Maths - Groupoid

There are different ways to define a groupoid such as:

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:

Algebraic definition

I am trying to visualise the algebraic definition: same definition as a group but with a partial function replacing the binary operation.

cayleyGraph This is a conventional Cayley graph.
cayley graph part Here I have made the functions 'f' and 'g' partial.

Catagorical definition

I am trying to visualise the catagorical definition: a category in which every morphism is invertible.

permutation group This is an attempt to represent a permutation group with 3 elements and permutations shown in red, green and blue.
groupoid So how can we show multiple nodes? Ill start by assuming that isomorphism requires the elements are one-to-one?

ω-groupoid

An ω-groupoid is an ω-category in which all higher morphisms are equivalences.

A ω-category (or ∞-category) is a higher order category.

Fundamental Groupoid - Topology

The fundamental groupoid of a space is a groupoid with:

ω-groupoid topological space  
type space  
term point  
path(x,y)

Id(x,y)

paths from x to y in the space.

 

 


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see also:

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

cover Modern Graph Theory (Graduate Texts in Mathematics, 184)

Terminology and Notation

Specific to this page here:

 

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