  In "Backprop as functor", the authors show that the fundamental elements of
deep learning -- gradient descent and backpropagation -- can be conceptualized
as a strong monoidal functor $\mathbf{Para}(\mathbf{Euc})\to\mathbf{Learn}$
from the category of parameterized Euclidean spaces to that of learners, a
category developed explicitly to capture parameter update and backpropagation.
It was soon realized that there is an isomorphism
$\mathbf{Learn}\cong\mathbf{Para}(\mathbf{SLens})$, where $\mathbf{SLens}$ is
the symmetric monoidal category of simple lenses as used in functional
programming.
  In this note, we observe that $\mathbf{SLens}$ is a full subcategory of
$\mathbf{Poly}$, the category of polynomial functors in one variable, via the
functor $A\mapsto Ay^A$. Using the fact that $(\mathbf{Poly},\otimes)$ is
monoidal closed, we show that a map $A\to B$ in $\mathbf{Para}(\mathbf{SLens})$
has a natural interpretation in terms of dynamical systems (more precisely,
generalized Moore machines) whose interface is the internal-hom type
$[Ay^A,By^B]$.
  Finally, we review the fact that the category $p\text{-}\mathbf{Coalg}$ of
dynamical systems on any $p\in\mathbf{Poly}$ forms a topos, and consider the
logical propositions that can be stated in its internal language. We give
gradient descent as an example, and we conclude by discussing some directions
for future work.
