Basic theory of abstract groupoids and their linear representations. Applications.
Laiachi El Kaoutit.
E-mail: [email protected] URL: https://www.ugr.es/~kaoutit/
Universidad de Granada. Campus Universitario de Ceuta. Spain
TETUÁN, December 2018.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 1 / 42 2 Groupoids and Symmetries. Abstract groupoids: Definition and examples. Left and right stars and the isotropy groups. Symmetries and groupoids. Homogeneous vector bundles
3 Linear representations of abstract groupoids. The context, motivations and overviews. Linear representations of groupoids. The representative functions functor. The contravariant adjunction.
Contents
1 Rappels on categories and equivalences. Categories, functors and natural transformations. Adjunctions and equivalence between categories. The pull-back in general categories.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 2 / 42 3 Linear representations of abstract groupoids. The context, motivations and overviews. Linear representations of groupoids. The representative functions functor. The contravariant adjunction.
Contents
1 Rappels on categories and equivalences. Categories, functors and natural transformations. Adjunctions and equivalence between categories. The pull-back in general categories.
2 Groupoids and Symmetries. Abstract groupoids: Definition and examples. Left and right stars and the isotropy groups. Symmetries and groupoids. Homogeneous vector bundles
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 2 / 42 Contents
1 Rappels on categories and equivalences. Categories, functors and natural transformations. Adjunctions and equivalence between categories. The pull-back in general categories.
2 Groupoids and Symmetries. Abstract groupoids: Definition and examples. Left and right stars and the isotropy groups. Symmetries and groupoids. Homogeneous vector bundles
3 Linear representations of abstract groupoids. The context, motivations and overviews. Linear representations of groupoids. The representative functions functor. The contravariant adjunction.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 2 / 42 1 Rappels on categories and equivalences. Categories, functors and natural transformations. Adjunctions and equivalence between categories. The pull-back in general categories.
2 Groupoids and Symmetries. Abstract groupoids: Definition and examples. Left and right stars and the isotropy groups. Symmetries and groupoids. Homogeneous vector bundles
3 Linear representations of abstract groupoids. The context, motivations and overviews. Linear representations of groupoids. The representative functions functor. The contravariant adjunction.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 3 / 42 A category (precisely a Hom-set category) consists of the following data:
A collections of objects C0 (or the ’class’ of object ob(C)),
for any two objects X, Y in C0, there is a set of arrows, or morphisms denoted by C(X, Y), where any of the sets C(X, X) contains a special
element denote 1X called the identity morphism of X. The set of all morphism is denoted by C1.
Together with an associative and unital composition law:
HomC(X, Y) × HomC(Y, Z) −→ HomC(X, Z), ( f, g) 7−→ g ◦ f
If the ’class’ of object C0 is actually a set, then C is said to be a small category.
Categories, functors and natural transformations.
Categories: definition and examples.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 4 / 42 A collections of objects C0 (or the ’class’ of object ob(C)),
for any two objects X, Y in C0, there is a set of arrows, or morphisms denoted by C(X, Y), where any of the sets C(X, X) contains a special
element denote 1X called the identity morphism of X. The set of all morphism is denoted by C1.
Together with an associative and unital composition law:
HomC(X, Y) × HomC(Y, Z) −→ HomC(X, Z), ( f, g) 7−→ g ◦ f
If the ’class’ of object C0 is actually a set, then C is said to be a small category.
Categories, functors and natural transformations.
Categories: definition and examples. A category (precisely a Hom-set category) consists of the following data:
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 4 / 42 for any two objects X, Y in C0, there is a set of arrows, or morphisms denoted by C(X, Y), where any of the sets C(X, X) contains a special
element denote 1X called the identity morphism of X. The set of all morphism is denoted by C1.
Together with an associative and unital composition law:
HomC(X, Y) × HomC(Y, Z) −→ HomC(X, Z), ( f, g) 7−→ g ◦ f
If the ’class’ of object C0 is actually a set, then C is said to be a small category.
Categories, functors and natural transformations.
Categories: definition and examples. A category (precisely a Hom-set category) consists of the following data:
A collections of objects C0 (or the ’class’ of object ob(C)),
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 4 / 42 Together with an associative and unital composition law:
HomC(X, Y) × HomC(Y, Z) −→ HomC(X, Z), ( f, g) 7−→ g ◦ f
If the ’class’ of object C0 is actually a set, then C is said to be a small category.
Categories, functors and natural transformations.
Categories: definition and examples. A category (precisely a Hom-set category) consists of the following data:
A collections of objects C0 (or the ’class’ of object ob(C)),
for any two objects X, Y in C0, there is a set of arrows, or morphisms denoted by C(X, Y), where any of the sets C(X, X) contains a special
element denote 1X called the identity morphism of X. The set of all morphism is denoted by C1.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 4 / 42 If the ’class’ of object C0 is actually a set, then C is said to be a small category.
Categories, functors and natural transformations.
Categories: definition and examples. A category (precisely a Hom-set category) consists of the following data:
A collections of objects C0 (or the ’class’ of object ob(C)),
for any two objects X, Y in C0, there is a set of arrows, or morphisms denoted by C(X, Y), where any of the sets C(X, X) contains a special
element denote 1X called the identity morphism of X. The set of all morphism is denoted by C1.
Together with an associative and unital composition law:
HomC(X, Y) × HomC(Y, Z) −→ HomC(X, Z), ( f, g) 7−→ g ◦ f
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 4 / 42 Categories, functors and natural transformations.
Categories: definition and examples. A category (precisely a Hom-set category) consists of the following data:
A collections of objects C0 (or the ’class’ of object ob(C)),
for any two objects X, Y in C0, there is a set of arrows, or morphisms denoted by C(X, Y), where any of the sets C(X, X) contains a special
element denote 1X called the identity morphism of X. The set of all morphism is denoted by C1.
Together with an associative and unital composition law:
HomC(X, Y) × HomC(Y, Z) −→ HomC(X, Z), ( f, g) 7−→ g ◦ f
If the ’class’ of object C0 is actually a set, then C is said to be a small category.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 4 / 42 A category with one object is a monoid. If every arrow is invertible, then we have a group, instead. Any directed graph can be considered as a category, The collection of all sets and maps between them form the category S ets. Abelian groups and their morphisms from the category Ab. More general for any ring A right A-modules and their morphisms form a
category ModA.
Given a group G, we have the category Repk(G) of k-representations of G, as well as the category of G-sets. Given a poset (P, ≤) one can consider the category whose collection of objects is the set P it self and P(p, q) contains one element if p ≤ q and empty otherwise. Take for instance a topological space X and consider it poset Open(X) of all open subsets with inclusion as partial order.
Categories, functors and natural transformations. Examples of categories.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 5 / 42 Any directed graph can be considered as a category, The collection of all sets and maps between them form the category S ets. Abelian groups and their morphisms from the category Ab. More general for any ring A right A-modules and their morphisms form a
category ModA.
Given a group G, we have the category Repk(G) of k-representations of G, as well as the category of G-sets. Given a poset (P, ≤) one can consider the category whose collection of objects is the set P it self and P(p, q) contains one element if p ≤ q and empty otherwise. Take for instance a topological space X and consider it poset Open(X) of all open subsets with inclusion as partial order.
Categories, functors and natural transformations. Examples of categories. A category with one object is a monoid. If every arrow is invertible, then we have a group, instead.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 5 / 42 The collection of all sets and maps between them form the category S ets. Abelian groups and their morphisms from the category Ab. More general for any ring A right A-modules and their morphisms form a
category ModA.
