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Introduction à l’analyse algébrique. II: Algèbres figuratives et esquisses. (French) Zbl 0649.18005

[For Part I see ibid. 96, 49-63 (1986; Zbl 0617.18001).]
The author considers the following generalization of the notion of an algebraic theory. A figuration is a 7-tuple \(T=(S,F,D,C,L,O,M)\), where S, F, C, O are categories and \(D: F^{op}\times S\to Set\), \(L: F\to C\), \(M: S\to O\) are functors; moreover, L and M are injective and bijective on objects. A T-algebra is a pair (s,A), where s is an object of S and \(A: C^{op}\to Set\) is a functor such that the composite of \(L^{op}: F^{op}\to C^{op}\) and A is \(D(-,s).\) These definitions are illustrated by many examples and pictures. Sketches in the sense of Ch. Ehresmann [Bul. Inst. Politeh. Iaşi, Nouv. Sér. 14(18), No.1/2, 1-14 (1968; Zbl 0196.031)] are also discussed in this context.
Reviewer: A.Wiweger

MSC:

18C10 Theories (e.g., algebraic theories), structure, and semantics
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