Symmetry

Groups , abstracted from transformation groups, axiomatize the symmetries of a geometric object. They are best suited to describing the symmetries of highly “uniform” or “homogeneous” objects.

Group objects can be defined in any cartesian category or, more generally, in any monoidal category with diagonals. However, the definition of a group object can be problematic when the group inverses are “anti-homomorphisms” rather than homomorphisms. For example, a partially ordered group is not a group object in \(\mathbf{Pos}\).

• Blohmann & Weinstein, 2008: Group-like objects in Poisson geometry and algebra (arxiv)
  • Ideas about how to go beyond groups and group objects
  • In Dito et al, eds., 2008: Poisson geometry in mathematics and physics (doi)
• Forrester-Barker, 2002: Group objects and internal categories (arxiv)
  • Exposition of equivalence between crossed modules , group objects in \(\mathbf{Cat}\), and internal categories in \(\mathbf{Grp}\)

Groupoids generalize groups to allow symmetries between different objects and to better describe the symmetries of “inhomogeneous” objects.

An important class of groupoids, called transformation groupoids or action groupoids, arise from group actions. If \(G\) is a group acting on a space \(X\), then there is a groupoid with objects \(x \in X\) and morphisms \(x \to y\) the group elements \(g \in G\) with \(y = g \cdot x\). In his Tale of Groupoidification , culled from This Week’s Finds, Baez explains that actions groupoids are the same as groupoids with a faithful functor to a group.

TODO: My MO question on Cencov and Erlangen program

General literature

• Brown, 1987: From groups to groupoids: A brief survey (doi, pdf)
• Weinstein, 1996: Groupoids: unifying internal and external symmetry (pdf, arxiv)
• Vistoli, 2011: Groupoids: a local theory of symmetry (pdf)
  • Brief introduction to groups and groupoids for philosophers
• Olver, 2016: The symmetry groupoid and weighted signature of a geometric object (online , pdf, slides)

Partial group actions and inverse semigroup actions

Generalizing group actions, partial group actions also give rise to action groupoids (Abadie, 2004) and are closely related to actions of inverse semigroups (see below).

• Paterson, 1999: Groupoids, inverse semigroups, and their operator algebras (doi)
• Abadie, 2004: On partial actions and groupoids (doi, pdf)
• Buss, Exel, Meyer, 2012: Inverse semigroup actions as groupoid actions (doi, arxiv)
• Clark & Hazrat, 2019: Étale groupoids and Steinberg algebras, a concise introduction (arxiv)

Haar systems

Haar measure on a group generalizes to systems of Haar measures on groupoids.

• Seda, 1986: On the continuity of Haar measure on topological groupoids (doi)
• Deitmar, 2016: On Haar systems for groupoids (arxiv)

Inverse semigroups and inverse monoids axiomatize “partial” symmetries. The Wagner-Preston theorem is the analogue of Cayley’s theorem for inverse semigroups, with the symmetric inverse semigroup playing the role of the symmetric group.

General literature

• Lawson, 1998: Inverse semigroups: The theory of partial symmetries (doi)
  • Ch. 10 follows: Lawson, 1999: Constructing inverse semigroups from category actions (doi, pdf)
• Hollings, 2015: Three approaches to inverse semigroups (pdf)

Examples

• Mills, 1993: Factorizable semigroup of partial symmetries of a regular polygon (doi)
• Everitt & Fountain, 2010: Partial symmetry, reflection monoids and Coxeter groups (doi, arxiv)
  • Everitt & Fountain, 2013: Partial mirror symmetry, lattice presentations and algebraic monoids (doi, arxiv)

Partial group actions

Actions of inverse semigroups are in one-to-one correspondence with partial group actions (Exel, 1998).

• Exel, 1998: Partial actions of groups and actions of inverse semigroups (doi, pdf)
  • Exel, 2013, lecture notes: Partial group actions (pdf)
  • Exel, 2017: Partial dynamical systems, Fell bundles, and applications (doi, arxiv)
• Kellendonk & Lawson, 2004: Partial actions of groups (doi)