Explicit model of the $E_{n}$-operad in simplicial sets The space of $k$ little $n$-disks, denoted $E_{n}(k)$, is usually constructed in the category of topological spaces as the space of $k$-tuples $(c_{d_{1},p_{1}},\dots, c_{d_{k},p_{k}})$ of disjoint $n$-disk embeddings. Explicitly, each $c_{d_{i},p_{i}}$ is a map
$$
c_{p_{i},d_{i}}:\mathbb{D}^{n}\to \mathbb{D}^{n},\ \ \ x\mapsto d_{i}x+p_{i},
$$
and disjointness means that for each $i,j\in\{1,\dots, k\}$,
$$
|p_{i}-p_{j}|\leq d_{i}+d_{j}. 
$$
By composing embeddings, these spaces can be assembled into the little $n$-disks operad $E_{n}$. 
I would like to understand how to explicitly define these operads in purely combinatorial terms as operads in the category of simplicial sets. Ideally I would like to express each $E_{n}(k)$ as a simplicial mapping space, which an explicit model of disjoint little $n$-disk embeddings in terms of simplicial sets. 
The central hurdle seems to be the following question:
How can we specify that two little $n$-disk embeddings are disjoint without reference to a metric? 
 A: This isn't exactly what you are asking about, but let me mention that there are nice simplicial models of the $E_n$ operad. One of them is the Barratt–Eccles operad, or rather its filtration components. It is defined as follows.
Let $\newcommand{\E}{\mathscr{E}}\E(r)_n = (\Sigma_r)^n$ be the set given by the $n$th power of the symmetric group on $r$ elements. Define simplicial maps by (where the hat means that the variable is omitted):
\begin{align}
d_i : \E(r)_k & \to \E(r)_{k-1} \\
(\sigma_1, \dots, \sigma_k) & \mapsto (\sigma_1, \dots, \widehat{\sigma}_i, \dots, \sigma_k) \\
s_j : \E(r)_k &\to \E(r)_{k+1} \\
(\sigma_1, \dots, \sigma_k) & \mapsto (\sigma_1, \dots, \sigma_i, \sigma_i, \dots, \sigma_k)
\end{align}
It is not hard to check that each $\E(r)_\bullet$ is a simplicial set. There is an action of $\Sigma_r$ on $\E(r)$ by the diagonal. (In fact, $\E(r)$ is the total space of the universal principal $\Sigma_r$-bundle, and $\E(r)/\Sigma_r = B\Sigma_r$.) Finally, one can define operad maps:
\begin{align}
\circ_i : \E(r)_k \times \E(s)_k & \to \E(r+s-1)_k \\
(\sigma_1, \dots, \sigma_k), (\tau_1, \dots, \tau_k) & \mapsto (\sigma_1 \circ_i \tau_1, \dots, \sigma_k \circ_i \tau_1)
\end{align}
where, given $\sigma \in \Sigma_r$ and $\tau \in \Sigma_s$, the permutation $\sigma \circ_i \tau \in \Sigma_{r+s-1}$ is defined in the obvious way (act by $\sigma$ on $\{1, \dots, i-1,i+s,\dots,r+s-1\}$ and by $\tau$ on $\{i, \dots, i+s-1\}$).
The operad $\E$ is an $E_\infty$-operad. Indeed, it's easy to see that $\E(r)$ is contractible. There is a contracting homotopy $H(\sigma_1, \dots, \sigma_k) = (1, \sigma_1, \dots, \sigma_k)$.
Moreover, there is a filtration by suboperads $F_1\E \subset F_2\E \subset \dots \subset \E$ such that each $F_n\E$ is an $E_n$-operad. These operads are defined as follows.Let $\sigma = (\sigma_1, \dots, \sigma_k) \in \E(r)_k$ and $i < j$. Define the sequence $\sigma_{ij} = ((\sigma_1)_{ij}, \dots, (\sigma_k)_{ij}) \in (\Sigma_2)^k$, where $\tau_{ij} = 1$ if $\tau(i) < \tau(j)$ and $-1$ otherwise. Then
$$F_n\E(r)_k = \{ \sigma \in \E(r)_k \mid \forall i, j,\, \sigma_{ij} \text{ has at most } n-1 \text{ variations} \}.$$
One can then check that $F_n\E(r)$ is a simplicial subset of $\E(r)$, and all together they define a suboperad $F_n\E \subset \E$. This operad is a completely combinatorial $E_n$-operad.
For references:


*

*M. G. Barratt and Peter J. Eccles. "$\Gamma^{+}$-structures. I. A free group functor for stable homotopy theory" (1974). Topology, 13, pp. 25--45.

*Clemens Berger. "Opérades cellulaires et espaces de lacets itérés" (1996). Annales de l'Institut Fourier, Tome 46 no. 4, pp. 1125–1157. doi:10.5802/aif.1543. numdam:AIF_1996__46_4_1125_0

*Clemens Berger and Benoit Fresse. "Combinatorial operad actions on cochains" (2004). Mathematical Proceedings of the Cambridge Philosophical Society, 137(1), pp. 135–174. doi:10.1017/S0305004103007138