Given a group G, we have the category Repk(G) of k-representations of G, as well as the category of G-sets. Given a poset (P, ≤) one can consider the category whose collection of objects is the set P it self and P(p, q) contains one element if p ≤ q and empty otherwise. Take for instance a topological space X and consider it poset Open(X) of all open subsets with inclusion as partial order.
Categories, functors and natural transformations. Examples of categories. A category with one object is a monoid. If every arrow is invertible, then we have a group, instead. Any directed graph can be considered as a category,
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 5 / 42 Abelian groups and their morphisms from the category Ab. More general for any ring A right A-modules and their morphisms form a
category ModA.
Given a group G, we have the category Repk(G) of k-representations of G, as well as the category of G-sets. Given a poset (P, ≤) one can consider the category whose collection of objects is the set P it self and P(p, q) contains one element if p ≤ q and empty otherwise. Take for instance a topological space X and consider it poset Open(X) of all open subsets with inclusion as partial order.
Categories, functors and natural transformations. Examples of categories. A category with one object is a monoid. If every arrow is invertible, then we have a group, instead. Any directed graph can be considered as a category, The collection of all sets and maps between them form the category S ets.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 5 / 42 More general for any ring A right A-modules and their morphisms form a
category ModA.
Given a group G, we have the category Repk(G) of k-representations of G, as well as the category of G-sets. Given a poset (P, ≤) one can consider the category whose collection of objects is the set P it self and P(p, q) contains one element if p ≤ q and empty otherwise. Take for instance a topological space X and consider it poset Open(X) of all open subsets with inclusion as partial order.
Categories, functors and natural transformations. Examples of categories. A category with one object is a monoid. If every arrow is invertible, then we have a group, instead. Any directed graph can be considered as a category, The collection of all sets and maps between them form the category S ets. Abelian groups and their morphisms from the category Ab.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 5 / 42 Given a group G, we have the category Repk(G) of k-representations of G, as well as the category of G-sets. Given a poset (P, ≤) one can consider the category whose collection of objects is the set P it self and P(p, q) contains one element if p ≤ q and empty otherwise. Take for instance a topological space X and consider it poset Open(X) of all open subsets with inclusion as partial order.
Categories, functors and natural transformations. Examples of categories. A category with one object is a monoid. If every arrow is invertible, then we have a group, instead. Any directed graph can be considered as a category, The collection of all sets and maps between them form the category S ets. Abelian groups and their morphisms from the category Ab. More general for any ring A right A-modules and their morphisms form a
category ModA.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 5 / 42 Given a poset (P, ≤) one can consider the category whose collection of objects is the set P it self and P(p, q) contains one element if p ≤ q and empty otherwise. Take for instance a topological space X and consider it poset Open(X) of all open subsets with inclusion as partial order.
Categories, functors and natural transformations. Examples of categories. A category with one object is a monoid. If every arrow is invertible, then we have a group, instead. Any directed graph can be considered as a category, The collection of all sets and maps between them form the category S ets. Abelian groups and their morphisms from the category Ab. More general for any ring A right A-modules and their morphisms form a
category ModA.
Given a group G, we have the category Repk(G) of k-representations of G, as well as the category of G-sets.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 5 / 42 Categories, functors and natural transformations. Examples of categories. A category with one object is a monoid. If every arrow is invertible, then we have a group, instead. Any directed graph can be considered as a category, The collection of all sets and maps between them form the category S ets. Abelian groups and their morphisms from the category Ab. More general for any ring A right A-modules and their morphisms form a
category ModA.
Given a group G, we have the category Repk(G) of k-representations of G, as well as the category of G-sets. Given a poset (P, ≤) one can consider the category whose collection of objects is the set P it self and P(p, q) contains one element if p ≤ q and empty otherwise. Take for instance a topological space X and consider it poset Open(X) of all open subsets with inclusion as partial order.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 5 / 42 Let X be a topological space and k a topological base field (e.g., k = R or C). A k-vector bundle E = (E, π) over X consists of:
1 a family {Ex}x ∈ X of finite-dimensional k-vector spaces,
2 U The disjoint union E = x ∈ X Ex admits a topology, which induces the natural
topology on each (fibre) Ex, such that the canonical projection π : E → X is continuous.
3 for every point x ∈ X, there exist a neighbourhood U of x, finite-dimensional k-vector space V and a homeomorphism ϕ : U × V → π−1(U) such that the diagram ϕ U × V / π−1(U)
pr1 π ' U w |π−1(U)
commutes, and such that for every point the obvious map ϕy : V → Ey is k-linear.
such that EU is a isomorphic to a trivial bundle.
Morphism of vector bundles are intuitively defined and the category VBk(X) so is obtained, is referred to as the category of k-vector bundle over X.
Categories, functors and natural transformations.
More examples of categories.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 6 / 42 Categories, functors and natural transformations.
More examples of categories. Let X be a topological space and k a topological base field (e.g., k = R or C). A k-vector bundle E = (E, π) over X consists of:
1 a family {Ex}x ∈ X of finite-dimensional k-vector spaces,
2 U The disjoint union E = x ∈ X Ex admits a topology, which induces the natural
topology on each (fibre) Ex, such that the canonical projection π : E → X is continuous.
3 for every point x ∈ X, there exist a neighbourhood U of x, finite-dimensional k-vector space V and a homeomorphism ϕ : U × V → π−1(U) such that the diagram ϕ U × V / π−1(U)
pr1 π ' U w |π−1(U)
commutes, and such that for every point the obvious map ϕy : V → Ey is k-linear.
such that EU is a isomorphic to a trivial bundle.
Morphism of vector bundles are intuitively defined and the category VBk(X) so is obtained, is referred to as the category of k-vector bundle over X.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 6 / 42 A functor is a kind of "morphism” between two categories. Precisely, a functor F : C → D consists of the following data:
An assignment F0 : C0 → D0,
A map F1 : C(X, Y) −→ D(F0(X), F0(Y)) which satisfies
F ◦ F ◦ F F 1(g f ) = 1(g) 1( f )(covariant), (1X ) = 1F0(X).
Examples of functors. The way in which we forget the structure of an abelian group and take its underlying set is a functor O : Ab → S ets. Similarly, we have:
O : Repk(G) → Vectk and O : ModA → Ab.
The identity functor idC of a category C. For an object X in C, we have the covariant functor C(X, −): C → S ets and the contravariant functor: C(−, X): C → S ets.
Let k|k0 be a field extension and V a k-vector space. We can then − ⊗ → consider the functor k0 V : Vectk0 Vectk.
Categories, functors and natural transformations. Functors between categories.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 7 / 42 Precisely, a functor F : C → D consists of the following data:
An assignment F0 : C0 → D0,
A map F1 : C(X, Y) −→ D(F0(X), F0(Y)) which satisfies
F ◦ F ◦ F F 1(g f ) = 1(g) 1( f )(covariant), (1X ) = 1F0(X).
Examples of functors. The way in which we forget the structure of an abelian group and take its underlying set is a functor O : Ab → S ets. Similarly, we have:
O : Repk(G) → Vectk and O : ModA → Ab.
The identity functor idC of a category C. For an object X in C, we have the covariant functor C(X, −): C → S ets and the contravariant functor: C(−, X): C → S ets.
Let k|k0 be a field extension and V a k-vector space. We can then − ⊗ → consider the functor k0 V : Vectk0 Vectk.
Categories, functors and natural transformations.
Functors between categories. A functor is a kind of "morphism” between two categories.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 7 / 42 An assignment F0 : C0 → D0,
A map F1 : C(X, Y) −→ D(F0(X), F0(Y)) which satisfies
F ◦ F ◦ F F 1(g f ) = 1(g) 1( f )(covariant), (1X ) = 1F0(X).
