Groupoid (category theory)

In mathematical category theory , a groupoid is a small category in which every morphism is an isomorphism . To put it in more detail, a groupoid consists of:

• A set of objects ; ${\displaystyle G_{0}}$
• For each pair of objects from a set of morphisms (or arrows ) from to . Instead of writing (based on the usual notation for functions ) ${\displaystyle x,y\in G_{0}}$ ${\displaystyle G(x,y)}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle f\in G(x,y)}$ ${\displaystyle f:x\rightarrow y}$
• For every object there is an excellent element ; ${\displaystyle x}$ ${\displaystyle \mathrm {id} _{x}\in G(x,x)}$
• For every three objects a mapping is given, called a concatenation; ${\displaystyle x,y,z\in G_{0}}$ ${\displaystyle \mathrm {comp} _{x,y,z}:G(y,z)\times G(x,y)\rightarrow G(x,z):(g,f)\mapsto g\circ f}$
• For every two objects there is a function called inversion. ${\displaystyle x,y\in G_{0}}$ ${\displaystyle \mathrm {inv} :G(x,y)\rightarrow G(y,x):f\mapsto f^{-1}}$

These structures must be compatible with one another in the following ways:

• The following applies to all (associativity); ${\displaystyle f:x\to y,g:y\to z,h:z\to w}$ ${\displaystyle (h\circ g)\circ f=h\circ (g\circ f)}$
• The following applies to all : and (neutral elements); ${\displaystyle f:x\to y}$ ${\displaystyle f\circ \mathrm {id} _{x}=f}$ ${\displaystyle \mathrm {id} _{y}\circ f=f}$
• The following applies to all : as well as (Inverse). ${\displaystyle f:x\to y}$ ${\displaystyle f\circ f^{-1}=\mathrm {id} _{y}}$ ${\displaystyle f^{-1}\circ f=\mathrm {id} _{x}}$

The three compatibility conditions are similar to the group axioms . It's not a coincidence. A groupoid with exactly one object is nothing more than a group. In this sense, the term groupoid is a generalization of the term group.

Application and examples

• In algebraic topology , the fundamental groupoid is associated to a topological space . The objects of the groupoid are the points of . The arrows are the homotopies classes (relative start and end points) of continuous imaging , where the beginning point is the source and the end point of the destination. See also the article on fundamental groups . As in this case, groupoids often carry additional structures such as a topology on the set of objects and arrows. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle f\colon [0,1]\longrightarrow X}$ ${\displaystyle f(0)}$ ${\displaystyle f(1)}$

• In loop quantum gravity , groupoids are used to describe the spin networks .

• In crystallography , groupoids are used to describe the symmetry of polytype structures.

• Each group is a groupoid with an object and the group elements as arrows.

• A groupoid arises from any small category if only the arrows are considered, which are isomorphisms.

• Every equivalence relation is a groupoid with the elements of the carrier set as objects, so that a morphism exists between two objects if and only if they are equivalent.

properties

• The category of all gruppoids with functors as morphisms is a subcategory of Cat , the category of all small categories.