It's often a great technical convenience to know that you can "do homotopy theory" with simplicial sets. But if you really get down to it geometrically, it can be awkward.

In general, if I have a CW complex that I want to study simplicially, I have to build up a simplicial structure cell-by-cell, and I need to keep applying barycentric subdivision to simplicially approximate my attaching maps.

There are a few interesting spaces with economical simplicial structures. For example spheres, or $K(G,1)$'s (for which one can take the nerve of $G$ as a model), and suspensions thereof. I suppose Moore spaces aren't too bad either. One trick suggested by this is to take classifying spaces of other categories, but when it comes down to it there really aren't that many categories whose classifying spaces I know.

As far as I can tell, that might be about it! For example, even to write down a simplicial structure on $\mathbb C \mathbb P^\infty$ that I could actually work with, I'm not sure what I'd do. I could imagine that if there were an "economical" model of $S^1$ which was a simplicial topological group, then one could perform a bar construction to get $\mathbb C \mathbb P^\infty$, but I don't even really know a good model for $S^1$ as a simplicial group, short of geometrically realizing and taking the singular complex, which I think is very far from economical. It's not even clear to me whether $Ex^\infty S^1$ is a simplicial topological group!

**Questions:**

1. Is there a general method to write down a simplicial model of a homotopy type which is more economical than just subdividing to simplicially aproximate each attaching map?

2. Are there at least some examples of homotopy types which are not suspensions of $K(G,1)$'s or Moore spaces which admit "simple" simplicial models? By "simple", I roughly mean that one might really think about the space "geometrically", rather than just on some formal level.

3. As a test case, what would be a good simplicial model for $\mathbb C \mathbb P^\infty$?

**EDIT:** I should clarify that when I talk about a "simplicial model" of a homotopy type, I'm asking for a simplical set whose geometric realization has that homotopy type. I'm not requiring a simplicial complex or anything.