Higher homotopical information in racks and quandles
$begingroup$
A quandle is defined to be a set $Q$ with two binary operations $star,barstarcolon Qtimes Qto Q$ for which the following axioms hold.
Q1. a $star$ a = a
Q2. (a $star$ b) $barstar$ b = (a $barstar$ 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 $a barstar b = bab^{−1}$), 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 $a' barstar b' = c'$.
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 $qin Q$ 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) $axrightarrow{a}a$ is replaced by $a$, or $axleftarrow{a}a$ is replaced by $a$.
(H2) $axrightarrow{b}a star bxleftarrow{b}a$ is replaced by $a$, or $axleftarrow{b}a barstar b xrightarrow{b}a$ is replaced by $a$.
(H3) $axrightarrow{b}a star bxrightarrow{c}(a star b) star c$ is replaced by $axrightarrow{c} a star c xrightarrow{bstar 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?
abstract-algebra simplicial-stuff higher-category-theory
asked Nov 24 '18 at 17:38
Nicola Di VittorioNicola Di Vittorio
165
$endgroup$
add a comment |
$begingroup$
A quandle is defined to be a set $Q$ with two binary operations $star,barstarcolon Qtimes Qto Q$ for which the following axioms hold.
Q1. a $star$ a = a
Q2. (a $star$ b) $barstar$ b = (a $barstar$ 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 $a barstar b = bab^{−1}$), 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 $a' barstar b' = c'$.
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 $qin Q$ 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) $axrightarrow{a}a$ is replaced by $a$, or $axleftarrow{a}a$ is replaced by $a$.
(H2) $axrightarrow{b}a star bxleftarrow{b}a$ is replaced by $a$, or $axleftarrow{b}a barstar b xrightarrow{b}a$ is replaced by $a$.
(H3) $axrightarrow{b}a star bxrightarrow{c}(a star b) star c$ is replaced by $axrightarrow{c} a star c xrightarrow{bstar 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?
abstract-algebra simplicial-stuff higher-category-theory
asked Nov 24 '18 at 17:38
Nicola Di VittorioNicola Di Vittorio
165
$endgroup$
add a comment |
$begingroup$
A quandle is defined to be a set $Q$ with two binary operations $star,barstarcolon Qtimes Qto Q$ for which the following axioms hold.
Q1. a $star$ a = a
Q2. (a $star$ b) $barstar$ b = (a $barstar$ 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 $a barstar b = bab^{−1}$), 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 $a' barstar b' = c'$.
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 $qin Q$ 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) $axrightarrow{a}a$ is replaced by $a$, or $axleftarrow{a}a$ is replaced by $a$.
(H2) $axrightarrow{b}a star bxleftarrow{b}a$ is replaced by $a$, or $axleftarrow{b}a barstar b xrightarrow{b}a$ is replaced by $a$.
(H3) $axrightarrow{b}a star bxrightarrow{c}(a star b) star c$ is replaced by $axrightarrow{c} a star c xrightarrow{bstar 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?
abstract-algebra simplicial-stuff higher-category-theory
asked Nov 24 '18 at 17:38
Nicola Di VittorioNicola Di Vittorio
165
$endgroup$
A quandle is defined to be a set $Q$ with two binary operations $star,barstarcolon Qtimes Qto Q$ for which the following axioms hold.
Q1. a $star$ a = a
Q2. (a $star$ b) $barstar$ b = (a $barstar$ 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 $a barstar b = bab^{−1}$), 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 $a' barstar b' = c'$.
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 $qin Q$ 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) $axrightarrow{a}a$ is replaced by $a$, or $axleftarrow{a}a$ is replaced by $a$.
(H2) $axrightarrow{b}a star bxleftarrow{b}a$ is replaced by $a$, or $axleftarrow{b}a barstar b xrightarrow{b}a$ is replaced by $a$.
(H3) $axrightarrow{b}a star bxrightarrow{c}(a star b) star c$ is replaced by $axrightarrow{c} a star c xrightarrow{bstar 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?
abstract-algebra simplicial-stuff higher-category-theory
abstract-algebra simplicial-stuff higher-category-theory
asked Nov 24 '18 at 17:38
Nicola Di VittorioNicola Di Vittorio
165
asked Nov 24 '18 at 17:38
Nicola Di VittorioNicola Di Vittorio
165
edited Nov 24 '18 at 21:45
Nicola Di Vittorio
asked Nov 24 '18 at 17:38
Nicola Di VittorioNicola Di Vittorio
165
asked Nov 24 '18 at 17:38
Nicola Di VittorioNicola Di Vittorio
165
asked Nov 24 '18 at 17:38
Nicola Di VittorioNicola Di Vittorio
165
165
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