Grasping a Concept as an Image or as a Word – A Categorical Formulation of Visual and Verbal Thinking Processes

Abstract

Aim: Categorical formulations for thought processes and communications are provided. Study Design: Category theory and Sheaf theory. Place and Duration of Study: June 2010 to April 2013 in the Department of Mathematics, California Polytechnic State University and the Institute of Economic Research, Kyoto University. Methodology: In order to capture the changing state of an individual over a time period, we introduce a state controlling "variable" in terms of objects (called generalized time) in a site (i.e., a category with a Grothendieck topology). We also capture communications (information flows) as observation morphisms between individuals. The interplay among the change of the state of an individual, the communication between individuals, and the thought processes of those individuals is unified as the commutativity of the fundamental diagram in Section 3 of [1]. Results: We provide formulations for communication diagrams (information flows) between the language dependency type and the image dependency type which are denoted as Type (L) and Type (I), respectively. Conclusion:We have used categorical notions to formulate to express Type (I) and Type (L) of thought patterns. Then we have formulated the information flows in terms of the induced morphisms between Type (I) and Type (L).