How to relate and generalize nerve, bar-cobar, necklaces, templicial and simplicial objects

It is known that the category of comonoids in $Sets$ is equivalent to the category $Sets$ itself. This can be rephrased as follows: the inclusion of the initial operad, whose only operation is the identity, into the operad $uAs$ of monoids induces equivalence of corresponding categories of coalgebras.

We will describe a new zig-zag $sO{p}_{uAs//uAs}(1)\to sO{p}_{uAs//uAs}\stackrel{}{\leftarrow }T{w}^{\prime \prime }(uAs)$ of inclusions of operads that induces equivalence of corresponding categories of coalgebras, where on the left side the coalgebras are simplicial sets, and on the right side the coalgebras are non-symmetric cooperads. The functor from one side to the other has already appeared in [1, Example 3.6.5]. In general this equivalence breaks in enriched setting, however the right side is still quite interesting, being related to the necklace category of Dugger and Spivak and to templicial objects.

The operad $uAs$ in the above zig-zag can be replaced by any other operad $P$. The constructions $sO{p}_{P//P}$ and $T{w}^{\prime \prime }(P)$ are both reasonable generalizations of the notion of twisted arrow category of a category to operads, and have appeared respectively in [2, 3] and in [4]. As a particular case we recover the category of dendroidal necklaces and some of the key categories from [5].

Additionally, we show that both constructions $sO{p}_{P//P}$ and $T{w}^{\prime \prime }(P)$ fit into a larger picture that also involves the root functor of [6], which is a generalization of the last vertex map to dendroidal setting.