Examples of functors. The way in which we forget the structure of an abelian group and take its underlying set is a functor O : Ab → S ets. Similarly, we have:
O : Repk(G) → Vectk and O : ModA → Ab.
The identity functor idC of a category C. For an object X in C, we have the covariant functor C(X, −): C → S ets and the contravariant functor: C(−, X): C → S ets.
Let k|k0 be a field extension and V a k-vector space. We can then − ⊗ → consider the functor k0 V : Vectk0 Vectk.
Categories, functors and natural transformations.
Functors between categories. A functor is a kind of "morphism” between two categories. Precisely, a functor F : C → D consists of the following data:
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 7 / 42 A map F1 : C(X, Y) −→ D(F0(X), F0(Y)) which satisfies
F ◦ F ◦ F F 1(g f ) = 1(g) 1( f )(covariant), (1X ) = 1F0(X).
Examples of functors. The way in which we forget the structure of an abelian group and take its underlying set is a functor O : Ab → S ets. Similarly, we have:
O : Repk(G) → Vectk and O : ModA → Ab.
The identity functor idC of a category C. For an object X in C, we have the covariant functor C(X, −): C → S ets and the contravariant functor: C(−, X): C → S ets.
Let k|k0 be a field extension and V a k-vector space. We can then − ⊗ → consider the functor k0 V : Vectk0 Vectk.
Categories, functors and natural transformations.
Functors between categories. A functor is a kind of "morphism” between two categories. Precisely, a functor F : C → D consists of the following data:
An assignment F0 : C0 → D0,
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 7 / 42 The way in which we forget the structure of an abelian group and take its underlying set is a functor O : Ab → S ets. Similarly, we have:
O : Repk(G) → Vectk and O : ModA → Ab.
The identity functor idC of a category C. For an object X in C, we have the covariant functor C(X, −): C → S ets and the contravariant functor: C(−, X): C → S ets.
Let k|k0 be a field extension and V a k-vector space. We can then − ⊗ → consider the functor k0 V : Vectk0 Vectk.
Categories, functors and natural transformations.
Functors between categories. A functor is a kind of "morphism” between two categories. Precisely, a functor F : C → D consists of the following data:
An assignment F0 : C0 → D0,
A map F1 : C(X, Y) −→ D(F0(X), F0(Y)) which satisfies
F ◦ F ◦ F F 1(g f ) = 1(g) 1( f )(covariant), (1X ) = 1F0(X).
Examples of functors.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 7 / 42 The identity functor idC of a category C. For an object X in C, we have the covariant functor C(X, −): C → S ets and the contravariant functor: C(−, X): C → S ets.
Let k|k0 be a field extension and V a k-vector space. We can then − ⊗ → consider the functor k0 V : Vectk0 Vectk.
Categories, functors and natural transformations.
Functors between categories. A functor is a kind of "morphism” between two categories. Precisely, a functor F : C → D consists of the following data:
An assignment F0 : C0 → D0,
A map F1 : C(X, Y) −→ D(F0(X), F0(Y)) which satisfies
F ◦ F ◦ F F 1(g f ) = 1(g) 1( f )(covariant), (1X ) = 1F0(X).
Examples of functors. The way in which we forget the structure of an abelian group and take its underlying set is a functor O : Ab → S ets. Similarly, we have:
O : Repk(G) → Vectk and O : ModA → Ab.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 7 / 42 Let k|k0 be a field extension and V a k-vector space. We can then − ⊗ → consider the functor k0 V : Vectk0 Vectk.
Categories, functors and natural transformations.
Functors between categories. A functor is a kind of "morphism” between two categories. Precisely, a functor F : C → D consists of the following data:
An assignment F0 : C0 → D0,
A map F1 : C(X, Y) −→ D(F0(X), F0(Y)) which satisfies
F ◦ F ◦ F F 1(g f ) = 1(g) 1( f )(covariant), (1X ) = 1F0(X).
Examples of functors. The way in which we forget the structure of an abelian group and take its underlying set is a functor O : Ab → S ets. Similarly, we have:
O : Repk(G) → Vectk and O : ModA → Ab.
The identity functor idC of a category C. For an object X in C, we have the covariant functor C(X, −): C → S ets and the contravariant functor: C(−, X): C → S ets.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 7 / 42 Categories, functors and natural transformations.
Functors between categories. A functor is a kind of "morphism” between two categories. Precisely, a functor F : C → D consists of the following data:
An assignment F0 : C0 → D0,
A map F1 : C(X, Y) −→ D(F0(X), F0(Y)) which satisfies
F ◦ F ◦ F F 1(g f ) = 1(g) 1( f )(covariant), (1X ) = 1F0(X).
Examples of functors. The way in which we forget the structure of an abelian group and take its underlying set is a functor O : Ab → S ets. Similarly, we have:
O : Repk(G) → Vectk and O : ModA → Ab.
The identity functor idC of a category C. For an object X in C, we have the covariant functor C(X, −): C → S ets and the contravariant functor: C(−, X): C → S ets.
Let k|k0 be a field extension and V a k-vector space. We can then − ⊗ → consider the functor k0 V : Vectk0 Vectk.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 7 / 42 Given two functors F , G : C → D, a natural transformation η : F → G is the following data and constraints:
A collection of morphisms ηX : F (X) → G(X), For any morphism f : X → Y in C the diagram commutes:
η F (X) X / G(X)
F ( f ) G( f ) ηY F (Y) / G(Y) Examples of natural transformation. Let F : C → D be a functor and consider the associated functor D(F(−), +): Cop × D → S ets. Then any arrow f : X → Y in C1 determines a natural transformation
ζ− : D(F(Y), −) 7−→ D(F(X), −), ζP : D(F(Y), P) 7−→ D(F(X), P), g 7−→ g ◦ F ( f )
Categories, functors and natural transformations.
Natural transformations.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 8 / 42 A collection of morphisms ηX : F (X) → G(X), For any morphism f : X → Y in C the diagram commutes:
η F (X) X / G(X)
F ( f ) G( f ) ηY F (Y) / G(Y) Examples of natural transformation. Let F : C → D be a functor and consider the associated functor D(F(−), +): Cop × D → S ets. Then any arrow f : X → Y in C1 determines a natural transformation
ζ− : D(F(Y), −) 7−→ D(F(X), −), ζP : D(F(Y), P) 7−→ D(F(X), P), g 7−→ g ◦ F ( f )
Categories, functors and natural transformations.
Natural transformations. Given two functors F , G : C → D, a natural transformation η : F → G is the following data and constraints:
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 8 / 42 For any morphism f : X → Y in C the diagram commutes:
η F (X) X / G(X)
F ( f ) G( f ) ηY F (Y) / G(Y) Examples of natural transformation. Let F : C → D be a functor and consider the associated functor D(F(−), +): Cop × D → S ets. Then any arrow f : X → Y in C1 determines a natural transformation
ζ− : D(F(Y), −) 7−→ D(F(X), −), ζP : D(F(Y), P) 7−→ D(F(X), P), g 7−→ g ◦ F ( f )
Categories, functors and natural transformations.
Natural transformations. Given two functors F , G : C → D, a natural transformation η : F → G is the following data and constraints:
A collection of morphisms ηX : F (X) → G(X),
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 8 / 42 Let F : C → D be a functor and consider the associated functor D(F(−), +): Cop × D → S ets. Then any arrow f : X → Y in C1 determines a natural transformation
ζ− : D(F(Y), −) 7−→ D(F(X), −), ζP : D(F(Y), P) 7−→ D(F(X), P), g 7−→ g ◦ F ( f )
Categories, functors and natural transformations.
