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.

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 complex given by pairs $(\alpha,\pi)$ such that $U_\alpha\cap V_\pi\neq\emptyset$. It is not even obvious to me that this would give a simplical complex however - what would it give? Whatever it is, it 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 "simplex" $\big\{\{a,b,c\},\{1,2\}\big\}$ for the second.

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