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Category Theory Applied to Neural Modeling and Graphical Representations
 in Proceedings of the International Joint Conference on Neural Networks (IJCNN 2000), IEEE
, 2000
"... Category theory can be applied to mathematically model the semantics of cognitive neural systems. Here, we employ colimits, functors and natural transformations to model the implementation of concept hierarchies in neural networks equipped with multiple sensors. 1 Introduction In this paper, we des ..."
Abstract

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Category theory can be applied to mathematically model the semantics of cognitive neural systems. Here, we employ colimits, functors and natural transformations to model the implementation of concept hierarchies in neural networks equipped with multiple sensors. 1 Introduction In this paper, we describe a mathematical scheme for the analysis and design of cognitive neural network architectures based upon functors and natural transformations, the structural mappings of category theory. In a previous paper[3], we described a mathematical scheme for representing the hierarchical structure of subconceptconcept relationships based upon colimits, a categorical construction for objects which represent entire diagrams, or structural graphs, of related objects. Functors map the concept colimits into a category of neural components; natural transformations between the functors unify singlesensor concept representations in a fused, multimode neural network architecture. This kind of mathemati...
ONTOLOGIES AND WORLDS IN CATEGORY THEORY: IMPLICATIONS FOR NEURAL SYSTEMS
"... ABSTRACT. We propose category theory, the mathematical theory of structure, as a vehicle for defining ontologies in an unambiguous language with analytical and constructive features. Specifically, we apply categorical logic and model theory, based upon viewing an ontology as a subcategory of a cate ..."
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ABSTRACT. We propose category theory, the mathematical theory of structure, as a vehicle for defining ontologies in an unambiguous language with analytical and constructive features. Specifically, we apply categorical logic and model theory, based upon viewing an ontology as a subcategory of a category of theories expressed in a formal logic. In addition to providing mathematical rigor, this approach has several advantages. It allows the incremental analysis of ontologies by basing them in an interconnected hierarchy of theories, with an operation on the hierarchy that expresses the formation of complex theories from simple theories that express first principles. Another operation forms abstractions expressing the shared concepts in an array of theories. The use of categorical model theory makes possible the incremental analysis of possible worlds, or instances, for the theories, and the mapping of instances of a theory to instances of its more abstract parts. We describe the theoretical approach by applying it to the semantics of neural networks.