I recently came across an unusual definition of fibration between simplicial complexes, and I need help to understand it (or possibly find a reference that can help me).
Before giving the definition, I shall recall some of the basic concepts involved:
- Given a simplicial map $f:\Gamma\rightarrow\Sigma$, the image of $f$ is the subcomplex $\mbox{im}(f)$ of $\Sigma$ defined by $$ \mbox{im}(f):=\{f^\rightarrow \gamma\mid\gamma\in\Gamma\}, $$ where $f^\rightarrow\gamma:=\{f(v)\mid v\in\gamma\}$.
A simplicial map $f:\Gamma\rightarrow\Sigma$ is a simplicial quotient if $\mbox{im}(f)=\Sigma$.
Given an abstract simplicial complex $\Sigma$, the (open) star of a face $\sigma\in \Sigma$ is defined by $$\mbox{St}_\Sigma(\sigma):=\{\tau\in \Sigma\mid\sigma\subseteq\tau\}.$$ The closed star of $\sigma$ is defined by $$ \overline{\mbox{St}}_\Sigma(\sigma):=\{\tau\in\Sigma\mid\sigma\cup\tau\in\Sigma\} $$ which is analogous to the definition of the neighborhood of a vertex in a graph.
Every simplicial map $f:\Gamma\rightarrow\Sigma$ induces a local simplicial map: given a face $\gamma\in\Gamma$, define $$ f_\gamma:\overline{\mbox{St}}_\Gamma(\gamma)\rightarrow\overline{\mbox{St}}_\Sigma(f^\rightarrow\gamma)::\tau\mapsto \bigcup_{v\in \tau}f(v). $$
Definition A simplicial map $f:\Gamma\rightarrow \Sigma$ between abstract simplicial complexes is said to be a local quotient if the maps $f_\gamma$ are all simplicial quotients. If, moreover, $f$ is also a simplicial quotient, it is called a fibration map.
Questions My questions are the following:
- Why is this object called a fibration? In topology, fibrations are maps satisfying the homotopy lifting property. Does this `simplicial fibration' verify any analogous property?
- Would the geometric realization of such a map be a fibration in a topological sense?
- Simplicial complexes do not have enough structure for the definition of homotopy groups. The usual notion of simplicial fibration is the one of Kan fibration which involves simplicial sets. Is there a connection between these two concepts?
- Where can I find a reference explaining this definition in detail?