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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 𝐏𝐚𝐫𝐚(𝐄𝐮𝐜)→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 𝐋𝐞𝐚𝐫𝐧≅𝐏𝐚𝐫𝐚(𝐒𝐋𝐞𝐧𝐬), where 𝐒𝐋𝐞𝐧𝐬 is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that 𝐒𝐋𝐞𝐧𝐬 is a full subcategory of 𝐏𝐨𝐥𝐲, the category of polynomial functors in one variable, via the functor A↦ Ay^A. Using the fact that (𝐏𝐨𝐥𝐲,⊗) is monoidal closed, we show that a map A→ B in 𝐏𝐚𝐫𝐚(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-𝐂𝐨𝐚𝐥𝐠 of dynamical systems on any p∈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.
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