Semi-strict 4-categories as enriched categories

We introduce several flavours of semi-strict 4-category as enriched categories. We give the axioms for semi-strict 4-categories in elementary terms and describe a wide range of examples. In particular, we answer the question of what exactly Gray-categories and the higher transformations between them form: namely, a left-sharp 4-category G-Cat${}^4$. In addition, we describe examples arising from braided monoidal 2-categories, monoidal Gray-categories (herein defined) and Gray-categories with structure. We use the concepts introduced here and the relationships between them to unify various concepts in the literature, such as Crans’ 4-teisi. In the second part of the paper, we investigate homotopical aspects. In particular, we classify the equivalence n-cells in the left-sharp 4-category G-Cat${}^4$ for each n. Using this, we show that the left-sharp skew monoidal structure on the category of Gray-categories induces a symmetric monoidal closed structure on the homotopy category of Gray-categories.