Inner automorphisms as 2-cells (Q6593817)

The category of groups is to be regarded as a 2-category in a nontrivial way, where 2-cells correspond to elements \(g\) of the codomain such that\N\[\Ng\psi\left( -\right) g^{-1}=\phi\N\]\NThis paper takes the view that the 2-cells between group homomorphisms come from inner automorphisms called \textit{extended inner automorphisms}, which make sense in any category \(\boldsymbol{C}\), so that one can define the group \(\mathcal{Z}\left( A\right) \) of them at an object \(A\) as the group of natural automorphisms of the projection \(A/\boldsymbol{C}\rightarrow \boldsymbol{C}\). This results in the notion of (covariant) \textit{isotropy group} investigated in [\textit{J. Funk} et al., Theory Appl. Categ. 33, 537--582 (2018; Zbl 1419.18002); \textit{P. Hofstra} and \textit{M. Karvonen}, J. Pure Appl. Algebra 228, No. 11, Article ID 107717, 44 p. (2024; Zbl 1544.18015); \textit{P. Hofstra} et al., Electron. Notes Theor. Comput. Sci. 341, 201--217 (2018; Zbl 1528.03255); \textit{P. Hofstra} et al., LIPIcs -- Leibniz Int. Proc. Inform. 195, Article 26, 17 p. (2021; Zbl 1541.03108); \textit{J. Parker}, Appl. Categ. Struct. 30, No. 4, 779--803 (2022; Zbl 1491.18014)]. \N\NThe starting point of this paper is that such extended inner automorphisms are to be used to promote any category \(\boldsymbol{C}\) into a 2-category \(\boldsymbol{C}_{\mathcal{Z}}\) by defining for\N\[\Nf,g:A\rightrightarrows B\N\]\N2-cells \(f\rightarrow g\) as elements \(\alpha\in\mathcal{Z}\left( B\right) \) such that\N\[\N\alpha_{\mathrm{id}}f=g\N\]\N\NThis paper is concerned with two-dimensional limits and colimits in 2-categories. It is shown (Theorem 3.1) that any connected colimits in the underlying category becomes a two-dimensional colimit. It is also shown (Theorem 3.5) that, when the presheaf of sets underlying the presheaf of groups is representable, many futher pleasant properties hold. These positive results are sharp in a sense (Theorems 3.6 and 3.7, Corollary 3.8). The paper concludes with discussing briefly when a functor \(\boldsymbol{C}\rightarrow\boldsymbol{D}\) extends to a 2-functor \(\boldsymbol{C}_{\mathcal{Z}}\rightarrow\boldsymbol{D}_{\mathcal{Z}}\).