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\(L\)-valued categories generalities and examples related to algebra and topology. (English) Zbl 1068.18001
Gähler, W. (ed.) et al., Categorical structures and their applications. Proceedings of the North-West European category seminar, Berlin, Germany, March 28–29, 2003 . River Edge, NJ: World Scientific (ISBN 981-256-053-X/hbk). 291-311 (2004).
Suppose \(\mathcal L = (L,\leq,\wedge,\vee,*)\) is a \(GL\)-monoid. An \(\mathcal L\)-valued category is essentially a triple \((\mathcal C, \omega, \mu)\), where \(\mathcal C\) is a category (the bottom frame in this paper), \(\omega\) is a function from the class of objects in \(\mathcal C\) to \(L\), and \(\mu\) is a function from the class of morphisms in \(\mathcal C\) to \(L\) such that (1) \(\omega(X) = \mu(1_X)\) for every object \(X\); (2) \(\mu(f)\leq\omega(X)\wedge\omega(Y)\) for all \(f: X\longrightarrow Y\); (3) \(\mu(g\circ f)\geq\mu(g)*\mu(f)\). In this paper, after a brief discussion of some basic concepts related to \(\mathcal L\)-valued categories, the author exhibits several \(\mathcal L\)-valued categories arising from fuzzy topology and fuzzy group theory.
For the entire collection see [Zbl 1051.54001].

18A05 Definitions and generalizations in theory of categories
03E72 Theory of fuzzy sets, etc.
54A40 Fuzzy topology
54E05 Proximity structures and generalizations
54E35 Metric spaces, metrizability
20A05 Axiomatics and elementary properties of groups