RDF (wikipedia entry) is a first order logic for the web built on the concept of Graphs.
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2009, PhD Thesis, Benjamin Braatz: Formal Modelling and Applicationof Graph Transformations in the Resource Description Framework. Chapter 2 of the thesis gives a faithful Category Theoretic interpretation of RDF.
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2017, Evan Patterson Knowledge Representation in Bicategories of Relations. This article shows how one gets a typed version of RDF and OWL using a bicategory of relations.
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