David I. Spivak

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In this paper we introduce the olog, or ontology log, a category-theoretic model for knowledge representation (KR). Grounded in formal mathematics, ologs can be rigorously formulated and cross-compared in ways that other KR models (such as semantic networks) cannot. An olog is similar to a relational database schema; in fact an olog can serve as a data(More)
In this paper we present a simple database definition language: that of categories and functors. A database schema is a category and a state is a set-valued functor. We show that morphisms of schemas induce three “data migration functors” that translate states from one schema to the other in canonical ways. Database states form a topos of which the(More)
Wiring diagrams, as seen in digital circuits, can be nested hierarchically and thus have an aspect of self-similarity. We show that wiring diagrams form the morphisms of an operad T , capturing this self-similarity. We discuss the algebra Rel of mathematical relations on T , and in so doing use wiring diagrams as a graphical language with which to structure(More)
We give a new construction for rigidifying a quasi-category into a simplicial category, and prove that it is weakly equivalent to the rigidification given by Lurie. Our construction comes from the use of necklaces, which are simplicial sets obtained by stringing simplices together. As an application of these methods, we use our model to reprove some basic(More)
The category Man of smooth manifolds is not closed under arbitrary fiber products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the non-transverse intersection of submanifolds is not a manifold. We describe a category dMan, called the category of derived manifolds with the following properties: 1. dMan(More)
We describe our experience implementing a broad categorytheory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share(More)
In this paper we describe a functorial data migration [6] scenario about the manufacturing service capability of a distributed supply chain. The scenario is a category-theoretic analog of an OWL ontologybased “semantic enrichment” scenario [5] developed at the National Institute of Standards and Technology (NIST). The scenario is presented using, and is(More)
An important objective in materials design is to develop a systematic methodology for replacing unavailable or expensive material building blocks by simpler and abundant ones, while maintaining or improving the functionality of the material. The mathematical field of category theory provides a formal specification language which lies at the heart of such a(More)