In homotopy theory there are lots of nice constructions that seem designed to have some effect on the homotopy of a space, i.e. completing, localizing, and taking various homotopy (co)limits. It seems like some of these (for instance Quillen's + construction) have moderately visual interpretations by attaching cells and things like that. In general, does anyone out there have interest in constructing such things geometrically, i.e. drawing a picture of what a sphere looks like after you, I don't know, complete it at a prime, or something like that? Are there sensible ways to visually represent mod p Moore spaces? Do such constructions usually cause things to have to leap into higher, non-visual, dimensions or something? Or is it simply that this kind of thing doesn't really serve any purpose, and we can just prove that it works and that it exists? In general it also seems very interesting to me to know what a sphere localized at a given homology theory might look like. 



Geometric Interpretations of Homotopy Theoretical Constructions
Though it is often satisfying and useful to have an appropriate way to visualize a construction, in fact the real innovation in homotopy theory that lead to the constructions you mention, among many others, was the realization that you need not limit your attention to geometric, visualizable operations and spaces but can be content knowing only certain relevant properties. 


This realization led to what Mike Hopkins (and perhaps others before him) called *designer homotopy theory*. The simplest example of a designer homotopy type is the Eilenberg-MacLane spaces---once you convince yourself they exist, you don't concern yourself with what they look like (a horrendous infinite CW complex, perhaps), but use only their characteristic relationship to homotopy groups. Needless to say this particular example led to spectacular advances in the hands of Serre and others.


