Given a (Hausdorff) topological group , the Milnor construction of the universal principal bundle for (also known as Milnor’s join construction) constructs the join of infinitely many copies of , i.e., the colimit of joins
and canonically equips it with a continuous and free right action of that yields the structure of a CW-complex such that the action of permutes the cells. Consequently, the natural projection is a model for the universal bundle of locally trivial principal -bundles over paracompact Hausdorff spaces, or equivalently, of numerable principal -bundles over all Hausdorff topological spaces.
There is a generalisation of Milnor’s construction that works for topological groupoids, and reduces to the above case when the groupoid is , the delooping of the group .
John Milnor, Construction of Universal Bundles, I, Ann. of Math. 63:2 (1956) 272-284 jstor; Construction of Universal Bundles, II, Ann. of Math. 63:3 (1956) 430-436, jstor; reprinted in Collected Works of John Milnor, gBooks.
Wikipedia: Classifying Space.
Dale Husemöller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, The Milnor Construction: Homotopy Classification of Principal Bundles, doi, in: Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Physics, Vol. 726 (2008) 75-81.