The Whitehead tower of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice spaces) as the connected component and the universal cover.

Can the remaining spaces be constructed functorially?

For the dual situation the answer is yes. I.e. for the Postnikov tower where we have a tower of spaces where the bottom homotopy groups are intact, but where we have killed off all the higher homotopy groups does have a functorial construction (again for nice spaces). The construction I know passes through simplicial sets. I'm wondering if something similar exists for the Whitehead tower?