A semi-strictly generated closed structure on \textbf{Gray}-\textbf{Cat} (Q6564647)

\(\boldsymbol{Gray}\)-categories are described via enrichment over a symmetric monoidal closed structure on 2-\(\boldsymbol{Cat}\), whose closed fragment consists of 2-functors, pseudonatural transformations and modifications. These higher dimensional maps between 2-categories are characterized by preserving operations, such as composition and identities, on the nose, but only abiding by naturality like conditions up to coherent higher dimensional constant data.\N\NThe author [\textit{A. Miranda}, ``Strictifying Operational Coherences and Weak Functor Classifiers in Low Dimensions'', Preprint, \url{arXiv:2307.01498}] identified those maps between \(\boldsymbol{Gray}\)-categories which preserve operations strictly but naturality like conditions weakly. This paper shows that the semi-strictly generated internal homs of \(\boldsymbol{Gray}\)-categories \(\left[ \mathfrak{A},\mathfrak{B}\right] _{\mathrm{ssg}}\)\ underlie a closed structure on the category \(\boldsymbol{Gray}\)-\(\boldsymbol{Cat}\)\ of \(\boldsymbol{Gray}\)-categories and \(\boldsymbol{Gray}\)-functors. The morphisms of \(\left[ \mathfrak{A},\mathfrak{B}\right] _{\mathrm{ssg}}\) are composites of those trinatural transformations abiding by the unit and composition conditions for pseudonatural transformations on the nose rather than up to an invertible 3-cell. Such trinatural transformations leverage three-dimensional strictification while overcoming the challenges caused by failure of middle four interchange to hold in \(\boldsymbol{Gray}\)-categories [\textit{J. Bourke} and \textit{N. Gurski}, Theory Appl. Categ. 30, 387--409 (2015; Zbl 1338.18029)]. A closed structure that is only partially monoidal with respect to [\textit{S. E. Crans}, Theory Appl. Categ. 5, 12--69 (1999; Zbl 0914.18006)] is obtained as a result. A slight strengthening of strictification results for braided monoidal bicategories [\textit{N. Gurski}, Adv. Math. 226, No. 5, 4225--4265 (2011; Zbl 1260.18008)] is obtained as a corollary, being further improved in a forthcoming paper.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] recalls the notion of a closed structure on a category, and of closed functors (\S 2.1), recalling the semi-strictly generated hom of Gray categories from [\textit{A. Miranda}, ``Strictifying Operational Coherences and Weak Functor Classifiers in Low Dimensions'', Preprint, \url{arXiv:2307.01498}] (\S 2.2).\N\N\item[\S 3] establishes the identity assigner (Proposition 3.1.4)\N\[\Ni_{\mathfrak{A}}:\boldsymbol{1}\rightarrow\left[ \mathfrak{A},\mathfrak{A} \right]\N\]\Nwhile establishing the constant map assigner (Proposition 3.1.5)\N\[\Nc_{\mathfrak{A}}:\mathfrak{A}\rightarrow\left[ \boldsymbol{1},\mathfrak{A} \right]\N\]\N\N\item[\S 4] deals with the axioms for the closed structure, establishing (Theorem 4.2.4) that the semi-strictly generated homs \(\left[ \mathfrak{A},\mathfrak{B}\right] _{\mathrm{ssg}}\), as defined in [\textit{A. Miranda}, ``Strictifying Operational Coherences and Weak Functor Classifiers in Low Dimensions'', Preprint, \url{arXiv:2307.01498}], equip the category \(\boldsymbol{Gray}\)-\(\boldsymbol{Cat}\) with a closed structure in the sense of \textit{S. Eilenberg} and \textit{G. M. Kelly} [in: Proc. Conf. Categor. Algebra, La Jolla 1965, 421--562 (1966; Zbl 0192.10604)] [\textit{M. L. Laplaza}, Trans. Am. Math. Soc. 233, 85--91 (1977; Zbl 0342.18003); \textit{O. Manzyuk}, Theory Appl. Categ. 26, 132--175 (2012; Zbl 1261.18009)].\N\N\item[\S 5] relates the semi-strictly generated closed structure to the closed structure \(\boldsymbol{Ps}\left( -,?\right) \)\ on \(\boldsymbol{Gray}\)-\(\boldsymbol{Cat}\)\ introduced in [\textit{J. Bourke} and \textit{G. Lobbia}, Adv. Math. 434, Article ID 109327, 92 p. (2023; Zbl 1524.18047)].\N\N\item[\S 6] relates the various interchangers between trinatural transformations and trimodifications in \S 3 to the generalized center construction for \(\boldsymbol{Gray}\)-monoids of [\textit{S. E. Crans}, Adv. Math. 136, No. 2, 183--223 (1998; Zbl 0908.18004)].\N\N\item[\S 7] is conclusion.\N\N\item[\S 8] is appendices concerning definitions \(\left( 3,k\right) \)-transfors between \(\boldsymbol{Gray}\)-functors (\S 8.1) and \(\boldsymbol{Gray}\)-categories of \(\left( 3,k\right) \)-transfors.\N\end{itemize}