Example [lcc-0017]

Let \mathcal {C} be symmetric monoidal. Consider the monoidal category where

1. Objects are lists X_n, n\geq 0 of objects from \mathcal {C}
2. A morphism is a sequence of objects S_n \in \mathcal {C}, and a family of morphisms X_n \otimes S_{n-1} \to Y_n \otimes S_n (with S_{-1} = I), up to the obvious sliding equivalence
3. Tensoring is given index-wise

This admits a feedback structure given by

1. F(X_\bullet )_n = X_{n-1}
2. Observe that a map X \otimes TS \to Y \otimes S amounts to a family of objects S'_n and a sequence of maps X_n \otimes S_{n-1} \otimes S'_{n-1} \to Y_n \otimes S_n \otimes S'_n - tracing is simply given by regarding this map as a map X \to Y with state space S \otimes S'.

This category seems to have been rediscovered many times, see eg. Monoidal Streams for Dataflow Programming, Definition 3.4