  We study 2d $\mathcal{N}=(0,2)$ supersymmetric quiver gauge theories that
describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY$_4$)
singularities. On general grounds, the holomorphic sector of these
theories---matter content and (classical) superpotential interactions---should
be fully captured by the topological $B$-model on the CY$_4$. By studying a
number of examples, we confirm this expectation and flesh out the dictionary
between B-brane category and supersymmetric quiver: the matter content of the
supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext
groups of coherent sheaves), while the superpotential interactions are encoded
in the $A_\infty$ algebra satisfied by the morphisms. This provides us with a
derivation of the supersymmetric quiver directly from the CY$_4$ geometry. We
also suggest a relation between triality of $\mathcal{N}=(0,2)$ gauge theories
and certain mutations of exceptional collections of sheaves. 0d $\mathcal{N}=1$
supersymmetric quivers, corresponding to D-instantons probing CY$_5$
singularities, can be discussed similarly.
