  We present a unified mathematical framework that elegantly describes
minimally SUSY gauge theories in even dimension, ranging from $6d$ to $0d$, and
their dualities. This approach combines recent developments on graded quiver
with potentials, higher Ginzburg algebras and higher cluster categories (also
known as $m$-cluster categories). Quiver mutations studied in the context of
mathematics precisely correspond to the order $(m+1)$ dualities of the gauge
theories. Our work suggests that these equivalences of quiver gauge theories
sit inside an infinite family of such generalized dualities, whose physical
interpretation is yet to be understood.
