Inner automorphisms of groupoids

Richard Garner

arxiv: 1907.10378 · v1 · pith:6ZL7NBIInew · submitted 2019-07-24 · 🧮 math.CT · math.GR

Inner automorphisms of groupoids

Richard Garner This is my paper

Pith reviewed 2026-05-24 16:33 UTC · model grok-4.3

classification 🧮 math.CT math.GR

keywords inner automorphismsgroupoidsbisectionscomorphismsconjugationcategory theoryinverse semigroups

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The pith

Inner automorphisms of groupoids are exactly those induced by conjugation by a bisection.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies an abstract definition of inner automorphisms, originally given for groups, to the setting of groupoids. The definition identifies an automorphism as inner when it extends functorially along any morphism to an automorphism of the target object. For groupoids the resulting inner automorphisms turn out to be precisely the maps obtained by conjugation with a bisection. The identification holds only when the arrows between groupoids are taken to be comorphisms rather than ordinary homomorphisms. A sympathetic reader would care because the result supplies a uniform, conjugation-style description of inner automorphisms for one more class of algebraic objects.

Core claim

The paper computes the inner automorphisms of groupoids, defined as those automorphisms that extend functorially along any morphism to an automorphism of the codomain, and shows that they coincide exactly with the automorphisms induced by conjugation by a bisection. This equality is false when morphisms are ordinary homomorphisms, but becomes true when the category is taken to consist of groupoids and comorphisms.

What carries the argument

The functorial-extension definition of inner automorphism, realized concretely by conjugation along bisections.

If this is right

Inner automorphisms of any groupoid are given explicitly by its bisections.
The same computation extends to topological groupoids and to Lie groupoids.
The approach also yields descriptions for ordinary categories and for partial automorphisms.
A link appears between these inner automorphisms and the theory of inverse semigroups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Comorphisms appear to be the arrows that make symmetry computations natural for groupoids.
The same pattern of adjusting the notion of morphism may be needed for other structures whose standard homomorphisms do not yield clean inner-automorphism results.
Concrete small groupoids can be enumerated by hand to verify that every extendable automorphism arises from a bisection.
The connection to inverse semigroups suggests a possible dictionary between bisections and certain idempotents or partial units.

Load-bearing premise

The result depends on equipping the category of groupoids with comorphisms rather than with ordinary homomorphisms.

What would settle it

An explicit groupoid together with an automorphism that extends functorially along every comorphism yet fails to arise from conjugation by any bisection would show the claimed identification is incorrect.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Garner computes inner automorphisms of groupoids as bisection conjugations, but only after switching to the comorphism category where the result holds.

read the letter

The punchline is that this short note takes Bergman's functorial definition of inner automorphisms and works out what it gives for groupoids. The answer is conjugation by bisections, but only when morphisms are comorphisms in the Higgins-Mackenzie sense; the same statement is false for ordinary groupoid homomorphisms, and the paper says so plainly. That category choice is the main technical move, and it lets the computation go through cleanly. The paper does the obvious next step after the Bergman and Hofstra-Parker-Scott results on other structures, and it records the explicit description without extra machinery. It also flags the link to inverse semigroups and sketches how the same idea might run for topological or Lie groupoids and for partial automorphisms. Those extensions are not fully proved here, but they are signposted. The argument rests on standard definitions plus Bergman's prior characterization, so there is no circularity or invented data. The only soft spot is that the note is exactly what it says it is: a computation rather than a broad theory, so readers wanting full proofs of the generalizations will need to do the work themselves. This is for people already working with groupoids or with Bergman's style of abstract inner automorphism. A category theorist who cares about these structures will find the explicit result useful and the category restriction clearly marked. It is worth sending to a referee who knows comorphisms and groupoid theory; the central claim is narrow but grounded and new.

Referee Report

0 major / 2 minor

Summary. The paper extends Bergman's abstract characterization of inner automorphisms to groupoids. It proves that, in the category of groupoids and comorphisms (Higgins-Mackenzie), the inner automorphisms are precisely the automorphisms induced by conjugation by a bisection. The result is shown to fail in the category of groupoids and ordinary homomorphisms. The manuscript also discusses generalizations to topological and Lie groupoids, to categories, to partial automorphisms, and connections to inverse semigroups.

Significance. If the central computation holds, the paper supplies a concrete new instance in the program of determining inner automorphisms for algebraic structures, with explicit credit to the functorial-extension definition and the choice of comorphisms. The explicit scoping to comorphisms, the link to inverse semigroups, and the sketched generalizations to topological/Lie settings add value for researchers working at the interface of category theory and groupoid theory.

minor comments (2)

The abstract and introduction state that the result fails for homomorphisms but hold for comorphisms; a brief sentence in §1 or §2 recalling the precise definition of comorphism (Higgins-Mackenzie) would help readers who are not already familiar with that notion.
The discussion of generalizations to topological and Lie groupoids is only sketched; if space permits, a short paragraph indicating which parts of the bisection-conjugation argument carry over verbatim and which require additional continuity or smoothness hypotheses would strengthen the section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation applies Bergman's external abstract characterization of inner automorphisms (cited as prior work by a different author) to groupoids, then computes that these coincide with bisection conjugation specifically in the category of groupoids and comorphisms (defined via the external Higgins-Mackenzie reference). The text explicitly states the result fails for ordinary homomorphisms, treating the category choice as a precondition rather than a hidden assumption. No equations, definitions, or claims reduce by construction to the paper's own inputs or fitted values; the computation is a direct verification against the cited external definitions with no self-citation load-bearing the result and no renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on Bergman's abstract characterization of inner automorphisms (invoked as the starting definition) and the choice to work in the category of comorphisms; no free parameters, invented entities, or additional ad-hoc axioms are indicated in the abstract. Review is abstract-only so ledger is necessarily incomplete.

axioms (1)

domain assumption Bergman's abstract characterisation of the inner automorphisms of a group G
The paper takes this as the definition to extend to groupoids.

pith-pipeline@v0.9.0 · 5727 in / 1371 out tokens · 66222 ms · 2026-05-24T16:33:48.754013+00:00 · methodology

Inner automorphisms of groupoids

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)