (groups) which has a given structure, as the mathematical theories always did it, the theory of the categories stresses the Morphisme S and the process which preserve the structure between two objects. It appears that by studying these morphisms one is able to learn some more on the structure of the objects.

In our example, the studied morphisms are homomorphisms of groups. A homomorphism of group between two groups preserves the structure of group in a very precise way; these is a process which with a group associates some another, while preserving all information on the structure of the first group within the second group. The study of homomorphisms of group then provides a tool to study the general properties of the groups and the consequences of the axioms relating to the groups.

There exists a similar external intervention in many mathematical theories. A category is an axiomatic formulation relating to the idea to connect mathematical structures to the functions which preserve their structure. A systematic study of the categories makes it possible to prove general results starting from the axioms of a category.

A category is itself a type of structure mathematics of which there exist processes which preserve its structure. Such processes are called Foncteur S.

Definition

A category $\backslash \; mathcal\; C$ is the data of four elements:

• of a collection, whose elements are called objects,
• of a collection $\backslash \; mathrm\; \{Hom\}\; (has,\; B)$, for each pair of objects $A$ and $B$, whose elements $f$ are called morphisms (or arrows ) between $A$ and $B$, and are sometimes noted $f:\; With\; \backslash \; to\; B$,
• of a morphism $\backslash \; mathrm\; \{id\}\; \_A:\; With\; \backslash \; to\; A$, for each $A$ object, called identity on $A$ ,
• of a morphism $g\; \backslash \; circ\; F:\; With\; \backslash \; to\; C$ for any pair of morphisms $f:\; With\; \backslash \; to\; B$ and $g:\; B\; \backslash \; to\; C$, called made up of $f$ and $g$ ,

who are such as

• the composition is associative: for all morphisms $f:\; With\; \backslash \; to\; B$, $g:\; B\; \backslash \; to\; C$ and $h:\; C\; \backslash \; to\; D$,

$(H\; \backslash \; circ\; G)\; \backslash \; circ\; f=h\; \backslash \; circ\; (G\; \backslash \; circ\; F)$,

• the identities are neutral elements of the composition: for any morphism $f:\; With\; \backslash \; to\; B$,

$\backslash \; mathrm\; \{id\}\; \_B\; \backslash \; circ\; f=f=f\; \backslash \; circ\; \backslash \; mathrm\; \{id\}\; \_\; \{has\}$. sometimes

• one requires (for safety) that: $\backslash \; mathrm\; \{Hom\}\; (has,\; B)\; \backslash \; course\; \backslash \; mathrm\; \{Hom\}\; (C,\; D)\; =\; \backslash \; emptyset$ if $(has,\; B)\; \backslash \; neq\; (C,\; D)$

When a category is current, certain give him like name the abbreviation of the name of its objects, between brackets to announce that it is about their category; we will follow this convention here.

Examples

• the category $\backslash \; mathcal\; (Ens)$, whose objects are the Ensemble S, and arrows the applications, with the usual composition of the applications. In particular, it is seen that the objects of a category do not form a unit inevitably!
• the category $\backslash \; mathcal\; (Signal)$, whose objects are the topological spaces, and arrows continuous applications, with the usual composition.
• the category $\backslash \; mathcal\; (Met)$, whose objects are the metric spaces, and arrows applications uniformly continuous, with the usual composition.
• the category $\backslash \; mathcal\; (My)$, whose objects are the Monoïde S and the arrows the morphisms, with the usual composition.
• the category $\backslash \; mathcal\; (Grp)$, whose objects are the group S and the arrows the morphisms, with the usual composition.
• the category $\backslash \; mathcal\; (Ab)$, whose objects are the abelian groups and the arrows the morphisms, with the usual composition.
• the category $\backslash \; mathcal\; (ACU)$, whose objects are the unit commutative rings and the arrows the morphisms, with the usual composition.
• the category $\backslash \; mathcal\; (Ord)$, whose objects are the ordered units and the arrows the increasing applications.

The preceding examples have a joint property: the arrows are always applications, and the objects of the units (they are concrete categories); this property is very particular. Here examples of categories without this property:

• One gives itself a Monoïde $(M,\; *,\; E)\; \backslash ,$, and one defines the category $M\; \backslash ,$ as follows:

* objects: only one (it does not matter which)

* arrows: the elements of the monoid, they leave all the single object to return there;

* composition: given by the law of the monoid (the identity is thus the arrow associated with $e\; \backslash ,$).

