  We construct a natural bijective correspondence between equivalence classes
of Pin$^-$ structures on a compact simplicial $n$-manifold $M^n$, possibly with
boundary, and $\mathbb{Z}/4$-valued 'quadratic functions' $Q$ defined on degree
$n-1$ relative $\mathbb{Z}/2$ cocycles, $Q \colon Z^{n-1}(M^n, \partial M^n ;
\mathbb{Z} /2) \to \mathbb{Z}/4$. The 'quadratic' property of $Q(p+q)$ and the
values $Q(dc)$ on coboundaries are expressed in terms of higher $\cup_i$
products of Steenrod. For $n = 2$ the results extend old results relating
Pin$^-$ structures on closed surfaces to quadratic refinements of the cup
product pairing on $H^1(M^n ; \mathbb{Z} /2)$. In the oriented case, that is,
for Spin manifolds, the results extend results of Kapustin, see
arXiv:1505.05856v2, and results in our previous paper on the Pontrjagin dual
4-dimensional Spin bordism, see arXiv:1803.08147. The extension of those
results to Pin$^-$ manifolds in this paper required a different approach,
involving some stable homotopy theory of Postnikov towers.
