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?

  • $\begingroup$ Have you seen Dwyer-Kan's hammock localization? Sec 2.1 here: web.math.rochester.edu/people/faculty/doug/otherpapers/… $\endgroup$ – Vidit Nanda Nov 24 '18 at 21:58
  • $\begingroup$ 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. $\endgroup$ – Nicola Di Vittorio Nov 24 '18 at 22:29

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