Unicity of enrichment over \(\mathcal Cat\) or \(\mathcal Gpd\)
The paper is motivated by the question posed by \textit{G.~M.~Kelly} on whether a monad on \(\mathcal Cat\) (the category of small categories) can be enriched over \(\mathcal Cat\) in a unique way. The author shows a sufficient condition (the existence of tensors or cotensors with the arrow category) for an ordinary monad on a 2-category to have at most one enrichment (\(\mathcal Cat\) can be easily replaced with the category \(\mathcal Gpd\) of groupoids). On the other hand, there exist monads on \(\mathcal Cat\), which have no enrichment and, moreover, enrichment over the categories different from \(\mathcal Cat\) and \(\mathcal Gpd\) can be not unique (the respective examples are provided in the manuscript).
scientific article; zbMATH DE number 5871640
scientific article