# New PDF release: Algebraic Numbers and Algebraic Functions

By Cohn P.M.

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A collective link from P towards an object A of K is deﬁned as a family F ¼ ( fi)ieI of individual links of K such that (Fig. 2): (i) associated to each index i of the pattern is a link fi from the component Pi to A; 54 Chapter 2 Fig. 2 A collective link. A collective link from the pattern P to an object A in K consists of a link from every component of the pattern to A, these links being correlated by the distinguished links of the pattern. Thus, for each arrow x from i to j in the sketch sP of P, we have fi ¼ P(x)fj, where P(x) is the corresponding distinguished link from Pi to Pj.

In the left ﬁgure, the elements of the ﬁeld of A are seen in the category K; the ﬁeld as a category is drawn on the right. 36 Chapter 1 complete view of the system) only if A is a final object, that is, an object to which any other object is linked by one and only one link. For example, in an ecosystem an animal recognizes only its Umwelt (in the terminology of von Uexku¨ll, 1956) in which he cannot see a predator which is not near enough. g. transmission of information or imposing a constraint.

Thus K has for objects the vertices of G, and an arrow is an equivalence class of paths with the same weights. Consequently, an arrow is entirely determined by the data of its source, its target and its weight. The category K is also labelled in M. The categories modelling natural systems are often labelled categories constructed by this process: weights are initially given on (a graph of) generators, and the category is constructed by generators and relations, as the category associated to this labelled graph.