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Split Lemma and First Isomorphism Theorem for groupoids

Davide Ferri

TL;DR

The paper examines why the classical First Isomorphism Theorem fails in the category ${ m Gpd}$ of groupoids and develops a universal lifting framework that uses virtual kernels to obtain a lifted First Isomorphism Theorem. It introduces crossed and semidirect products for groupoids, establishing a Split Lemma in ${ m Gpd}$ and its fixed-vertex variant ${ m Gpd}_ Λ$, and connects these constructions with universal properties for short exact sequences. The work unifies several approaches to semidirect-type constructions for groupoids, clarifies when these notions coincide with classical groupoid-split structures, and provides a robust toolkit (virtual kernels, crossed products, and balanced tensor products) to handle images, kernels, and quotients in a setting where standard kernels and quotients may be inadequate. The results have implications for representation theory, topology, and categorical algebra by enabling a principled way to factor groupoid morphisms and to realize split extensions via universal constructions. Overall, the methods deliver a coherent framework for First Isomorphism Theorems, semidirect-type decompositions, and Split Lemmas in the broader context of groupoids and their vertex-fixed variants.

Abstract

Groupoids are the oidification of groups, and they are largely used in topology and representation theory. We consider here the category $\mathsf{Gpd}$ of all groupoids with all morphisms, and the category $\mathsf{Gpd}_Λ$ of groupoids over a fixed set of vertices $Λ$, with morphisms fixing $Λ$. In $\mathsf{Gpd}_Λ$, a First Isomorphism Theorem is already well known; see Ávila, Marín, and Pinedo (2020). Famously, the First Isomorphism Theorem fails to hold in $\mathsf{Gpd}$. However, we retrieve here a universally lifted version of the First Isomorphism Theorem in $\mathsf{Gpd}$, through the definition of virtual kernels. Semidirect products of a group by a groupoid are well known. We define crossed products in $\mathsf{Gpd}$, and prove that they are equivalent to split epimorphisms, i.e. that they are the `categorial' notion of semidirect product in $\mathsf{Gpd}$ in the sense of Bourn and Janelidze (1998). We observe that in $\mathsf{Gpd}_Λ$ crossed products and semidirect products are essentially equivalent, under mild assumptions, and our Split Lemma in $\mathsf{Gpd}$ collapses to a much simpler Split Lemma in $\mathsf{Gpd}_Λ$ that appears in Metere and Montoli (2010) and Ibort and Marmo (2023).

Split Lemma and First Isomorphism Theorem for groupoids

TL;DR

The paper examines why the classical First Isomorphism Theorem fails in the category of groupoids and develops a universal lifting framework that uses virtual kernels to obtain a lifted First Isomorphism Theorem. It introduces crossed and semidirect products for groupoids, establishing a Split Lemma in and its fixed-vertex variant , and connects these constructions with universal properties for short exact sequences. The work unifies several approaches to semidirect-type constructions for groupoids, clarifies when these notions coincide with classical groupoid-split structures, and provides a robust toolkit (virtual kernels, crossed products, and balanced tensor products) to handle images, kernels, and quotients in a setting where standard kernels and quotients may be inadequate. The results have implications for representation theory, topology, and categorical algebra by enabling a principled way to factor groupoid morphisms and to realize split extensions via universal constructions. Overall, the methods deliver a coherent framework for First Isomorphism Theorems, semidirect-type decompositions, and Split Lemmas in the broader context of groupoids and their vertex-fixed variants.

Abstract

Groupoids are the oidification of groups, and they are largely used in topology and representation theory. We consider here the category of all groupoids with all morphisms, and the category of groupoids over a fixed set of vertices , with morphisms fixing . In , a First Isomorphism Theorem is already well known; see Ávila, Marín, and Pinedo (2020). Famously, the First Isomorphism Theorem fails to hold in . However, we retrieve here a universally lifted version of the First Isomorphism Theorem in , through the definition of virtual kernels. Semidirect products of a group by a groupoid are well known. We define crossed products in , and prove that they are equivalent to split epimorphisms, i.e. that they are the `categorial' notion of semidirect product in in the sense of Bourn and Janelidze (1998). We observe that in crossed products and semidirect products are essentially equivalent, under mild assumptions, and our Split Lemma in collapses to a much simpler Split Lemma in that appears in Metere and Montoli (2010) and Ibort and Marmo (2023).

Paper Structure

This paper contains 29 sections, 25 theorems, 62 equations, 11 figures.

Table of Contents

  1. Quivers
  2. Quivers, morphism, and strong morphisms
  3. Monomorphism and epimorphisms
  4. Equivalence pairs and quotients
  5. Twisted fibre product
  6. Monoidal structure on $\maybebm{{\mathsf{Quiv}}_\Lambda}$
  7. On the category of groupoids
  8. Groupoids, morphisms, and strong morphisms
  9. Describing connected groupoids as a product
  10. On the geometry of groupoids
  11. Subgroupoids
  12. On the images of groupoid morphisms
  13. Mo-no-mor-phisms and epimorphisms in $\maybebm{{\sf Gpd}}$
  14. Normal subgroupoids
  15. Quotients
  16. ...and 14 more sections

Key Result

Lemma 1.2

A morphism $f$ is a monomorphism, resp. epimorphism in ${\mathsf{Quiv}}$, if and only if $f^1$ and $f^0$ are both monomorphisms, resp. epimorphisms in ${\sf Set}$. A strong morphism $f$ is a monomorphism, resp. an epimorphism in ${\mathsf{Quiv}}_\Lambda$, if and only if it is a monomorphism, resp. a

Figures (11)

  • Figure 1: A morphism of groupoids whose image is not a subgroupoid. Since the underlying quivers are Schurian, the groupoid structures are unambiguous. One has $x = f^1(a)f^1(b)$, but $x$ does not lie in the image of $f^1$.
  • Figure 2: An example of a morphism $\mathscr{G}\to \mathscr{H}$ with $\mathscr{G}$ connected, such that the image is not a subgroupoid of $\mathscr{H}$. Here $\mathscr{H} = \mathbb{Z}/4\mathbb{Z}$, and $f^1(a) = 1$, $f^1(a^{-1})= 3$. Observe that $a^2$ is not defined in $\mathscr{G}$, while $f^1(a)^2 =2$ is defined in $\mathscr{H}$.
  • Figure 3: When $\mathscr{N}$ is not a subgroup bundle, the implication $x\sim_Ly \implies x\sim y$ fails in general.
  • Figure 4: The morphism $f\colon \mathscr{G}\to \mathscr{H}$ identifies $\mu$ and $\mu'$, thus we would like to take the groupoid on the left as its kernel; but the dashed arrows do not exist in $\mathscr{G}$.
  • Figure 5: The map $f$ from \ref{['fig:case_study_1']}, with $\ker(f)$, the groupoids $\tilde{\mathscr{G}}$ and $\tilde{\mathscr{N}}$, the induced morphism $\tilde{f}$, and the various inclusions.
  • ...and 6 more figures

Theorems & Definitions (92)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Definition 2.1
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5: see e.g. Brown GroupsToGroupoidsBrown
  • ...and 82 more