Descent of a function through a surjection

Given a surjection $p:X\to Y$ , a function $f:X\to Z$ descends through $p$ if there is a function $F:Y\to Z$ such that $f=F\circ p$ .

Criterion and uniqueness

Descent holds exactly when $p(x)=p(x')$ implies $f(x)=f(x')$ . In that case define $F(y)=f(x)$ for any $x$ over $y$ ; constancy on fibers makes the value independent of the choice, and surjectivity gives uniqueness.

For a smooth covering, a smooth descending function has a smooth descended representative by the local inverse charts. In the torus covering $p_A$ , the criterion becomes invariance under all deck translations $a$ with $Aa\in\mathbb Z^n$ . Invariance under an arbitrary smaller collection of translations need not suffice.