This report presents a mathematical model of the semantics, or meaning, of the connec-tionist structure and stimulus activity of a neural network, whether artificial or biologi-cal. The mathematical model associates concepts about sensed objects with the neuron-like nodes in a neural network and composable concept relationships with the connec-tion pathways in the network. Category-theoretic constructs, specifically colimits, limits, and functors, organize the concept structure and map it to a formal neural network in a structure-preserving manner. Starting with a simple example of a neural vision system, we show that this mathematical model of neural network structure and activity can be used to derive connectionist architectures that work as intended. We also claim an additional advantage of this approach: A properly-functioning connectionist architecture has an ac-companying concept representation and this representation is both local and distributed. These properties are derived from the category-theoretic formalism described here
