simplicial complex of two covers

Question

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.

Answer 1

Given a single cover $\mathcal{W}=\{W_i\}_{i\in I}$ of a space $X$ by nonempty sets, the nerve complex $\mathcal{N}(\mathcal{W})$ has $I$ as its vertex set, and it has $[i_0,\ldots,i_k]$ as a $k$-simplex if $\cap_{j=0}^k W_{i_j}\neq\emptyset$.

To see if I have understand your definitions correctly, is it correct to say that your complex in (1) is the nerve complex $\mathcal{N}(\mathcal{U}\cap\mathcal{V})$, where $\mathcal{U}\cap\mathcal{V}$ is the cover of $X$ given by $\mathcal{U}\cap\mathcal{V}=\{U_\alpha\cap V_j~|~U_\alpha\in\mathcal{U}\text{ and }V_j\in\mathcal{V}\}$? [For this nerve complex, if $U_\alpha\cap V_j=\emptyset$, then it would not be included as a vertex.]

Also, is it correct to say that your complex in (2) is the nerve complex $\mathcal{N}(\mathcal{U}\cup\mathcal{V})$, where $\mathcal{U}\cup\mathcal{V}$ is the cover of $X$ given by $\mathcal{U}\cup\mathcal{V}=\{U_a,U_b,\ldots\}\cup\{V_1,V_2,\ldots\}$?