Universal problem

The concept of universal problem , one of most fundamental of the Theory of the categories, was introduced by the mathematicians Samuel Eilenberg and Saunders MacLane about 1943. It upset the vision of mathematics by introducing a dimension “behaviorist”, of which it should be heard that one not defines the mathematical objects by constructions (like one does it in Set theory), but by a characterization of their total behavior with respect to the other mathematical objects.

Initial object and final object.

We give a category $\ mathcal {C}$ . A $I$ object of $\ mathcal {C}$ is known as initial so for any $E$ object of $\ mathcal {C}$ , there exists one and only one arrow of $I$ towards $E$ . In the same way, a $F$ object is known as final so for any $E$ object, there exists one and only one arrow of $E$ towards $F$ .

This definition could appear extrèmement naive if it did not involve the following property:

Two initial objects (respectively final) in a category are canonically isomorphous.

Otherwise-known as if $I$ and $J$ are both initial in $\ mathcal {C}$ , the single arrow $f$ of $I$ towards $J$ is an isomorphism. Indeed, as $J$ is initial, there exist the same a single arrow $g$ of $J$ towards $I$ , and the compound $g \ circ f$ can be only the arrow identity of $I$ , always because $I$ is initial. For the same reason $f \ circ g$ can be only the identity of $J$ .

It is thus seen that the simple fact of asking that an object be initial defines it perfectly in canonical isomorphism close (i.e., as the data processing specialists would say, with the details of implementation near). In other terms, such definitions make it possible to concentrate on essence (the behavior of the definite object) without being concerned with details of its construction.

Of course, such a definition does not prove the existence of the object, which must possibly be proven by a construction. It does nothing but remove the definition from the object of all that is contingent. In against part, it obliges to integrate in the definition the necessary tools and sufficient for the handling of the object.

When a mathematical object is defined in this way, it is said that it is defined by a universal problem .

Examples.

Each following sentence constitutes a definition of what appears in fatty there.

the empty set is an initial object in the category of the units.

All singleton (together with only one element) is a final object in the category of the units.

the ring of the relative entireties $\ mathbb {Z}$ is initial in the category of the unit rings (commutative or not).

the quotient $\ pi \ colonist E \ to E/F$ (provided with its canonical projection) of a vector space $E$ by the vectorial subspace $F$ is initial in the category whose objects are the linear applications $f \ colonist E \ to G$ whose core contains $F$ . The arrows of this category of the object $f \ colonist E \ to G$ towards the object $g \ colonist E \ to H$ are the linear applications $\ varphi \ colonist G \ to H$ such as $g = \ varphi \ circ f$ .

the diagram $1 \ to^ {0} \ mathbb {NR} \ to^ {S} \ mathbb {NR}$ (where $1$ is a singleton, $0$ the single application of image $\ {0 \}$ and $S$ the function successor ), is initial in the category of the diagrams of the form $1 \ to X \ to^ {H} X$ . The arrows of this category of the object $1 \ to X \ to^ {H} X$ towards the object $1 \ to Y \ to^ {K} Y$ , are the applications $\ varphi \ colonist X \ to Y$ , such as $\ varphi \ circ 0 = 0$ and $\ varphi \ circ H = K \ circ \ varphi$ . (Definition of William Lawvere of the natural whole ).

the free group on the $E$ unit is initial in the category whose objects are the applications $a \ colonist E \ to G$ , where $G$ a group. The arrows of this category, of the object $a \ colonist E \ to G$ towards the object $b \ colonist E \ to H$ are the morphisms of groups $h \ colonist G \ to H$ such as $h \ circ has = b$ .

the produces tensorial $\ otimes \ colonist M \ times NR \ to M \ otimes_ {has} N$ of two modules $M$ (module on the right) and $N$ (module on the left) on the ring $A$ is initial in the category of the bilinear applications of source $M \ times N$ . The arrows of this category of the object $f \ colonist M \ times NR \ to A$ towards the object $g \ colonist M \ times NR \ to B$ are the linear applications $\ varphi \ colonist has \ to B$ , such as $g= \ varphi \ circ f$ .

the compactifié of Stone-Cech $\ gamma \ colonist X \ to \ check X$ of topological space $X$ is initial in the category whose objects are the continuous applications $f \ colonist X \ to Y$ , where $Y$ is a compact space. The arrows of this category of the object $f \ colonist X \ to Y$ towards the object $g \ colonist X \ to Z$ are the continuous applications $h \ colonist Y \ to Z$ such as $g=h \ circ f$ .

One could multiply the examples. It is not very probable that there exists a mathematical concept escaping a definition from this type.

Other formulations

This concept of universal problem can be expressed in a way more sophisticated (leading to automatic obtaining certain theorems) through that of Foncteur assistant.

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