  Consider the case of many sensors, each returning very different types of
data (e.g., a camera returning images, a thermometer returning probability
distributions, a newspaper returning articles, a traffic counter returning
numbers). Additionally we have a set of questions, or variables, that we wish
to use these sensors to inform (e.g., temperature, location, crowd size,
topic). Rather than using one sensor to inform each variable we wish to
integrate 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 a
number, a newspaper article, and an image. How do we integrate these very
different types of information? Michael Robinson proposes that sheaf theory is
the canonical answer. Moreover, one of the axioms in Robinson's paper which
makes sheaf theory work for data integration is that all data sources have the
structure of a vector space. Therefore, the motivating question for everything
in this report is "How do we interpret arbitrary sensor output as a vector
space with the intent to integrate?"