• One gives itself a Ensemble $E\; \backslash ,$ provided with a relation reflexive and transitive $R\; \backslash ,$, and one defines the category associated as follows:

* objects: elements of the unit;

* arrows: for all objects $e\; \backslash ,$ and $f\; \backslash ,$, there exists an arrow of $e\; \backslash ,$ towards $f\; \backslash ,$ if and only if $eRf\; \backslash ,$ (and not of arrow if not);

* composition: the made up one of two arrows is the only arrow which joins together the two ends (the relation is transitive!) ; the identity is the only arrow which connects an object to itself (the relation is reflexive!).

This example is particularly interesting in the following case: the unit is the whole of open of a topological Espace, and the relation is inclusion; that makes it possible to define the concepts of Préfaisceau and Faisceau, via the Foncteur S.

Dual category

Starting from a category $\backslash \; mathcal\; C$, one can define another category $\backslash \; mathcal\; C^\; \{COp\}$ (or $\backslash \; mathcal\; C\; ^\; o$), known as opposite or dual , by taking the same objects, but by reversing the direction of the arrows.

More precisely: $Hom\_\; \{\backslash \; mathcal\; C^\; \{COp\}\}\; (has,\; B)=Hom\_\; \{\backslash \; mathcal\; C\}\; (B,\; A)$, and the composition of two opposite arrows is the opposite one of their composition:

$f^\; \{COp\}\; \backslash \; circ\; g^\; \{COp\}\; =\; (G\; \backslash \; circ\; F)\; ^\; \{COp\}$

It is clear that the dual category of the dual category is the starting category: $(\backslash \; mathcal\; C^\; \{COp\})\; ^\; \{COp\}\; =\; \backslash \; mathcal\; C$.

This extremely simple dualisation makes it possible to symmetrize the majority of the statements, which can be painful for the beginners…

Properties of the arrows

Definitions

An arrow $f:\; With\; \backslash \; rightarrow\; B$ a Monomorphisme is known as when it checks the following property: for any couple $g,\; H\; \backslash ,$ of arrows $E\; \backslash \; rightarrow\; A$ (and thus also for all $E\; \backslash ,$), if $f\; \backslash \; circ\; g=f\; \backslash \; circ\; h$, then $g=h\; \backslash ,$.

An arrow $f:\; With\; \backslash \; rightarrow\; B$ a epimorphism is known as when it checks the following property: for any couple $g,\; H\; \backslash ,$ of arrows $B\; \backslash \; rightarrow\; E$ (and thus also for all $E\; \backslash ,$), if $g\; \backslash \; circ\; f=h\; \backslash \; circ\; f$, then $g=h\; \backslash ,$.

The concepts of monomorphism and epimorphism are dual one of the other: an arrow is a monomorphism if and only if it is a epimorphism in the dual category.

An arrow $f:\; With\; \backslash \; rightarrow\; B$ a isomorphism is known as if there exists an arrow $g:\; B\; \backslash \; rightarrow\; A$ such as $g\; \backslash \; circ\; f=I\_A$ and $f\; \backslash \; circ\; g=I\_B$. This concept is autoduale.

Examples

• In the category of the Together S, monomorphisms is the injections, the epimorphisms are the surjections and isomorphisms are the bijections.
• an important counterexample in theory of the categories: a morphism can at the same time be a monomorphism and a epimorphism, without being for as much an isomorphism; to see this counterexample, it is necessary to be placed in the category of the unit commutative rings, and to consider the arrow (single!) $\backslash \; mathbb\; Z\; \backslash \; rightarrow\; \backslash \; mathbb\; Q$: it is a monomorphism because comes from an injective mapping, a epimorphism by localization, but is clearly not an isomorphism!
• One finds also such épimorphisme-monomorphism non-isomorphic in the categories of topological spaces: any injection is there a monomorphism, any surjection is a epimorphism, isomorphisms are the Homéomorphisme S, but there are at the same time injective and surjective continuous functions which are not homeomorphisms: for example identity on a unit provided with two different topologies, one coarser than the other.
• In the category of the ordered units isomorphisms are the increasing bijections (they are necessarily strictly increasing).

See too

• Functor.

Notice

Sometimes it happens that one forgets the objects of a category and that one is not interested any more but in the arrows, in substituent the arrow identity at the object.

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