Natural transformations. Given two functors F , G : C → D, a natural transformation η : F → G is the following data and constraints:
A collection of morphisms ηX : F (X) → G(X), For any morphism f : X → Y in C the diagram commutes:
η F (X) X / G(X)
F ( f ) G( f ) ηY F (Y) / G(Y) Examples of natural transformation.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 8 / 42 Then any arrow f : X → Y in C1 determines a natural transformation
ζ− : D(F(Y), −) 7−→ D(F(X), −), ζP : D(F(Y), P) 7−→ D(F(X), P), g 7−→ g ◦ F ( f )
Categories, functors and natural transformations.
Natural transformations. Given two functors F , G : C → D, a natural transformation η : F → G is the following data and constraints:
A collection of morphisms ηX : F (X) → G(X), For any morphism f : X → Y in C the diagram commutes:
η F (X) X / G(X)
F ( f ) G( f ) ηY F (Y) / G(Y) Examples of natural transformation. Let F : C → D be a functor and consider the associated functor D(F(−), +): Cop × D → S ets.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 8 / 42 Categories, functors and natural transformations.
Natural transformations. Given two functors F , G : C → D, a natural transformation η : F → G is the following data and constraints:
A collection of morphisms ηX : F (X) → G(X), For any morphism f : X → Y in C the diagram commutes:
η F (X) X / G(X)
F ( f ) G( f ) ηY F (Y) / G(Y) Examples of natural transformation. Let F : C → D be a functor and consider the associated functor D(F(−), +): Cop × D → S ets. Then any arrow f : X → Y in C1 determines a natural transformation
ζ− : D(F(Y), −) 7−→ D(F(X), −), ζP : D(F(Y), P) 7−→ D(F(X), P), g 7−→ g ◦ F ( f )
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 8 / 42 An adjunction between two functors F : C → D and G : D → C is a natural isomorphism (on both components)
ΦX, P D(F (X), P) / C(X, G(P)) for any pair of objects X in C and P in D. Notation: F a G
Example of Adjunction.
Consider a field extension k|k0 and V a k-vector space. Then − ⊗ a − k0 V Vectk(V, ). Let Top denote the category of topological spaces and their continuous maps and S S ets the category of simplicial sets. The geometric realization | − | : S S ets → Top and the singular S(−): Top → S sets functors define the adjunction | − | a S. For a given small category C the associated simplicial set / ···C o / C o / C is called the nerve of C. 2 o / 1 / 0
Adjunctions and equivalence between categories.
Adjunction between functors.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 9 / 42 Notation: F a G
Example of Adjunction.
Consider a field extension k|k0 and V a k-vector space. Then − ⊗ a − k0 V Vectk(V, ). Let Top denote the category of topological spaces and their continuous maps and S S ets the category of simplicial sets. The geometric realization | − | : S S ets → Top and the singular S(−): Top → S sets functors define the adjunction | − | a S. For a given small category C the associated simplicial set / ···C o / C o / C is called the nerve of C. 2 o / 1 / 0
Adjunctions and equivalence between categories.
Adjunction between functors. An adjunction between two functors F : C → D and G : D → C is a natural isomorphism (on both components)
ΦX, P D(F (X), P) / C(X, G(P)) for any pair of objects X in C and P in D.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 9 / 42 Example of Adjunction.
Consider a field extension k|k0 and V a k-vector space. Then − ⊗ a − k0 V Vectk(V, ). Let Top denote the category of topological spaces and their continuous maps and S S ets the category of simplicial sets. The geometric realization | − | : S S ets → Top and the singular S(−): Top → S sets functors define the adjunction | − | a S. For a given small category C the associated simplicial set / ···C o / C o / C is called the nerve of C. 2 o / 1 / 0
Adjunctions and equivalence between categories.
Adjunction between functors. An adjunction between two functors F : C → D and G : D → C is a natural isomorphism (on both components)
ΦX, P D(F (X), P) / C(X, G(P)) for any pair of objects X in C and P in D. Notation: F a G
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 9 / 42 Consider a field extension k|k0 and V a k-vector space. Then − ⊗ a − k0 V Vectk(V, ). Let Top denote the category of topological spaces and their continuous maps and S S ets the category of simplicial sets. The geometric realization | − | : S S ets → Top and the singular S(−): Top → S sets functors define the adjunction | − | a S. For a given small category C the associated simplicial set / ···C o / C o / C is called the nerve of C. 2 o / 1 / 0
Adjunctions and equivalence between categories.
Adjunction between functors. An adjunction between two functors F : C → D and G : D → C is a natural isomorphism (on both components)
ΦX, P D(F (X), P) / C(X, G(P)) for any pair of objects X in C and P in D. Notation: F a G
Example of Adjunction.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 9 / 42 Let Top denote the category of topological spaces and their continuous maps and S S ets the category of simplicial sets. The geometric realization | − | : S S ets → Top and the singular S(−): Top → S sets functors define the adjunction | − | a S. For a given small category C the associated simplicial set / ···C o / C o / C is called the nerve of C. 2 o / 1 / 0
Adjunctions and equivalence between categories.
Adjunction between functors. An adjunction between two functors F : C → D and G : D → C is a natural isomorphism (on both components)
ΦX, P D(F (X), P) / C(X, G(P)) for any pair of objects X in C and P in D. Notation: F a G
Example of Adjunction.
Consider a field extension k|k0 and V a k-vector space. Then − ⊗ a − k0 V Vectk(V, ).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 9 / 42 For a given small category C the associated simplicial set / ···C o / C o / C is called the nerve of C. 2 o / 1 / 0
Adjunctions and equivalence between categories.
Adjunction between functors. An adjunction between two functors F : C → D and G : D → C is a natural isomorphism (on both components)
ΦX, P D(F (X), P) / C(X, G(P)) for any pair of objects X in C and P in D. Notation: F a G
Example of Adjunction.
Consider a field extension k|k0 and V a k-vector space. Then − ⊗ a − k0 V Vectk(V, ). Let Top denote the category of topological spaces and their continuous maps and S S ets the category of simplicial sets. The geometric realization | − | : S S ets → Top and the singular S(−): Top → S sets functors define the adjunction | − | a S.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 9 / 42 Adjunctions and equivalence between categories.
Adjunction between functors. An adjunction between two functors F : C → D and G : D → C is a natural isomorphism (on both components)
ΦX, P D(F (X), P) / C(X, G(P)) for any pair of objects X in C and P in D. Notation: F a G
Example of Adjunction.
Consider a field extension k|k0 and V a k-vector space. Then − ⊗ a − k0 V Vectk(V, ). Let Top denote the category of topological spaces and their continuous maps and S S ets the category of simplicial sets. The geometric realization | − | : S S ets → Top and the singular S(−): Top → S sets functors define the adjunction | − | a S. For a given small category C the associated simplicial set / ···C o / C o / C is called the nerve of C. 2 o / 1 / 0
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 9 / 42 The pull-back of this diagram, if it exists, is an object of C denoted by
X f × g Y with the following universal property:
U
' X f × g Y / X f g . Y / Z
Pull-backs
Let C be a category and consider a diagram
X f g Y / Z
of morphisms in C.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 10 / 42 U
' X f × g Y / X f g . Y / Z
Pull-backs
Let C be a category and consider a diagram
X f g Y / Z
of morphisms in C. The pull-back of this diagram, if it exists, is an object of C denoted by
X f × g Y with the following universal property:
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 10 / 42 Pull-backs
Let C be a category and consider a diagram
X f g Y / Z
of morphisms in C. The pull-back of this diagram, if it exists, is an object of C denoted by
X f × g Y with the following universal property:
U
' X f × g Y / X f g . Y / Z
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 10 / 42 1 Rappels on categories and equivalences. Categories, functors and natural transformations. Adjunctions and equivalence between categories. The pull-back in general categories.
