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This work is the first one in a series, in which we develop a mathematicaltheory of enriched (braided) monoidal categories and their representations. Inthis 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, andcompute the centers explicitly when the enriched (braided monoidal or monoidal)categories are obtained from the canonical constructions. These centers haveimportant applications in the mathematical theory of gapless boundaries of 2+1Dtopological orders and that of topological phase transitions in physics. Theyalso play very important roles in the higher representation theory, which isthe focus of the second work in the series.

Enriched monoidal categories I: centers