Semi-strict 4-categories as enriched categories
Abstract
We introduce several flavours of semi-strict 4-category as enriched categories. This is done using the framework of enrichment over skew monoidal categories of Gray-categories and their associated multicategories, as developed in our previous work. 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-Cat4. 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-Cat4 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.
Related
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BibTeX
@misc{bourke2026semistrict,
title = {{Semi-strict 4-categories as enriched categories}},
author = {John Bourke and Gabriele Lobbia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31448},
eprint = {2610.31448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31448},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Semi-strict 4-categories as enriched categories
AU - John Bourke
AU - Gabriele Lobbia
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.31448
UR - https://arxiv.org/abs/2610.31448
AB - We introduce several flavours of semi-strict 4-category as enriched categories. This is done using the framework of enrichment over skew monoidal categories of Gray-categories and their associated multicategories, as developed in our previous work. 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.
ER -