Préfaisceau

In Mathematical, and more particularly in the Theory of the categories , a préfaisceau on a topological Espace $X$ is a functor contravariant of the category of open the of $X$ in another category. One can thus have the préfaisceaux ones of Ensemble S, of groups, rings or any other type of mathematical structures. The préfaisceaux ones precede the beams. In Geometry , as well besides in algebraic Geometry as in differential Geometry, the concept of beam is a generalization of the sections of a vectorial Fibré. Within this framework, X is a algebraic Variété or a differential Variété.

The beams were introduced into the Années 1940 for the needs for the Géométrie complexes by Henri Cartan, then by Jean Leray in Topologie. The beams took a considerable importance thereafter.

Vocabulary

Préfaisceau

The definition of a préfaisceau is the following one: Another point of view is to affirm that the application $\backslash \; mathcal\; \{F\}:\; U\; \backslash \; mapsto\; \backslash \; mathcal\; \{F\}\; (U)$ is a functor contravariant of the category of open of X in the category of the units.

When the units $\backslash \; mathcal\; \{F\}\; (U)$ are groups (resp. algebras, vector spaces,…) and that the applications of restriction are morphisms of groups (resp. morphisms of algebras, linear applications,…), one speaks about préfaisceau of groups ( préfaisceau of algebras , préfaisceau of vector spaces ,…).

If X is provided with the coarse Topologie, a préfaisceau of units $\backslash \; mathcal\; F$ on X is the data of a salaried Ensemble $(E,\; E)\; =\; (\backslash \; mathcal\; \{F\}\; (X),\; \backslash \; mathcal\; \{F\}\; (\backslash \; emptyset))$. A préfaisceau of groups is the data of a group, etc…

Examples

• On a differential Variety X , the data $C^\; \{\backslash \; infty\}\; (U)$ of the real functions $C^\; \{\backslash \; infty\}$ on open a U of X defines a préfaisceau $C^\; \{\backslash \; infty\}$ on X . The applications of restrictions are precisely the restrictions in the usual sense.

• Is Y a unit. For X a topological space, the data $Y\; (U)$ of the functions locally constant on U defines a préfaisceau Y on X .
• In the complex plan, the data, for each open U , of the holomorphic functions on this opened, form a préfaisceau (and even a beam).
• Always in the complex plan, an ordinary, linear differential equation and with holomorphic coefficients, being given, spaces of solutions on the open ones avoiding the singular points of the equation form a préfaisceau (and even a beam) of vector spaces of size equal to the order of the equation.
• In any category, is $X$ a variety (or object) of this category, then $Hom\; (*,\; X)$ is a beam on the category, it is even the canonical example because one always plunges a category in his Topos and any beam, F, are represented in the topos (or category of the beams) by $Hom\; (*,\; F)$. You can notice that all the preceding examples fall under its cut:
• In the category of the differential varieties where the arrows are the functions $C^\; \backslash \; infty$ the beam is $Hom\; (*,\; \backslash \; mathbb\; R)$.
• One takes $Y$ provided with his separate topology and then the beam becomes $Hom\; (*,\; Y)$.
• $Hom\; (*,\; \backslash \; mathbb\; C)$ in the category of open of $\backslash \; mathbb\; C$ where the arrows are the holomorphic functions.
• (I have the flemme to think of it).

It will be noticed that if the category admits a final object, $pt$ (Pt for point), then $Hom\; (*,\; Pt)$ is the final object of the topos (thus noted $pt$) and that if the category admits an initial object, $\backslash \; emptyset$ (this notation is not alleviating), then $Hom\; (*,\; \backslash \; emptyset)$ is the initial object of the topos (thus noted $\backslash \; emptyset$).

• (more examples?)

Beam

Concerning the continuous functions or the functions $C^\; \{\backslash \; infty\}$, the property is local. It is thus possible of " recoller" continuous functions or $C^\; \{\backslash \; infty\}$ coinciding on their field of definition in a continuous function or $C^\; \{\backslash \; infty\}$ total. It is this property which one wishes here to generalize in the world of the préfaisceaux one: For a family of open $\backslash \; \{U\_i\; \backslash \}\; \_I$ as in the definition, one notes:

$V\_\; \{ij\}\; =V\_i\; \backslash \; course\; V\_j$; $V\_\; \{ijk\}\; =V\_i\; \backslash \; course\; V\_j\; \backslash \; course\; V\_k$; …

Examples

• the préfaisceau of the constant functions is not a beam, because if one considers two open disjoined, and two constant functions on these open, one cannot define a constant function on both opened, which coincides with them in general. It is due to the fact that a constant function is defined by a total property.

• the functions locally constant, on the other hand, form a beam well, just as the functions derivable, $C^\; \backslash \; infty$, holomorphic… It is due to the fact that the definition of these functions is local.

Morphisms the préfaisceaux one

The préfaisceaux ones on a unit X can be regarded as Objets of a Catégorie. Which are the arrows?

Stems

The stem (or germinates) of a préfaisceau F on X is defined by:

$\backslash \; mathcal\; \{F\}\; \_x=\; \backslash \; lim\; \backslash \; mathcal\; \{F\}\; (U)$

limit being taken on all the open ones containing $x$. An element of $\backslash \; mathcal\; \{F\}\; \_x$ is thought like a germ of a section of F on an open vicinity of X .

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