I am trying to understand what delta complexes are: Is the below definition correct?
$X\text{ is a }\Delta\text{-complex if }X=\bigcup_{i=0}^\infty X^i\text{ where }X^i=\bigcup_{\alpha\in A_i}\Delta_\alpha\text{ where }\Delta_\alpha\approx\Delta^i \text{such that }\forall\Delta_\alpha\in A^i,\Delta_\beta\in A^j,~\exists\Delta_{\gamma_1}\in X^{k_1},\ldots\Delta_{\gamma_n}\in X^{k_n}:~\Delta_\alpha\cap\Delta_\beta=\Delta_{\gamma_1}\cup\ldots\cup\Delta_{\gamma_n} $
Here $\Delta^i$ is the standard i - simplex. In the simplicial complex the above definition works with $n = 1$.
I don't get how this is related to this formal definition:
A $\Delta$ complex is
- For $ n \geq 0$, a sequence of sets $\{X_n\}_{n=0}^{\infty}$
- For each $n>0$ each $ 0\leq i \leq n$, a map $d_i: X_n \rightarrow X_{n-1}$ such that
- For each $0\leq j < i \leq n$ with $n > 1$, $d_i \circ d_j = d_j \circ d_{i+1}$
If someone could explain what is the correspondence between the two definitions or where the first definition fails with an example would be very helpful for me.