2023/11/19 by Adrian Miranda, Miranda, Adrian · 1 voice
Mathematics · Medicine · #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications
paper · pdf · doi:10.48550/arxiv.2311.11403
openalex publication_date 2023/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the semi-strictly generated internal homs of Gray-categories [\mathfrakA, \mathfrakB]ssg defined in \citeMiranda strictifying operational coherences underlie a closed structure on the category Gray-Cat of Gray-categories and Gray-functors. The morphisms of [\mathfrakA, \mathfrakB]ssg are composites of those trinatural transformations which satisfy 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 \citeMiranda strictifying operational coherences while overcoming the challenges posed by failure of middle four interchange to hold in Gray-categories \citeBourke Gurski Cocategorical Obstructions to a Tensor Product of Gray Categories. As a result we obtain a closed structure that is only partially monoidal with respect to \citecrans tensor of gray categories. As a corollary we obtain a slight strengthening of strictification results for braided monoidal bicategories \citeGurski Loop Spaces, which will be improved further in a forthcoming paper \citeMiranda weak interchange 4-categories.