  We provide an explicit formula for localizing $A^1$-homotopy invariants of
topological Fukaya categories of marked surfaces. Following a proposal of
Kontsevich, this differential $\mathbb Z$-graded category is defined as global
sections of a constructible cosheaf of dg categories on any spine of the
surface. Our theorem utilizes this sheaf-theoretic description to reduce the
calculation of invariants to the local case when the surface is a
boundary-marked disk. At the heart of the proof lies a theory of localization
for topological Fukaya categories which is a combinatorial analog of
Thomason-Trobaugh's theory of localization in the context of algebraic K-theory
for schemes.
