Sheaving - A universal construction for semantic compositionality

Abstract

Semantic compositionality-the way that meanings of complex entities obtain from meanings of constituent entities and their structural relations-is supposed to explain certain concomitant cognitive capacities, such as systematicity. Yet, cognitive scientists are divided on mechanisms for compositionality: E.g. a language of thought on one side versus a geometry of thought on the other. Category theory is a field of (meta)mathematics invented to bridge formal divides. We focus on sheaving-a construction at the nexus of algebra and geometry/topology, alluding to an integrative view, to sketch out a category theory perspective on the semantics of compositionality. Sheaving is a universal construction for making inferences from local knowledge, where meaning is grounded by the underlying topological space. Three examples illustrate how topology conveys meaning, in terms of the inclusion relations between the open sets that constitute the space, though the topology is not regarded as the only source of semantic information. In this sense, category (sheaf ) theory provides a general framework for semantic compositionality.