Inner autoequivalences in general and those of monoidal categories in particular
type: Item
instance of: scholarly article
title: Inner autoequivalences in general and those of monoidal categories in particular
zbMATH Open document ID: 1544.18015
author: Pieter Hofstra
author: Martti Karvonen
published in: Journal of Pure and Applied Algebra
publication date: 2024-07-01T00:00:00Z
review text: An inner automorphism of a group \(G\) is an automorphism that is of the form \(g \cdot (-) \cdot g^{-1}\), i.e., one that equals conjugation by an element of \(G\). For an inner automorphism of \(G\), there might be several elements that induce it -- for instance, all central elements induce the identity. However, a fixed \(g \in G\) induces more automorphisms than just one: in fact, for any homomorphism \(\varphi \colon G \to H\) it induces an inner automorphism of \(H\) by conjugation with \(\varphi(g)\). This family of automorphisms constitutes what is called an extended inner automorphism. Formally, this can be defined as a natural automorphism of the projection functor \(G/\mathrm{Grp}\to \mathrm{Grp}\) (the projection functor from the co-slice category). Every such extended inner automorphism is induced in the above manner by a unique element of \(G\), so that there is a natural isomorphism between a group and extend inner automorphisms of the projection functor out of its co-slice category.\N\NThis motivates the definition of the isotropy group of a category at an object, that is the group of natural automorphisms of the projection functor out of the co-slice category of this object. This is functorial: there is a functor from a category to the category of groups that assigns to an object its isotropy group. In particular, when considering the category of strict monoidal categories and strict functors, the isotropy group of a strict monoidal category is its Picard group.\N\NIn this paper, a technique is introduced to compute isotropy groups for categories with binary coproducts together with a choice of convenient dense subcategory. This unifies several particular approaches.\N\NThen, the theory of isotropy 2-groups of bicategories is developed, and the technique mentioned above is extended to it. Particular examples of isotropy 2-groups are considered:\N\begin{itemize}\N\item For the 2-category of groupoids, its isotropy 2-groups vanish, thus groupoids have…
reviewed by: Nicolas Blanco
Mathematics Subject Classification ID: 18G45
Mathematics Subject Classification ID: 18M05
Mathematics Subject Classification ID: 18N10
zbMATH DE Number: 7873749
zbMATH Keywords: Picard 2-group
zbMATH Keywords: 2-category
zbMATH Keywords: 2-group
zbMATH Keywords: inner autoequivalence
zbMATH Keywords: dense pseudofunctor
MaRDI profile type: Publication
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DOI: 10.1016/J.JPAA.2024.107717
description: scientific article; zbMATH DE number 7873749