higher category theory
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Tetracategories are the algebraic definition of higher category for the most general (i.e. weak) 4-categories.
By the discussion at k-tuply monoidal (n,r)-category a tetracategory with a single object may be regarded as the delooping of a monoidal tricategory. This is discussed in (Hoffnung).
The structure of a tetracategories was given by Todd Trimble.
Todd Trimble Notes on Tetracategories
U52 unit condition
U53 unit condition
U54 unit condition
A polished writeup of the definition appears in section 3.2 of