2 Groupoids and Symmetries. Abstract groupoids: Definition and examples. Left and right stars and the isotropy groups. Symmetries and groupoids. Homogeneous vector bundles
3 Linear representations of abstract groupoids. The context, motivations and overviews. Linear representations of groupoids. The representative functions functor. The contravariant adjunction.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 11 / 42 A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Whenever pull-backs are allowed groupoid objects are allowed as well:
Categories Groupoid objects Top Topological groupoids Diff-manifolds Lie groupoids Algebraic Varieties Algebraic groupoids Groups Crossed modules (pre) Sheaves (pre) Stacks
Abstract groupoids: Definition and examples.
Abstract groupoid.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Whenever pull-backs are allowed groupoid objects are allowed as well:
Categories Groupoid objects Top Topological groupoids Diff-manifolds Lie groupoids Algebraic Varieties Algebraic groupoids Groups Crossed modules (pre) Sheaves (pre) Stacks
Abstract groupoids: Definition and examples.
Abstract groupoid. A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 Whenever pull-backs are allowed groupoid objects are allowed as well:
Categories Groupoid objects Top Topological groupoids Diff-manifolds Lie groupoids Algebraic Varieties Algebraic groupoids Groups Crossed modules (pre) Sheaves (pre) Stacks
Abstract groupoids: Definition and examples.
Abstract groupoid. A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 Categories Groupoid objects Top Topological groupoids Diff-manifolds Lie groupoids Algebraic Varieties Algebraic groupoids Groups Crossed modules (pre) Sheaves (pre) Stacks
Abstract groupoids: Definition and examples.
Abstract groupoid. A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Whenever pull-backs are allowed groupoid objects are allowed as well:
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 Top Topological groupoids Diff-manifolds Lie groupoids Algebraic Varieties Algebraic groupoids Groups Crossed modules (pre) Sheaves (pre) Stacks
Abstract groupoids: Definition and examples.
Abstract groupoid. A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Whenever pull-backs are allowed groupoid objects are allowed as well:
Categories Groupoid objects
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 Diff-manifolds Lie groupoids Algebraic Varieties Algebraic groupoids Groups Crossed modules (pre) Sheaves (pre) Stacks
Abstract groupoids: Definition and examples.
Abstract groupoid. A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Whenever pull-backs are allowed groupoid objects are allowed as well:
Categories Groupoid objects Top Topological groupoids
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 Algebraic Varieties Algebraic groupoids Groups Crossed modules (pre) Sheaves (pre) Stacks
Abstract groupoids: Definition and examples.
Abstract groupoid. A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Whenever pull-backs are allowed groupoid objects are allowed as well:
Categories Groupoid objects Top Topological groupoids Diff-manifolds Lie groupoids
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 Groups Crossed modules (pre) Sheaves (pre) Stacks
Abstract groupoids: Definition and examples.
Abstract groupoid. A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Whenever pull-backs are allowed groupoid objects are allowed as well:
Categories Groupoid objects Top Topological groupoids Diff-manifolds Lie groupoids Algebraic Varieties Algebraic groupoids
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 (pre) Sheaves (pre) Stacks
Abstract groupoids: Definition and examples.
Abstract groupoid. A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Whenever pull-backs are allowed groupoid objects are allowed as well:
Categories Groupoid objects Top Topological groupoids Diff-manifolds Lie groupoids Algebraic Varieties Algebraic groupoids Groups Crossed modules
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 Abstract groupoids: Definition and examples.
Abstract groupoid. A groupoid is a small category G where every arrow is invertible (or any morphism is an isomorphism).
That is a pair of sets (G1, G0) and a diagram
/ × s / G o · / G o ι G 2 o / 1 t / 0
· where G2 := G1 s× t G1 / G1 is the multiplication (opposite to the composition), and the rest of the maps are the obvious ones.
Whenever pull-backs are allowed groupoid objects are allowed as well:
Categories Groupoid objects Top Topological groupoids Diff-manifolds Lie groupoids Algebraic Varieties Algebraic groupoids Groups Crossed modules (pre) Sheaves (pre) Stacks
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 12 / 42 Any set can be considered as a discrete category (only identities arrow). This what is known as a trivial groupoid. Any group is a groupoid with one object. The multiplication is that of the group. Thus every arrow is a loop. The groupoid of pairs is a groupoid of the form (X × X, X) with source an target the first and second projection. The multiplication goes as follows: (x, y)(y, z) = (x, z), (x, y)−1 = (y, x). Any equivalence relation R ⊆ X × X defines what is known as the equivalence relation groupoid whose structure is analogue to the previous one. The action groupoid is a groupoid of the form (X × G, X) where X a right G-set. The source is the action while the target is the first projection. The multiplication and the inverse are given by: (x, g)(y, h) = (x, gh), (x, g)−1 = (gx, g−1).
Abstract groupoids: Definition and examples.
Some examples of groupoids.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 13 / 42 Any group is a groupoid with one object. The multiplication is that of the group. Thus every arrow is a loop. The groupoid of pairs is a groupoid of the form (X × X, X) with source an target the first and second projection. The multiplication goes as follows: (x, y)(y, z) = (x, z), (x, y)−1 = (y, x). Any equivalence relation R ⊆ X × X defines what is known as the equivalence relation groupoid whose structure is analogue to the previous one. The action groupoid is a groupoid of the form (X × G, X) where X a right G-set. The source is the action while the target is the first projection. The multiplication and the inverse are given by: (x, g)(y, h) = (x, gh), (x, g)−1 = (gx, g−1).
Abstract groupoids: Definition and examples.
Some examples of groupoids. Any set can be considered as a discrete category (only identities arrow). This what is known as a trivial groupoid.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 13 / 42 The groupoid of pairs is a groupoid of the form (X × X, X) with source an target the first and second projection. The multiplication goes as follows: (x, y)(y, z) = (x, z), (x, y)−1 = (y, x). Any equivalence relation R ⊆ X × X defines what is known as the equivalence relation groupoid whose structure is analogue to the previous one. The action groupoid is a groupoid of the form (X × G, X) where X a right G-set. The source is the action while the target is the first projection. The multiplication and the inverse are given by: (x, g)(y, h) = (x, gh), (x, g)−1 = (gx, g−1).
Abstract groupoids: Definition and examples.
Some examples of groupoids. Any set can be considered as a discrete category (only identities arrow). This what is known as a trivial groupoid. Any group is a groupoid with one object. The multiplication is that of the group. Thus every arrow is a loop.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 13 / 42 Any equivalence relation R ⊆ X × X defines what is known as the equivalence relation groupoid whose structure is analogue to the previous one. The action groupoid is a groupoid of the form (X × G, X) where X a right G-set. The source is the action while the target is the first projection. The multiplication and the inverse are given by: (x, g)(y, h) = (x, gh), (x, g)−1 = (gx, g−1).
Abstract groupoids: Definition and examples.
Some examples of groupoids. Any set can be considered as a discrete category (only identities arrow). This what is known as a trivial groupoid. Any group is a groupoid with one object. The multiplication is that of the group. Thus every arrow is a loop. The groupoid of pairs is a groupoid of the form (X × X, X) with source an target the first and second projection. The multiplication goes as follows: (x, y)(y, z) = (x, z), (x, y)−1 = (y, x).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 13 / 42 The action groupoid is a groupoid of the form (X × G, X) where X a right G-set. The source is the action while the target is the first projection. The multiplication and the inverse are given by: (x, g)(y, h) = (x, gh), (x, g)−1 = (gx, g−1).
