The derivative of a combinatorial species $S: core(FinSet) \to core(FinSet)$ is given by $S^\prime [N] = S[N\sqcup 1]$. Intuitively, an $S^\prime$-structure is built by introducing a "hole" to the set of building blocks and then building an $S$-structure incorporating the hole. If one builds a cyclic order with one vertex being a "hole", this can be identified with a linear order on the rest of the vertices. One says that the species of linear orders is the derivative of the species of cyclic orders.
The augmented simplex category admits strikingly natural characterizations. For example, the augmented simplex category is the free (monoid)-pointed monoidal category: it co-classifies monoids in strict monoidal categories via monoidal functors.
The augmented simplex 2-category is freely generated by the walking span under the operations of taking pushouts and adjoints.
These characterizations are quite useful in some applications. It would be nice to have a one-liner characterizing the Connes cycle category $\Lambda$, too. The self-duality of $\Lambda$ gives me the feeling that it might admit an even more natural universal characterization than its would-be "derivative" $\Delta_{\alpha}$. I am sweeping some details under the rug concerning the difference between $\Lambda$ and "$\Lambda_{\alpha}$", if an augmented cycle category has even been described. I think I have painted enough of a picture.
Is this idea buried in the literature somewhere? I would love a sharper compression of $\Lambda$. Thank you!
