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Egalisateur et coégalisateur

Publié le 14 décembre 2015

1. Egalisateur ( equalizer)

22-egalisateur-equalizer : Let X and Y be sets. Let f and g be functions, both from X to Y. Then the equaliser of f and g is the set of elements x of X such that f(x) equals g(x) in Y. Symbolically: \mathrm{Eq}(f,g) := \{x \in X \mid f(x) = g(x)\}\mbox{.}\! The equaliser may be denoted Eq(f,g) or a variation on that theme (such as with lowercase letters "eq"). In informal contexts, the notation {f = g} is common. The definition above used two functions f and g, but there is no need to restrict to only two functions, or even to only finitely many functions. In general, if F is a set of functions from X to Y, then the equaliser of the members of F is the set of elements x of X such that, given any two members f and g of F, f(x) equals g(x) in Y. Symbolically: \mathrm{Eq}(\mathcal{F}) := \{x \in X \mid \forall{f,g \,}{\in}\, \mathcal{F}, \; f(x) = g(x)\}\mbox{.}\! This equaliser may be written as Eq(f,g,h,...) if \mathcal{F} is the set {f,g,h,...}. In the latter case, one may also find {f = g = h = ···} in informal contexts.

2. Coégalisateur ( coequalizer )

23-coégalisateur-coequalizer : Equalisers can be defined by a universal property, which allows the notion to be generalised from the category of sets to arbitrary categories. In the general context, X and Y are objects, while f and g are morphisms from X to Y. These objects and morphisms form a diagram in the category in question, and the equaliser is simply the limit of that diagram. In more explicit terms, the equaliser consists of an object E and a morphism eq : E → X satisfying f \circ eq = g \circ eq, and such that, given any object O and morphism m : O → X, if f \circ m = g \circ m, then there exists a unique morphism u : O → E such that eq \circ u = m.

3. Noyaux et conoyaux

24-noyaux-conoyaux : Difference kernels A binary equaliser (that is, an equaliser of just two functions) is also called a difference kernel. This may also be denoted DiffKer(f,g), Ker(f,g), or Ker(f − g). The last notation shows where this terminology comes from, and why it is most common in the context of abstract algebra: The difference kernel of f and g is simply the kernel of the difference f − g. Furthermore, the kernel of a single function f can be reconstructed as the difference kernel Eq(f,0), where 0 is the constant function with value zero. Of course, all of this presumes an algebraic context where the kernel of a function is its preimage under zero; that is not true in all situations. However, the terminology "difference kernel" has no other meaning.

 

Younes Derfoufi
CRMEF OUJDA

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