Inner autoequivalences in general and those of monoidal categories in particular (Q6564635)

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 no nontrivial inner autoequivalences in this sense.\N\item For the 2-category of indexed categories on \(\mathbf C\), i.e., pseudofunctors \(F \colon \mathbf{C}^{\mathrm{op}}\to \mathbf{Cat}\), the isotropy 2-group at \(F\) is the 2-group of pseudonatural autoequivalences of the identity. This generalizes the characterization of the covariant isotropy group of a presheaf topos.\N\item For the 2-category of monoidal categories, strong monoidal functors and monoidal natural transformations, then the isotropy 2-group of a monoidal category is equivalent to its Picard 2-group.\N\end{itemize}