Given two covers $\{U_a,U_b,\dots\}$ and $\{V_1,V_2,\dots\}$ of a space $X$, what is the appropriate idea of simplicial complex? As far as I see there are two ideas, and I was wondering where these were discussed and what their mapping properties are with respect to the complexes for the single covers. Apologies if this is well known, I am not an expert in this topic, and would grateful appreciate any help or references. 

1) The sub complex of the categorical product complex, given by vertices $(a,1)$ etc. and corresponding to the cover $U_a\cap V_1$. The sub complex comes about by deleting empty intersections, so $\{(a,1),(b,2)\}$ would be deleted if $U_a\cap V_1 \cap U_b\cap V_2=\emptyset$. 

2) Using multi-indices we have $U_\alpha=U_{\alpha_1}\cap \dots\cap U_{\alpha_k}$ and $V_\pi=U_{\pi_1}\cap \dots\cap U_{\pi_s}$. Now take the cover given by pairs $(\alpha,\pi)$ such that $U_\alpha\cap V_\pi\neq\emptyset$. This is smaller than the previous complex as, for example $\{(a,1),(b,2),(c,1)\}$ and $\{(a,1),(b,1),(c,2)\}$ for the first complex both correspond to the single vertex $\big\{\{a,b,c\},\{1,2\}\big\}$ for the second complex.

This work is motivated by considering simplicial complexes as data structures in computer science.