I would like to compute the homology of certain low dimensional CW complexes and I am hoping to take advantage of software that handles simplicial sets as input. Thus, I would like to convert a CW complex into a homologically equivalent simplicial set.

In the case where the CW complex is regular, everything is easy. I just build the order complex of the face partial order on the cells of $\mathcal{X}$. What should I do when confronted with a non-regular CW complex?

More precisely, consider a (not necessarily regular) finite CW complex $\mathcal{X}$ and define the following combinatorial structure $(X,a)$. $X$ is a graded set $X = \sqcup_k X_k$ where $X_k$ is the set of all $k$-cells of $\mathcal{X}$. The maps $a_k: X_k \times X_{k-1} \to \mathbb{Z}$ are defined as follows: $a(\xi,\eta)$ is the degree of the attaching map from the boundary of $\xi$ onto $\eta$.

Question:

Is there a standard method for taking $(X,a)$ as input and producing a simplicial set whose homology (and more ideally, homotopy type!) coincides with that of $\mathcal{X}$?

If not, can we come up with such a method?