Directed Binary Categorical Splices, Duality Principle, and Categorical Model of Neural Networks

The theory of categorical systems developed by the author allows to naturally model traditional artificial neural networks of arbitrary topology, networks of living neurons, which in addition to spike communication have several dozen other types of cellular communication, as well as network structures similar to higher categories. The mathematical apparatus of categorical systems is the theory of categorical splices, this work is devoted to the following issues of this theory. Directed binary categorical splices are introduced and studied, which are a generalization of ordinary categories, in the theory of which, as is known, the concept of duality of categories and the principle of duality based on duality play a large role. In the theory of categorical splices, in addition to duality similar to duality generated by replacing the direction of category arrows, there is a new type of duality associated with replacing the names of arrows with the names of convolutions generalizing the usual operation of composition. The construction of dual in both senses categorical splices of the studied type is carried out, theorems corresponding to the principles of duality for the specified two types of duality are proved. The theorems are given within the framework of the theory of proofs, for the special case of ordinary categories giving new proofs of the principle of duality. Generalization of the approach to convolutional analogs of multicategories find applications in neural networks, in particular, for the well-known formulas of S. Osovsky in the method of backpropagation of errors.