I have recently been tasked with reading up on simplicial sets and I'm trying to define some concrete maps in the category to try to get my head around how things relate to stuff I already know. In particular, I have been trying to translate the usual multiplication $\mu:\mathbb{S}^1\times \mathbb{S}^1 \to \mathbb{S}^1$ into a map of simplicial sets. Here, I would like to have $\mathbb{S}^1 \in \mathsf{sSet}$ defined by a single vertex $\bullet$ and a single $1$-cell $\circlearrowleft$. I proceed like this:
On vertices, there is only one choice for $\mu_0 : \mathbb{S}^1_0 \times \mathbb{S}^1_0 \to \mathbb{S}^1_0$ -- everything gets mapped to $\bullet$.
On $1$-cells, I now need to provide a map: $$\mu_1 : \{\bullet,\circlearrowleft\} \times \{\bullet,\circlearrowleft\} \to \{\bullet,\circlearrowleft\}$$ Since this is to mimic the usual $\mathbb{S}^1$ multiplication (or any multiplication for that matter), I want to define \begin{align*} \mu_1(\bullet,x) &= x\\ \mu_1(\circlearrowleft,\bullet) &= \circlearrowleft \end{align*} However, there doesn't seem to be much I can do at this point. Both $\mu_1(\circlearrowleft,\circlearrowleft) = \bullet$ and $\mu_1(\circlearrowleft,\circlearrowleft) = \circlearrowleft$ appears to require non-degenerate 2-cells in $\mathbb{S}^1$.
So, to my question(s): Is there a concrete way to define this map? Is my choice of definition of $\mathbb{S}^1$ appropriate?
You will likely need to replace the target $\mathbb{S}^1$ by a different simplicial set. The simplicial set you are using does model the topological circle, but it does not have enough simplices to model all maps into the circle. In the jargon of homotopy theory, it is not fibrant.
To better illustrate this non-fibrancy, try to write down a map that models the double cover $\mathbb{S}^1 \to \mathbb{S}^1$ using your model for the circle. Generally, a map $\mathbb{S}^1 \to \mathbb{S}^1$ is determined by what happens to the edge in the domain. For the double cover, you'll want to "wrap the edge around itself twice", but you find that the codomain simply does not have enough simplices to express this. To fix this, you may want to look into "edgewise subdivision" and a functor by the name $\mathrm{Ex}$.