Pursuing Classifying Topoi

(This post will outline the haze of thoughts and visions as we pursue Classifying Topoi. All facts stated are true modulo small lies . Allegories to be taken with a liberal pinch of bath salts. It will be wordy to boot, TeXing reserved for better days and better keyboards.)

1. We want to “classify” certain kinds of “structures” over a space/category/topos. Classifying means the following:
   • For every kind of structure we are interested in, there exists a special space, called (and denoted here) the Classifying Spåce for that structure.

   • Q: What kind of structures can be classified?
     A: Something something representable.

   • Probing the Classifying Space with a map from a Test Space corresponds to/yields a structure over the Test Space.
   • examples of structures: cohomology operations, nth cohomology, principal G-bundles, vector bundles etc. Each of these has a glorified Classifying Space.