A *quandle* is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold.

 Q1. a $\star$ a = a

 Q2. (a $\star$ b) $\bar\star$ b = (a $\bar\star$ b) $\star$ b = a 

 Q3. (a $\star$ b) $\star$ c = (a $\star$ c) $\star$ (b $\star$ c) 

When we drop out the first axiom we obtain a *rack*, by definition. Quandles generalize basic properties of the conjugation in a group (where $a \star b = b^{-1}ab$ and $a\ \bar\star \ b = bab^{−1}$), but they are also useful in knot theory.

Nevertheless, I'm interested mainly in the homotopy theory of these objects. In fact, following [this](https://arxiv.org/abs/math/0612459) article by Eisermann, we can define arrows

- $a \xrightarrow{b}c$  for each triple $a,b,c \in Q$ with $a \star b = c$. 

- $a' \xleftarrow{b'}c'$  for each triple $a',b',c' \in Q$ with $a'\ \bar\star \ b' = c'$. 

Then we have a notion of homotopy, build in the following way (see the article for details).

First define a *combinatorial path* between two elements $q,q'\in Q$ to be a sequence of arrows going in both ways, such that the first arrow is given by the action of an element of the quandle on $q$ and the last is given by the action of another one on $q'$.

> **Definition 1** Let $P(Q)$ be the category having as objects the elements $q\in Q$ and as morphisms from $q$ to $q'$ the set of combinatorial paths from $q$ to $q'$. Composition is given by juxtaposition:
$$(a_0 \to \cdots \to a_m) \circ (a_m \to \cdots \to a_n) = (a_0 \to \cdots \to a_m \to \cdots \to a_n).$$

Then we can construct an homotopy as in the following definition. 

> **Definition 2** Two combinatorial paths are *homotopic* if they can be transformed one into the other by a sequence of the following local moves and their inverses:

>(H1) $a\xrightarrow{a}a$ is replaced by $a$, or $a\xleftarrow{a}a$ is replaced by $a$.

>(H2) $a\xrightarrow{b}a \star b\xleftarrow{b}a$ is replaced by $a$, or $a\xleftarrow{b}a \ \bar\star \ b \xrightarrow{b}a$ is replaced by $a$.

> (H3) $a\xrightarrow{b}a \star b\xrightarrow{c}(a \star b) \star c$ is replaced by $a\xrightarrow{c} a \star c \xrightarrow{b\star c} (a \star  c) \star (b \star c) $

It seems to me that this data can be extended to a simplicial set, whose 0-simplices are elements of $Q$, 1-simplices are arrows between them and higher simplices witness these homotopical information.

My question is

> Does $P(Q)$ embed in a simplicial set which keeps track of these information? Is it possible that this simplicial set is actually an $\infty$-category having $P(Q)$ as homotopy category? Is there an analogous construction for racks?