Closed categories vs. closed multicategories (Q2884476)

One can equip the category of \(A_\infty\)-categories with the structure of a closed category. It turns out that \(A_\infty\)-categories also form a multicategory (or colored operad). This is one of many relations between closed categories and multicategories that in part motivated the present work. It was proved [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)] that the 2-category of monoidal categories, strict monoidal functors and monoidal transformations is 2-equivalent to the 2-category of representable multicategories, multifunctors preserving universal arrows and multinatural transformations.NEWLINENEWLINE As a first generalization of this result, \textit{Yu. Bespalov} et al. proved [Pretriangulated \(A_\infty\)-categories. Kyïv: Instytut Matematyky NAN Ukraïny (2008; Zbl 1199.18001)] that the 2-category of lax monoidal categories, lax monoidal functors and monoidal transformations is 2-equivalent to the 2-category of lax representable multicategories, multifunctors and multinatural transformations. The present paper completes the multicategorical interpretation of \textit{S. Eilenberg} and \textit{G. M. Kelly} [Proc. Conf. Categor. Algebra, La Jolla 1965, 421--562 (1966; Zbl 0192.10604)]. The author first proves that his definition of closed categories, which is none other than an extension system with one object [\textit{R. Street}, Lect. Notes Math. 420, 134--180 (1974; Zbl 0325.18005)], leads to a theory that is isomorphic to the one of Eilenberg and Kelly. He then goes on straightforwardly constructing a Cat-functor from the 2-category of closed multicategories with a unit object, multifunctors and multinatural transformations to the 2-category of closed categories, closed functors and closed natural transformations which he proves to be a categorical equivalence.