Comonads and enrichment
Richard Garner – 20 November 2019
A result due to Bourke, subsequently developed further by Miranda, states that the 2-category of finitely complete 2-categories is pseudocomonadic over the 2-category of finitely complete ordinary categories, via the pseudocomonad E |-> Cat(E).We explain a more general version of this result: if V is locally finitely presentable as a symmetric monoidal category, then the 2-category of finitely complete V-categories is comonadic over the 2-category of finitely complete ordinary categories. Besides the motivating case V = Cat, this applies, for example, when V = Ch or V = sSet.We also mention a result in a similar spirit, due to Rosolini, Emennegger and Pasquali, stating that the 2-category of finite product fibrations with equality is comonadic over the 2-category of finite product fibrations, and speculate as to how it might fit into the same pattern.
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