  We introduce and initiate the study of a general class of $2d$
$\mathcal{N}=(0,2)$ quiver gauge theories, defined in terms of certain
2-dimensional CW complexes on oriented 3-manifolds. We refer to this class of
theories as BFT$_2$$\mbox{'}$s. They are natural generalizations of Brane Brick
Models, which capture the gauge theories on D1-branes probing toric Calabi-Yau
4-folds. The dynamics and triality of the gauge theories translate into simple
transformations of the underlying CW complexes. We introduce various
combinatorial tools for analyzing these theories and investigate their
connections to toric Calabi-Yau manifolds, which arise as their master and
moduli spaces. Invariance of the moduli space is indeed a powerful criterion
for identifying theories in the same triality class. We also investigate the
reducibility of these theories.
