  We present our position on the elusive quest for a general-purpose framework
for specifying and studying deep learning architectures. Our opinion is that
the key attempts made so far lack a coherent bridge between specifying
constraints which models must satisfy and specifying their implementations.
Focusing on building a such a bridge, we propose to apply category theory --
precisely, the universal algebra of monads valued in a 2-category of parametric
maps -- as a single theory elegantly subsuming both of these flavours of neural
network design. To defend our position, we show how this theory recovers
constraints induced by geometric deep learning, as well as implementations of
many architectures drawn from the diverse landscape of neural networks, such as
RNNs. We also illustrate how the theory naturally encodes many standard
constructs in computer science and automata theory.