Abstract groupoids: Definition and examples.
Some examples of groupoids. Any set can be considered as a discrete category (only identities arrow). This what is known as a trivial groupoid. Any group is a groupoid with one object. The multiplication is that of the group. Thus every arrow is a loop. The groupoid of pairs is a groupoid of the form (X × X, X) with source an target the first and second projection. The multiplication goes as follows: (x, y)(y, z) = (x, z), (x, y)−1 = (y, x). Any equivalence relation R ⊆ X × X defines what is known as the equivalence relation groupoid whose structure is analogue to the previous one.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 13 / 42 Abstract groupoids: Definition and examples.
Some examples of groupoids. Any set can be considered as a discrete category (only identities arrow). This what is known as a trivial groupoid. Any group is a groupoid with one object. The multiplication is that of the group. Thus every arrow is a loop. The groupoid of pairs is a groupoid of the form (X × X, X) with source an target the first and second projection. The multiplication goes as follows: (x, y)(y, z) = (x, z), (x, y)−1 = (y, x). Any equivalence relation R ⊆ X × X defines what is known as the equivalence relation groupoid whose structure is analogue to the previous one. The action groupoid is a groupoid of the form (X × G, X) where X a right G-set. The source is the action while the target is the first projection. The multiplication and the inverse are given by: (x, g)(y, h) = (x, gh), (x, g)−1 = (gx, g−1).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 13 / 42 This seems to be one of the first example of groupoids (in mathematics) which was perhaps discovered by Henri Poincaré. Of course, there are more other examples of groupoids, specially, in differential geometry, and were mostly promoted by Charles Ehresmann. Let X be a topological space, and denoted by [γ] the homotopy equivalence class of a path γ : [0, 1] → X whose source is denoted by s([γ]) := γ(0) and its target by t([γ]) := γ(1). Consider, for every x ∈ X, the class ιx := [ix] of the constant ix path on x. The partial multiplication and the inverse of homotopy-classes are given by: [γ][δ] = [γ · δ] and [γ]−1 = [−γ], where γ(2t) if 0 ≤ t ≤ 1 γ · δ := 2 (−γ)(t) = γ(1 − t) 1 δ(2t − 1) if 2 ≤ t ≤ 1 The groupoid so is constructed is denoted by π(X).
Abstract groupoids: Definition and examples.
The Poincaré groupoid.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 14 / 42 Of course, there are more other examples of groupoids, specially, in differential geometry, and were mostly promoted by Charles Ehresmann. Let X be a topological space, and denoted by [γ] the homotopy equivalence class of a path γ : [0, 1] → X whose source is denoted by s([γ]) := γ(0) and its target by t([γ]) := γ(1). Consider, for every x ∈ X, the class ιx := [ix] of the constant ix path on x. The partial multiplication and the inverse of homotopy-classes are given by: [γ][δ] = [γ · δ] and [γ]−1 = [−γ], where γ(2t) if 0 ≤ t ≤ 1 γ · δ := 2 (−γ)(t) = γ(1 − t) 1 δ(2t − 1) if 2 ≤ t ≤ 1 The groupoid so is constructed is denoted by π(X).
Abstract groupoids: Definition and examples.
The Poincaré groupoid. This seems to be one of the first example of groupoids (in mathematics) which was perhaps discovered by Henri Poincaré.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 14 / 42 Let X be a topological space, and denoted by [γ] the homotopy equivalence class of a path γ : [0, 1] → X whose source is denoted by s([γ]) := γ(0) and its target by t([γ]) := γ(1). Consider, for every x ∈ X, the class ιx := [ix] of the constant ix path on x. The partial multiplication and the inverse of homotopy-classes are given by: [γ][δ] = [γ · δ] and [γ]−1 = [−γ], where γ(2t) if 0 ≤ t ≤ 1 γ · δ := 2 (−γ)(t) = γ(1 − t) 1 δ(2t − 1) if 2 ≤ t ≤ 1 The groupoid so is constructed is denoted by π(X).
Abstract groupoids: Definition and examples.
The Poincaré groupoid. This seems to be one of the first example of groupoids (in mathematics) which was perhaps discovered by Henri Poincaré. Of course, there are more other examples of groupoids, specially, in differential geometry, and were mostly promoted by Charles Ehresmann.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 14 / 42 The partial multiplication and the inverse of homotopy-classes are given by: [γ][δ] = [γ · δ] and [γ]−1 = [−γ], where γ(2t) if 0 ≤ t ≤ 1 γ · δ := 2 (−γ)(t) = γ(1 − t) 1 δ(2t − 1) if 2 ≤ t ≤ 1 The groupoid so is constructed is denoted by π(X).
Abstract groupoids: Definition and examples.
The Poincaré groupoid. This seems to be one of the first example of groupoids (in mathematics) which was perhaps discovered by Henri Poincaré. Of course, there are more other examples of groupoids, specially, in differential geometry, and were mostly promoted by Charles Ehresmann. Let X be a topological space, and denoted by [γ] the homotopy equivalence class of a path γ : [0, 1] → X whose source is denoted by s([γ]) := γ(0) and its target by t([γ]) := γ(1). Consider, for every x ∈ X, the class ιx := [ix] of the constant ix path on x.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 14 / 42 Abstract groupoids: Definition and examples.
The Poincaré groupoid. This seems to be one of the first example of groupoids (in mathematics) which was perhaps discovered by Henri Poincaré. Of course, there are more other examples of groupoids, specially, in differential geometry, and were mostly promoted by Charles Ehresmann. Let X be a topological space, and denoted by [γ] the homotopy equivalence class of a path γ : [0, 1] → X whose source is denoted by s([γ]) := γ(0) and its target by t([γ]) := γ(1). Consider, for every x ∈ X, the class ιx := [ix] of the constant ix path on x. The partial multiplication and the inverse of homotopy-classes are given by: [γ][δ] = [γ · δ] and [γ]−1 = [−γ], where γ(2t) if 0 ≤ t ≤ 1 γ · δ := 2 (−γ)(t) = γ(1 − t) 1 δ(2t − 1) if 2 ≤ t ≤ 1 The groupoid so is constructed is denoted by π(X).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 14 / 42 That is, φ = (φ0, φ1)), where φ0 : H0 → G0 and
φ1 : H1 → G1 satisfying the pertinent compatibility conditions:
φ1 ◦ ι = ι ◦ φ0, φ0 ◦ s = s ◦ φ1, φ0 ◦ t = t ◦ φ1, φ1( f g) = φ1( f )φ1(g),
whenever the multiplication f g in H1 is permitted. Some examples. For any groupoid G then inclusion G(i) ,→ G is a morphism of groupoids. Let ϕ : G → H be a morphism of groups, consider X and Y respectively a right G and H sets with equivariant map f : X → Y. Then we have a morphism of groupoids × / X G o / X f ×ϕ f × / Y H o / Y
Morphism of groupoids.
A morphism of groupoids φ : H → G is a functor between the underlying categories.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 15 / 42 Some examples. For any groupoid G then inclusion G(i) ,→ G is a morphism of groupoids. Let ϕ : G → H be a morphism of groups, consider X and Y respectively a right G and H sets with equivariant map f : X → Y. Then we have a morphism of groupoids × / X G o / X f ×ϕ f × / Y H o / Y
Morphism of groupoids.
