Abstract
Stone dualities are dual equivalences between certain categories of algebras and those of topological spaces. A Stone duality is often derived from a dual adjunction between such categories by cutting down unnecessary objects. This dual adjunction is called the fundamental adjunction of the duality, but building it often requires concrete topological arguments. The aim of this paper is to construct fundamental adjunctions generically using (co)fibered category theory. This paper defines an abstract notion of formal spaces (including ordinary topological spaces as the leading example), and gives a construction of a fundamental adjunction between the category of algebras and the category of corresponding formal spaces.
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Notes
- 1.
Grothendieck originally called it cofibred categories, but here we use the word opfibration to avoid confusion with cofibration in homotopy theory.
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Acknowledgments
This work was the supported by JSPS KAKENHI Grant Number JP24700017. The second and third authors carried out this research under the support of JST ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603).
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Nishizawa, K., Katsumata, Sy., Komorida, Y. (2020). Stone Dualities from Opfibrations. In: Fahrenberg, U., Jipsen, P., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2020. Lecture Notes in Computer Science(), vol 12062. Springer, Cham. https://doi.org/10.1007/978-3-030-43520-2_14
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