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Consider the case of many sensors, each returning very different types ofdata (e.g., a camera returning images, a thermometer returning probabilitydistributions, a newspaper returning articles, a traffic counter returningnumbers). Additionally we have a set of questions, or variables, that we wishto use these sensors to inform (e.g., temperature, location, crowd size,topic). Rather than using one sensor to inform each variable we wish tointegrate these sources of data to get more robust and complete information.The problem, of course, is how to inform a variable, e.g., crowd size, using anumber, a newspaper article, and an image. How do we integrate these verydifferent types of information? Michael Robinson proposes that sheaf theory isthe canonical answer. Moreover, one of the axioms in Robinson's paper whichmakes sheaf theory work for data integration is that all data sources have thestructure of a vector space. Therefore, the motivating question for everythingin this report is "How do we interpret arbitrary sensor output as a vectorspace with the intent to integrate?"

A Category Theoretical Investigation of the Type Hierarchy for Heterogeneous Sensor Integration