# Higher homotopical information in racks and quandles

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 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, built 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?

• Have you seen Dwyer-Kan's hammock localization? Sec 2.1 here: web.math.rochester.edu/people/faculty/doug/otherpapers/… – Vidit Nanda Nov 24 '18 at 21:58
• Yes, I've read something about that (although not in full detail) and it came to my mind. Even if a suitable class of weak equivalences could be found, I think that keeping track of (H3) at the level of simplicially enriched categories might be not manageable. – Nicola Di Vittorio Nov 24 '18 at 22:29