A morphism of groupoids φ : H → G is a functor between the
underlying categories. That is, φ = (φ0, φ1)), where φ0 : H0 → G0 and
φ1 : H1 → G1 satisfying the pertinent compatibility conditions:
φ1 ◦ ι = ι ◦ φ0, φ0 ◦ s = s ◦ φ1, φ0 ◦ t = t ◦ φ1, φ1( f g) = φ1( f )φ1(g),
whenever the multiplication f g in H1 is permitted.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 15 / 42 For any groupoid G then inclusion G(i) ,→ G is a morphism of groupoids. Let ϕ : G → H be a morphism of groups, consider X and Y respectively a right G and H sets with equivariant map f : X → Y. Then we have a morphism of groupoids × / X G o / X f ×ϕ f × / Y H o / Y
Morphism of groupoids.
A morphism of groupoids φ : H → G is a functor between the
underlying categories. That is, φ = (φ0, φ1)), where φ0 : H0 → G0 and
φ1 : H1 → G1 satisfying the pertinent compatibility conditions:
φ1 ◦ ι = ι ◦ φ0, φ0 ◦ s = s ◦ φ1, φ0 ◦ t = t ◦ φ1, φ1( f g) = φ1( f )φ1(g),
whenever the multiplication f g in H1 is permitted. Some examples.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 15 / 42 Morphism of groupoids.
A morphism of groupoids φ : H → G is a functor between the
underlying categories. That is, φ = (φ0, φ1)), where φ0 : H0 → G0 and
φ1 : H1 → G1 satisfying the pertinent compatibility conditions:
φ1 ◦ ι = ι ◦ φ0, φ0 ◦ s = s ◦ φ1, φ0 ◦ t = t ◦ φ1, φ1( f g) = φ1( f )φ1(g),
whenever the multiplication f g in H1 is permitted. Some examples. For any groupoid G then inclusion G(i) ,→ G is a morphism of groupoids. Let ϕ : G → H be a morphism of groups, consider X and Y respectively a right G and H sets with equivariant map f : X → Y. Then we have a morphism of groupoids × / X G o / X f ×ϕ f × / Y H o / Y
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 15 / 42 Given G a groupoid and an object x ∈ G0. We define the left star of x as the set of all incoming arrow to x and the right star of x as the set of all out-coming arrows from x:
−1 −1 n o t {x} := g ∈ G1| t(g) = x , s {x} := g ∈ G1| s(g) = x Clearly there is a bijection t−1 {x} −→ s−1 {x}, g 7−→ g−1 . The isotropy group of x is the group of loops above x:
x n o G := g ∈ G1| s(g) = t(g) = x
The fundamental groups π1(X, x) are the isotropy groups of π(X) at the point x. G(i) S Gx → G The isotropy groupoid is the bundle of groups = x ∈ G0 0.
Left and right stars and the isotropy groups.
Left and right stars.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 16 / 42 We define the left star of x as the set of all incoming arrow to x and the right star of x as the set of all out-coming arrows from x:
−1 −1 n o t {x} := g ∈ G1| t(g) = x , s {x} := g ∈ G1| s(g) = x Clearly there is a bijection t−1 {x} −→ s−1 {x}, g 7−→ g−1 . The isotropy group of x is the group of loops above x:
x n o G := g ∈ G1| s(g) = t(g) = x
The fundamental groups π1(X, x) are the isotropy groups of π(X) at the point x. G(i) S Gx → G The isotropy groupoid is the bundle of groups = x ∈ G0 0.
Left and right stars and the isotropy groups.
Left and right stars. Given G a groupoid and an object x ∈ G0.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 16 / 42 Clearly there is a bijection t−1 {x} −→ s−1 {x}, g 7−→ g−1 . The isotropy group of x is the group of loops above x:
x n o G := g ∈ G1| s(g) = t(g) = x
The fundamental groups π1(X, x) are the isotropy groups of π(X) at the point x. G(i) S Gx → G The isotropy groupoid is the bundle of groups = x ∈ G0 0.
Left and right stars and the isotropy groups.
Left and right stars. Given G a groupoid and an object x ∈ G0. We define the left star of x as the set of all incoming arrow to x and the right star of x as the set of all out-coming arrows from x:
−1 −1 n o t {x} := g ∈ G1| t(g) = x , s {x} := g ∈ G1| s(g) = x
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 16 / 42 The isotropy group of x is the group of loops above x:
x n o G := g ∈ G1| s(g) = t(g) = x
The fundamental groups π1(X, x) are the isotropy groups of π(X) at the point x. G(i) S Gx → G The isotropy groupoid is the bundle of groups = x ∈ G0 0.
Left and right stars and the isotropy groups.
Left and right stars. Given G a groupoid and an object x ∈ G0. We define the left star of x as the set of all incoming arrow to x and the right star of x as the set of all out-coming arrows from x:
−1 −1 n o t {x} := g ∈ G1| t(g) = x , s {x} := g ∈ G1| s(g) = x Clearly there is a bijection t−1 {x} −→ s−1 {x}, g 7−→ g−1 .
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 16 / 42 The fundamental groups π1(X, x) are the isotropy groups of π(X) at the point x. G(i) S Gx → G The isotropy groupoid is the bundle of groups = x ∈ G0 0.
Left and right stars and the isotropy groups.
Left and right stars. Given G a groupoid and an object x ∈ G0. We define the left star of x as the set of all incoming arrow to x and the right star of x as the set of all out-coming arrows from x:
−1 −1 n o t {x} := g ∈ G1| t(g) = x , s {x} := g ∈ G1| s(g) = x Clearly there is a bijection t−1 {x} −→ s−1 {x}, g 7−→ g−1 . The isotropy group of x is the group of loops above x:
x n o G := g ∈ G1| s(g) = t(g) = x
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 16 / 42 G(i) S Gx → G The isotropy groupoid is the bundle of groups = x ∈ G0 0.
Left and right stars and the isotropy groups.
Left and right stars. Given G a groupoid and an object x ∈ G0. We define the left star of x as the set of all incoming arrow to x and the right star of x as the set of all out-coming arrows from x:
−1 −1 n o t {x} := g ∈ G1| t(g) = x , s {x} := g ∈ G1| s(g) = x Clearly there is a bijection t−1 {x} −→ s−1 {x}, g 7−→ g−1 . The isotropy group of x is the group of loops above x:
x n o G := g ∈ G1| s(g) = t(g) = x
The fundamental groups π1(X, x) are the isotropy groups of π(X) at the point x.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 16 / 42 Left and right stars and the isotropy groups.
Left and right stars. Given G a groupoid and an object x ∈ G0. We define the left star of x as the set of all incoming arrow to x and the right star of x as the set of all out-coming arrows from x:
−1 −1 n o t {x} := g ∈ G1| t(g) = x , s {x} := g ∈ G1| s(g) = x Clearly there is a bijection t−1 {x} −→ s−1 {x}, g 7−→ g−1 . The isotropy group of x is the group of loops above x:
x n o G := g ∈ G1| s(g) = t(g) = x
The fundamental groups π1(X, x) are the isotropy groups of π(X) at the point x. G(i) S Gx → G The isotropy groupoid is the bundle of groups = x ∈ G0 0.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 16 / 42 The ancient notion of symmetry.
(Vitruvius, 1st Century BC): ”Symmetry is proportioned correspondence of the elements of the work itself, a response, in any given part, of the separate parts to the appearance of the entire figure as a whole. Just as in the human body there is a harmonious quality of shapeliness expressed in terms of the cubit, foot, palm, digit, and other small units, so it is in completing the work of architecture”.
The modern notion of symmetry.
(Hermann Weyl, 1952): ”Given a spatial configuration F, those automorphisms of space which leave F unchanged form a group Γ, and this group describes exactly the symmetry possessed by F”.
Symmetry and groups.
