Groupoid

A groupoid, in mathematics, especially in category theory and in homotopy, is a concept that simultaneously generalizes groups, equivalence relations in sets, and actions of groups in sets.
Frequently, they are used to capture information about geometric objects such as manifolds.

The term "groupoid" it is also used for a magma: an array with any kind of binary operation on it. We will not use that term for such a concept in this article.

Definitions

From a category point of view, a groupoid is simply one of them in which every morphism is an isomorphism (that is, it is inversible).

Alternatively it is possible to give the following equivalent definition: a groupid G M{displaystyle Grightarrows M} consists of

• Two sets G{displaystyle G^{}}the group and M{displaystyle M}The base.
• s,t:G→ → M{displaystyle s,t:Gto M} super-yctive functions. s{displaystyle s} is called source or source projection and t{displaystyle t} is called the final projection or destination.
• An application 1:M→ → G{displaystyle 1:Mto G}, x 1x{displaystyle xmapsto 1_{x}}the application of inclusion or identity.
• Yeah. G↓ ↓ G:={(MIL MIL ,roga roga )한 한 G× × G:t(roga roga )=s(MIL MIL )!{displaystyle G*G:={(etaxi)in Gtimes G:t(xi)=s(eta)}}, then there is a partial multiplication G↓ ↓ G→ → G{displaystyle G*Gto G} which meets the following conditions

• s(hg)=s(g){displaystyle s(hg)=s(g)}, t(hg)=t(h){displaystyle t(hg)=t(h)}For everything (h,g)한 한 G↓ ↓ G{displaystyle (h,g)in G*G}.
• Associative.
• s(1x)=t(1x)=x{displaystyle s(1_{x})=t(1_{x})=x}For everything x한 한 M{displaystyle xin M}.
• g1s(g)=1t(g)g=g{displaystyle g1_{s(g)}=1_{t(g)}g=g}For everything g한 한 G{displaystyle gin G}.
• For everything g한 한 G{displaystyle gin G}exists g− − 1한 한 G{displaystyle g^{-1}in G}, such that g− − 1g=1s(g){displaystyle g^{-1}g=1_{s(g)}} and gg− − 1=1t(g){displaystyle gg^{-1}=1_{t(g)}}.

Examples

• Groups are trivial-based groupids.

• Sea M{displaystyle M} set, G{displaystyle G} group, s:M× × G× × M→ → M{displaystyle s:Mtimes Gtimes Mto M} the projection to the third coordinate, t:M× × G× × M→ → M{displaystyle t:Mtimes Gtimes Mto M} the projection to the first coordinate, 1:M→ → M× × G× × M{displaystyle 1:Mto Mtimes Gtimes M} given x (x,1,x){displaystyle xmapsto (x,1,x)}. Partial and reverse multiplication given by (z,h,and)(and,g,x)=(z,hg,x){displaystyle (z,h,y)(y,g,x)=(z,hg,x)}, (and,g,x)− − 1=(x,g− − 1,and){displaystyle ,,(y,g,x)^{-1}=(x,g^{-1},y)}respectively. This turns out to be a denouncing group M× × G× × M M{displaystyle Mtimes Gtimes Mrightarrows M} and is called trivial group on M{displaystyle M} group G{displaystyle G}.

• In topology, the fundamental group of a topological space X{displaystyle X,} is the set of curve homotopia classes with the operation juxtapose classes (when it is possible to do so). It is represented with the expression π π 1(X){displaystyle pi _{1}(X)}.

The homotopia classes are the equivalence classes determined by the relationship of being homotopic, that is, two curves α α ,β β :[chuckles]0,1]→ → X{displaystyle alphabeta:[0.1]to X,} such as α α (0)=β β (0){displaystyle alpha (0)=beta (0),} and α α (1)=β β (1){displaystyle alpha (1)=beta (1),}; they are homotopic if there is a continuous application H:[chuckles]0,1]× × [chuckles]0,1]→ → X{displaystyle H:[0.1]times [0.1]to X,} such as

H(k,0)=α α (k){displaystyle H(k,0)=alpha (k),}, H(k,1)=β β (k){displaystyle H(k,1)=beta (k),}

H(0,r)=α α (0)=β β (0){displaystyle H(0,r)=alpha (0)=beta (0),}, H(1,r)=α α (1)=β β (1){displaystyle H(1,r)=alpha (1)=beta (1),}.

In this case the base is the space X{displaystyle X,}, the origin and end applications are the origin and end of each curve. Identity application is 1x(r)=x{displaystyle 1_{x}(r)=x}, i.e. the kind of equivalence of the constant curve in x{displaystyle x} and the reverse is to walk the curve in the opposite direction.

It is clear that the fundamental groupide includes all the fundamental groups and integrates them into a single structure, which ultimately becomes more natural for the study of homotopia.

• Yeah. X{displaystyle X} It's a set and {displaystyle simeq } is an equivalence ratio X{displaystyle X}Then we can form a group which represents this equivalence ratio as follows: the base is X{displaystyle X}and for any two elements x,and{displaystyle x,,y} in X{displaystyle X}There's only one morphism since x{displaystyle x} until and{displaystyle and} Yes and only if x and{displaystyle xsimeq y}.

Lie groupoids and Lie algebroids

When studying geometric objects, the groupoids that occur often carry some differentiable structure, becoming Lie groupoids. These can be studied in terms of Lie algebroids, in analogy to the relationship between Lie groups and Lie algebras.