  This work is the first one in a series, in which we develop a mathematical
theory of enriched (braided) monoidal categories and their representations. In
this work, we introduce the notion of the $E_0$-center ($E_1$-center or
$E_2$-center) of an enriched (monoidal or braided monoidal) category, and
compute the centers explicitly when the enriched (braided monoidal or monoidal)
categories are obtained from the canonical constructions. These centers have
important applications in the mathematical theory of gapless boundaries of 2+1D
topological orders and that of topological phase transitions in physics. They
also play very important roles in the higher representation theory, which is
the focus of the second work in the series.
