  Without the axiom of choice, the free exact completion of the category of
sets (i.e. the category of setoids) may not be complete or cocomplete. We will
show that nevertheless, it can be enhanced to a derivator: the formal structure
of categories of diagrams related by Kan extension functors. Moreover, this
derivator is the free cocompletion of a point in a class of "1-truncated
derivators" (which behave like a 1-category rather than a higher category).
  In classical mathematics, the free cocompletion of a point relative to all
derivators is the homotopy theory of spaces. Thus, if there is a homotopy
theory that can be shown to have this universal property constructively, its
1-truncation must contain not only sets, but also setoids. This suggests that
either setoids are an unavoidable aspect of constructive homotopy theory, or
more radical modifications to the notion of homotopy theory are needed.
