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Some fundamental algebraic tools for the semantics of computation. III: Indexed categories. (English) Zbl 0755.18004
From the authors’ abstract and introduction: “For each many-sorted algebraic signature \(\Sigma\), there is a category \(\text{Alg}(\Sigma)\) of \(\Sigma\)-algebras, and a signature morphism \(\sigma: \Sigma\to\Sigma'\) induces a functor \(\text{Alg}(\sigma): \text{Alg}(\Sigma')\to\text{Alg}(\Sigma)\), which we call a \(\sigma\)- reduct. Thus, there is a functor \(\text{Alg}: \text{Alg\;Sig}^{op}\to\text{Cat}\) from the (index) category of signatures to the category of categories.” The authors therefore study “strict” indexed categories which appear to be “a useful tool for the working computer scientist. An indexed category gives rise to a single flattened category as a disjoint union of its component categories plus some additional morphisms. Similarly, an indexed functor (which is a uniform family of functors between the components categories) induces a flattened functor between the corresponding flattened categories. Under certain assumptions, flattened categories are (co)complete if all their components are, and flattened functors have left adjoints if all their components do.”
[The present Part III is entirely independent of Parts I and II (to appear)].

18D30 Fibered categories
68Q55 Semantics in the theory of computing
08C05 Categories of algebras
08A70 Applications of universal algebra in computer science
Full Text: DOI
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