Return to Question

-vertex $v \in X_0$ lies in exactly one open set $U_v$
-every edge $e_{vw}$ with vertices $v$ and $w$ lies in $U_v \cup U_w$ for unique pair of vertices $v,w \in X_0$ and the intersection $e_{vw} \cap U_v \cap U_w$ is isomorphic to an open interval

-vertex $v \in X_0$ lies in exactly one open set $U_v$
-every edge $e_{vw}$ with vertices $v$ and $w$ (note: there might be several edges with identical vertices) lies in $U_v \cup U_w$ for unique pair of vertices $v,w \in X_0$ and the intersection $e_{vw} \cap U_v \cap U_w$ is isomorphic to an open interval

My question is if it's also possible to write down the classifying map purely combinatorically /simplicially: Assume that our base space $X$ is
a Delta set, and in following we work with Delta set model of $EG$ and $BG$ from Hatcher book. Hatcher's bar model of $BG$ is a Delta set with $n$-simplices $[g_1 | g_2 |... |g_n]$ with boundaries $[g_1 | ... | g_i g_{i+1} | ... |g_n]$.

-vertex $v \in X_0$ lies in exactly one open set $U_v$
-every edge $e_{vw}$ with vertices $v$ and $w$ lies in $U_v \cup U_w$ for unique pair of vertices $v,w \in X_0$ and the intersection $e_{vw} \cap U_v \cap U_w is isomorphic to an open interval

My question is if it's also possible to write down the classifying map purely combinatorically or simplicially: Assume that our base space $X$ is
a Delta set, and in following we work with Delta set model of $EG$ and $BG$ from Hatcher book. Hatcher's bar model of $BG$ is a Delta set with $n$-simplices $[g_1 | g_2 |... |g_n]$ with boundaries $[g_1 | ... | g_i g_{i+1} | ... |g_n]$.

-vertex $v \in X_0$ lies in exactly one open set $U_v$
-every edge $e_{vw}$ with vertices $v$ and $w$ lies in $U_v \cup U_w$ for unique pair of vertices $v,w \in X_0$ and the intersection $e_{vw} \cap U_v \cap U_w$ is isomorphic to an open interval

This covering family $(U_v)_{v \in X_0}$ would give us a cocycle $(g_vw: U_v \cap U_w \to G)_{v,w \in X_0}$ and we could try to construct from this cocycle combinatorically the classifying map $f_p: X \to BG$ which maps each vertex to the unique vertex of $BG$, and every edge/ $1$-simplex of $e_{vw} $ to the $1$-simplex of $BG$ determined by $g_{vw} \in G$ from the cocycle. Note that the $1$-simplices of $BG$ it Hatcher's bar construction correspond to elements of $G$.

This covering family $(U_v)_{v \in X_0}$ would give us a cocycle $(g_{vw}: U_v \cap U_w \to G)_{v,w \in X_0}$ and we could try to construct from this cocycle combinatorically the classifying map $f_p: X \to BG$ which maps each vertex to the unique vertex of $BG$, and every edge/ $1$-simplex of $e_{vw} $ to the $1$-simplex of $BG$ determined by $g_{vw} \in G$ from the cocycle. Note that the $1$-simplices of $BG$ it Hatcher's bar construction correspond to elements of $G$.