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The topos structure theory of computing

John Bernier
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In this paper, the foundational role of topos theory in computer science is described. This story goes back at least to the work of Hartmanis and Stearns and their groundbreaking book Algebraic Structure Theory of Sequential Machines (1966), which described a model of information flows in sequential machines that made use of partition pairs. We arrive at the same formalism by studying quotient objects in the Sierpinski topos. We further generalize their work so that the foundational idea of an information flow is defined with the highest level of generality. The result is a universal structure theory of computation. This demonstrates the foremost role of topos theory not only in computer science but, by extension, all math.


The topos theory of computing: introduction to the mathematics of dataflow