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Toward formalizing ologs: Linguistic structures, instantiations, and mappings

We define the notion of linguistic structure on a small category, in order to provide a more formal description of ontology logs, also known as ologs, introduced by R. E. Kent and D. I. Spivak in their paper "Ologs: A categorical framework for knowledge representation." In particular, we construct a bicategory $\mathsf{Eng}$, of English noun phrases and verb phrases, endorsed as functional by varying sets of authors. An olog is then defined as a lax functor to $\mathsf{Eng}$. We then present a new notion of linguistic functor, which extends Spivak's notion of meaningful functors. Finally, we discuss the relationship between ologs and databases in this context.