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Register a new userInner automorphisms of groupoids
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Richard Garner
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Bergman has given the following abstract characterisation of the inner automorphisms of a group $G$: they are exactly those automorphisms of $G$ which can be extended functorially along any homomorphism $G rightarrow H$ to an automorphism of $H$. This leads naturally to a definition of inner automorphism applicable to the objects of any category. Bergman and Hofstra--Parker--Scott have computed these inner automorphisms for various structures including $k$-algebras, monoids, lattices, unital rings, and quandles---showing that, in each case, they are given by an obvious notion of conjugation. In this note, we compute the inner automorphisms of groupoids, showing that they are exactly the automorphisms induced by conjugation by a bisection. The twist is that this result is false in the category of groupoids and homomorphisms; to make it true, we must instead work with the less familiar category of groupoids and comorphisms in the sense of Higgins and Mackenzie. Besides our main result, we also discuss generalisations to topological and Lie groupoids, to categories and to partial automorphisms, and examine the link with the theory of inverse semigroups.
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Any group $G$ gives rise to a 2-group of inner automorphisms, $mathrm{INN}(G)$. It is an old result by Segal that the nerve of this is the universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{(2)}$ there is a 3-group $mathrm{INN}(G_{(2)})$ and a slightly smaller 3-group $mathrm{INN}_0(G_{(2)})$ of inner automorphisms. We describe these for $G_{(2)}$ any strict 2-group, discuss how $mathrm{INN}_0(G_{(2)})$ can be understood as arising from the mapping cone of the identity on $G_{(2)}$ and show that its underlying 2-groupoid structure fits into a short exact sequence $G_{(2)} to mathrm{INN}_0(G_{(2)}) to Sigma G_{(2)}$. As a consequence, $mathrm{INN}_0(G_{(2)})$ encodes the properties of the universal $G_{(2)}$ 2-bundle.
VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation groupoid/algebroid.
We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids.
When $mathbb C$ is a semi-abelian category, it is well known that the category $mathsf{Grpd}(mathbb C)$ of internal groupoids in $mathbb C$ is again semi-abelian. The problem of determining whether the same kind of phenomenon occurs when the property of being semi-abelian is replaced by the one of being action representable (in the sense of Borceux, Janelidze and Kelly) turns out to be rather subtle. In the present article we give a sufficient condition for this to be true: in fact we prove that the category $mathsf{Grpd}(mathbb C)$ is a semi-abelian action representable algebraically coherent category with normalizers if and only if $mathbb C$ is a semi-abelian action representable algebraically coherent category with normalizers. This result applies in particular to the categories of internal groupoids in the categories of groups, Lie algebras and cocommutative Hopf algebras, for instance.
In this paper, we introduce the notion of a topological groupoid extension and relate it to the already existing notion of a gerbe over a topological stack. We further study the properties of a gerbe over a Serre, Hurewicz stack.
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