A semi-strictly generated closed structure on Gray-Cat
A semi-strictly generated closed structure on Gray-Cat
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Gray-Cat 上的半严格生成闭合结构
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发表时间：
2023
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通讯作者：
Adrian Miranda
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文献类型：
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作者：
Adrian Miranda
We show that the semi-strictly generated internal homs of $mathbf{Gray}$-categories $[mathfrak{A}, mathfrak{B}]_ ext{ssg}$ defined in cite{Miranda strictifying operational coherences} underlie a closed structure on the category $mathbf{Gray}$-$mathbf{Cat}$ of $mathbf{Gray}$-categories and $mathbf{Gray}$-functors. The morphisms of $[mathfrak{A}, mathfrak{B}]_ ext{ssg}$ 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 cite{Miranda strictifying operational coherences} while overcoming the challenges posed by failure of middle four interchange to hold in $mathbf{Gray}$-categories cite{Bourke 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 cite{crans tensor of gray categories}. As a corollary we obtain a slight strengthening of strictification results for braided monoidal bicategories cite{Gurski Loop Spaces}, which will be improved further in a forthcoming paper cite{Miranda weak interchange 4-categories}.
We show that the semi-strictly generated internal homs of $mathbf{Gray}$-categories $[mathfrak{A}, mathfrak{B}]_ ext{ssg}$ defined in cite{Miranda strictifying operational coherences} underlie a closed structure on the category $mathbf{Gray}$-$mathbf{Cat}$ of $mathbf{Gray}$-categories and $mathbf{Gray}$-functors. The morphisms of $[mathfrak{A}, mathfrak{B}]_ ext{ssg}$ 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 cite{Miranda strictifying operational coherences} while overcoming the challenges posed by failure of middle four interchange to hold in $mathbf{Gray}$-categories cite{Bourke 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 cite{crans tensor of gray categories}. As a corollary we obtain a slight strengthening of strictification results for braided monoidal bicategories cite{Gurski Loop Spaces}, which will be improved further in a forthcoming paper cite{Miranda weak interchange 4-categories}.