There are different ways to define a groupoid such as:
- Algebraic definition: same definition as a group but with a partial function replacing the binary operation.
- Catagorical definition: a category in which every morphism is invertible. That is, add a unary operation to the category.
- As an oriented graph (if <x,y> is an edge then <y,x> is not).
I get the impression that some older textbooks use the term 'groupoid' in an incompatible way to mean a group that is not associative?
Special cases include:
- Setoids: sets with an equivalence relation;
- G-sets, sets with an action of a group G.
I am trying to visualise the algebraic definition: same definition as a group but with a partial function replacing the binary operation.
|This is a conventional Cayley graph.|
|Here I have made the functions 'f' and 'g' partial.|
I am trying to visualise the catagorical definition: a category in which every morphism is invertible.
|This is an attempt to represent a permutation group with 3 elements and permutations shown in red, green and blue.|
|So how can we show multiple nodes? Ill start by assuming that isomorphism requires the elements are one-to-one?|
An ω-groupoid is an ω-category in which all higher morphisms are equivalences.
A ω-category (or ∞-category) is a higher order category.
- all composition operations are strictly associative.
- all composition operations strictly commute with all others (strict exchange laws).
- all identity higher morphisms are strict identities under all compositions.
Fundamental Groupoid - Topology
The fundamental groupoid of a space is a groupoid with:
- Objects - points of a space.
- Morphisms - paths in X, identified up to endpoint-preserving homotopy.
paths from x to y in the space.