Symmetric coloured operads as monoids?

Question

A symmetric operad in a symmetric monoidal category $\mathcal{C}$ can be defined as a monoid in the category of presheaves $[\mathbb{P}^{op},\mathcal{C}]$ with the substitution product $\circ$. Here $\mathbb{P}$ denotes the permutation category. Does an analogous slick definition exist for symmetric $I$-coloured operad? References welcome.

Answer 1

I don't know of a nice presentation along those lines, but here's my attempt to work out the construction based on the data in Yau's book, Colored Operads. Feel free to replace the braid groups with symmetric groups.

Let $C$ be a set of colors and consider the braid groupoid $\beta := \coprod_{n \in \mathbb{N}} B_n.$ The braid group on $n$ strands induces an obvious action on $n$ (possibly non-distinct) letters, and we denote such a group as $\beta_{\underline{c}},$ where $\underline{c} = (c_1, \dots, c_n).$ We define the $C-$colored braid groupoid as the action groupoid $\beta_C := \coprod_{n \in \mathbb{N}} C^n//\beta$ (using the obvious componentwise action). Recall that $C^0 = \varnothing,$ which contains only the empty color. We can equip this with the monoidal product of concatenating sequences of colors.

We define the category of $C-$colored species of braidings in $Mod_k$, $\mathrm{Species}^\beta_C(\mathrm{Mod}_k) := \prod_{c \in C} \mathrm{Fun}(\beta_C^{op}, \mathrm{Mod}_k),$ and denote the components of a $C-$colored species of braidings as $F_{c_0}^{c_1, \dots, c_n}= F_{c_0}^{\underline{c}}.$ First we define the componentwise Day convolution product \begin{align*} (F_{c_0} \otimes G_{c_0'})^= &:= \int^{\underline{c},\underline{c}' \in \beta_C} k \beta_C(=, \underline{c}\underline{c}') \otimes F_{c_0}^{\underline{c}} \otimes G_{c_0'}^{\underline{c}'}\\ \end{align*} with unit $\prod_{c_0 \in C} k \delta_{-,c_0}\beta_C(=,\varnothing).$ Next we define the maps $G_{\otimes -} : \beta_C \to \mathrm{Fun}(\beta_C^{op}, \mathrm{Mod}_k): \underline{c} \mapsto G_{\otimes \underline{c}} = G_{c_1} \otimes \dots \otimes G_{c_n}.$ Now we can equip $\mathrm{Species}^\beta_C(\mathrm{Mod}_k)$ with the substitution/plethysm monoidal product \begin{align*} F \circ G &:= F \otimes_{\beta_C} G_{\otimes}\\ &= \int^{\underline{c} \in \beta_C} F_-^{\underline{c}} \otimes_{\beta_{\underline{c}}} G_{\otimes \underline{c}}^= \end{align*} which has unit $k\delta_-^=,$ where $\delta_{c_0}^=$ is the indicator function for $c_0.$

Define the category of $k-$linear colored braided operads as monoids in $k-$linear colored species of braidings, $\mathrm{Operad}^{\beta}_C(\mathrm{Mod}_k) := \mathrm{Mon}(\mathrm{Species}^\beta_C(\mathrm{Mod}_k), \circ, k\delta).$