Cyclic category

We know, that $\B\bZ$ is homotopy equivalent to $S^1$. Consider a question: what is the simplicial set $C_{\bullet}$ whose geometric realization is circle with the cell structure consisting of one 0-cell and one 1-cell (not up to homotopy)?

The 0-cell $*\in C_0$ generate only one element, still denoted by $*$ in each $C_n$. Suppose we add additional element $\tau$ to $C_1$. Then we get
$$\begin{align*} C_0 &= \{*\} \\ C_1 &= \{*, \tau\}\\ C_1 &= \{*, s_0\tau, s_1\t... ...= \{*, \hdots, s_{n-1}\hdots \Hat{s_i}, \hdots s_0\tau, \hdots\}\\ \end{align*}$$
The faces are obvious to find. In particular $d_0(\tau)=*=d_1(\tau)$. Then $\vert C_{\bullet}\vert$ is a circle with its simplest cell structure. We can identify

$$C_n = \{*, \hdots, s_{n-1}\hdots \Hat{s_i}, \hdots s_0\tau, \hdots\}$$

with the cyclic group $\bZ/ (n+1)\bZ =: C_n$ by sending $*$ to 0, and $s_{n-1}\hdots \Hat{s_i}, \hdots s_0\tau$ to $i+1$. Denote the generator of $C_n$ by $t_n$.

There exists a cyclic category $\Dl C$ whose objects are finite ordered sets $[n]=\{0, \hdots, n\}$, and morphism $\Mor([n], [m])$ are generated by $\dl_i$, $\sg_j$ as in simplicial category, and additional morphisms $\tau_n\:[n]\to [n]$ for all $n\geq 0$ satisfying the relations
$$\begin{align*} \tau_n^{n+1} &= \id_{[n]} \\ \tau_n\dl_i &= \dl_{i-1}\tau_{n-1},... ...au_{n+1}, \quad 1\leq j \leq n \\ \tau_n\sg_0 &= \sg_n\tau_{n+1}^2 \end{align*}$$
If in this presentation we omit the relation $\tau_n^{n+1}=\id_{[n]}$, then we get a different category, denoted $\Dl \bZ$.





Every morphism of $\Dl C$ can be written uniquely as $\phi\circ g$, where $\phi\in\Mor_{\Dl}([n], [m])$, $g\in C_n=\Mor_{\Dl C}([n], [n])$. As sets

$$\Hom_{\Dl C}([n], [m])\isom \Hom_{\Dl}([n], [m])\times C_n$$

The compositon of two morphisms $(g\circ\phi)$ and $(h\circ \psi)$ is in $\Dl C$, so there exist $\phi^*(h)\in C_n$ and $h_*(\phi)\in \Mor_{\Dl}([n], [m])$ such that the following diagram commutes.

$$\xymatrix{ [n] \ar[r]^{g} & [n] \ar[r]^{\phi} \ar[rd]^{\phi^... ...^h & [m] \ar[r]^{\psi} & [r] \\ && [n] \ar[ru]_{h_*(\phi)} &&}$$

Analogously, suppose we have two subgroups $A,B<G$ such that every element of $G$ can be written uniquely as $g=ab$, $a\in A$, $b\in B$. In this situation

$$gg'=aba'b'=a\underbrace{b^*(a')}_{\in A} \underbrace{a_*'(b)}_{\in B}b'$$

The relations satisfied by $\phi^*$ and $h_*$ are exactly the same as the relations satisfied by $b^*\: A\to A$ and $a_*\: B\to B$.

We can ask what kind of structure on the geometric realization of the underlying simplicial set $X_{\bullet}$, that is $\vert X_{\bullet}\vert$, does the cyclic structure give? The answer is a structure of $S^1$-space. An open question is can we discretize analogously $S^3=\SU(2)$?