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Localizations of Transfors

Authors:	Sjoerd E Crans

Institution:	(1) School of Mathematics, Physics, Computing and Electronics, Macquarie University, NSW, 2109, Australia;(2) Present address: Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 7 rue René-Descartes, 67084 Strasbourg, France

Abstract:	Let $\mathbb{C}{\text{, }}\mathbb{D} {\text{and }}\mathbb{E}$ be n-dimensional teisi, i.e., higher-dimensional Gray-categorical structures. The following questions can be asked. Does a left q-transfor $\mathbb{C} \to \mathbb{D}$ , i.e., a functor 2_q $\otimes {\text{ }}\mathbb{C} \to \mathbb{D}$ , induce a right q-transfor $\mathbb{C} \to \mathbb{D}$ , i.e., a functor $\mathbb{C} \otimes {\text{ 2}}q \to \mathbb{D}{\text{?}}$ More generally, does a functor $\mathbb{C} \otimes {\text{ }}\mathbb{D} \to \mathbb{E}$ induce a functor $\mathbb{C} \otimes {\text{ }}\mathbb{D} \to \mathbb{E}{\text{?}}$ For k-arrows c and $c' \in \mathbb{C}$ whose (k – 1)-sources and targets agree, does a q-transfor $\mathbb{C} \to \mathbb{D}$ induce a q-transfor $\mathbb{C}(c,c') \to \mathbb{D}(d,d')$ , for appropriate k-arrows $d{\text{ and }}d' \in \mathbb{D}{\text{?}}$ For k-arrows c and $c' \in \mathbb{C}{\text{ and }}d{\text{ and }}d' \in \mathbb{D}$ whose (k – 1)-sources and targets agree, does a q-transfor $\mathbb{C} \otimes {\text{ }}\mathbb{D} \to \mathbb{E}$ induce a (q + k + 1)-transfor $\mathbb{C}(c,c') \otimes \mathbb{D}(d,d') \to \mathbb{E}(e,e')$ , for appropriate k-arrows $e{\text{ and }}e' \in \mathbb{E}{\text{?}}$ I give answers to these questions in the cases where n-dimensional teisi and their tensor product have been defined, i.e., for n 3, and for n up to 5 in some cases that do not need all data and axioms of n-dimensional teisi.I apply the above to compositions in teisi, in particular to braidings and syllepses. One of the results is that a braiding on a monoidal 2-category induces a pseudo-natural transformation $\widetilde{{\text{?}} \otimes - } \to {\text{?}} \otimes -$ , where $\widetilde{{\text{?}} \otimes - }$ is the reverse of ? –, which is almost, but not quite, equal to – ?. However, in higher dimensions need not be reversible, so a braiding on a higher-dimensional tas can not be seen as a transfor A B B A.

Keywords:	braiding natural transformation Gray-category symmetry tas transfor
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