Some classical definition.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 17 / 42 (Vitruvius, 1st Century BC): ”Symmetry is proportioned correspondence of the elements of the work itself, a response, in any given part, of the separate parts to the appearance of the entire figure as a whole. Just as in the human body there is a harmonious quality of shapeliness expressed in terms of the cubit, foot, palm, digit, and other small units, so it is in completing the work of architecture”.
The modern notion of symmetry.
(Hermann Weyl, 1952): ”Given a spatial configuration F, those automorphisms of space which leave F unchanged form a group Γ, and this group describes exactly the symmetry possessed by F”.
Symmetry and groups.
Some classical definition.
The ancient notion of symmetry.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 17 / 42 The modern notion of symmetry.
(Hermann Weyl, 1952): ”Given a spatial configuration F, those automorphisms of space which leave F unchanged form a group Γ, and this group describes exactly the symmetry possessed by F”.
Symmetry and groups.
Some classical definition.
The ancient notion of symmetry.
(Vitruvius, 1st Century BC): ”Symmetry is proportioned correspondence of the elements of the work itself, a response, in any given part, of the separate parts to the appearance of the entire figure as a whole. Just as in the human body there is a harmonious quality of shapeliness expressed in terms of the cubit, foot, palm, digit, and other small units, so it is in completing the work of architecture”.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 17 / 42 (Hermann Weyl, 1952): ”Given a spatial configuration F, those automorphisms of space which leave F unchanged form a group Γ, and this group describes exactly the symmetry possessed by F”.
Symmetry and groups.
Some classical definition.
The ancient notion of symmetry.
(Vitruvius, 1st Century BC): ”Symmetry is proportioned correspondence of the elements of the work itself, a response, in any given part, of the separate parts to the appearance of the entire figure as a whole. Just as in the human body there is a harmonious quality of shapeliness expressed in terms of the cubit, foot, palm, digit, and other small units, so it is in completing the work of architecture”.
The modern notion of symmetry.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 17 / 42 Symmetry and groups.
Some classical definition.
The ancient notion of symmetry.
(Vitruvius, 1st Century BC): ”Symmetry is proportioned correspondence of the elements of the work itself, a response, in any given part, of the separate parts to the appearance of the entire figure as a whole. Just as in the human body there is a harmonious quality of shapeliness expressed in terms of the cubit, foot, palm, digit, and other small units, so it is in completing the work of architecture”.
The modern notion of symmetry.
(Hermann Weyl, 1952): ”Given a spatial configuration F, those automorphisms of space which leave F unchanged form a group Γ, and this group describes exactly the symmetry possessed by F”.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 17 / 42 By considering all possible transformation interchanging the equivalent part, the symmetry of this spatial configuration is governed by the dihedral group D6.
Symmetries and groupoids.
Example of Weyl’s symmetry: The snowflake
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 18 / 42 By considering all possible transformation interchanging the equivalent part, the symmetry of this spatial configuration is governed by the dihedral group D6.
Symmetries and groupoids.
Example of Weyl’s symmetry: The snowflake
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 18 / 42 Symmetries and groupoids.
Example of Weyl’s symmetry: The snowflake
By considering all possible transformation interchanging the equivalent part, the symmetry of this spatial configuration is governed by the dihedral group D6.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 18 / 42 Spectral lines of the Hydrogen Atom
Symmetries and groupoids.
The Hydrogen Transition.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 19 / 42 Spectral lines of the Hydrogen Atom
Symmetries and groupoids.
The Hydrogen Transition.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 19 / 42 Symmetries and groupoids.
The Hydrogen Transition.
Spectral lines of the Hydrogen Atom
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 19 / 42 The different levels of energies E(n)1≤ n ≤7, form a groupoids of pairs, this seems was first observed by Alain Connes and was perhaps one of his motivation to formulate his non commutative geometry.
Symmetries and groupoids.
Groupoid and the birth of non-commutative geometry.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 20 / 42 Symmetries and groupoids.
Groupoid and the birth of non-commutative geometry.
The different levels of energies E(n)1≤ n ≤7, form a groupoids of pairs, this seems was first observed by Alain Connes and was perhaps one of his motivation to formulate his non commutative geometry.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 20 / 42 Consider the space of motions of Carbon Tetrachloride. At equilibrium the carbon atom lies at the center, and the four chlorine atoms at the vertices of a regular tetrahedron.
4 v 4
∈
E
4
v c 3 ∈ E E C ∈ 3 C v 3
v
2 1 ∈ E
1 2 E
∈
1 v 2
Figura: Molecular model of Carbon Tetrachloride. In a small displacement from equilibrium, each of the atoms moves in its own
three-dimensional vector space: E1, E2, E3, E4 and EC .
Symmetries and groupoids. Molecular vibrations and vector bundle.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 21 / 42 4 v 4
∈
E
4
v c 3 ∈ E E C ∈ 3 C v 3
v
2 1 ∈ E
1 2 E
∈
1 v 2
Figura: Molecular model of Carbon Tetrachloride. In a small displacement from equilibrium, each of the atoms moves in its own
three-dimensional vector space: E1, E2, E3, E4 and EC .
Symmetries and groupoids.
Molecular vibrations and vector bundle. Consider the space of motions of Carbon Tetrachloride. At equilibrium the carbon atom lies at the center, and the four chlorine atoms at the vertices of a regular tetrahedron.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 21 / 42 In a small displacement from equilibrium, each of the atoms moves in its own
three-dimensional vector space: E1, E2, E3, E4 and EC .
Symmetries and groupoids.
Molecular vibrations and vector bundle. Consider the space of motions of Carbon Tetrachloride. At equilibrium the carbon atom lies at the center, and the four chlorine atoms at the vertices of a regular tetrahedron.
4 v 4
∈
E
4
v c 3 ∈ E E C ∈ 3 C v 3
v
2 1 ∈ E
1 2 E
∈
1 v 2
Figura: Molecular model of Carbon Tetrachloride.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 21 / 42 Symmetries and groupoids.
Molecular vibrations and vector bundle. Consider the space of motions of Carbon Tetrachloride. At equilibrium the carbon atom lies at the center, and the four chlorine atoms at the vertices of a regular tetrahedron.
4 v 4
∈
E
4
v c 3 ∈ E E C ∈ 3 C v 3
v
2 1 ∈ E
1 2 E
∈
1 v 2
Figura: Molecular model of Carbon Tetrachloride. In a small displacement from equilibrium, each of the atoms moves in its own
three-dimensional vector space: E1, E2, E3, E4 and EC .
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 21 / 42 A displacement of the molecule as a whole moves each of the atoms, and so is a function f such that f (C) ∈ EC and f (i) ∈ Ei, for i = 1, 2, 3, 4, which tells how each atom has been displaced from its equilibrium.
Now, let us see how the group S 4 acts on the set of displacements. Consider, for example, the action of the element (123) ∈ S 4. On the molecule itself, at equilibrium, (123) leaves C fixed, rotates the chlorine atoms 1, 2 and 3 and leaves 4 fixed:
4 4
(123) C −→ C 3 3 1 1
2 2
Figura: The action of the element (123) ∈ S 4 on the displacements of Carbon Tetrachloride.
Symmetries and groupoids. Molecular vibrations and vector bundle.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 22 / 42 Now, let us see how the group S 4 acts on the set of displacements. Consider, for example, the action of the element (123) ∈ S 4. On the molecule itself, at equilibrium, (123) leaves C fixed, rotates the chlorine atoms 1, 2 and 3 and leaves 4 fixed:
4 4
(123) C −→ C 3 3 1 1
2 2
Figura: The action of the element (123) ∈ S 4 on the displacements of Carbon Tetrachloride.
Symmetries and groupoids.
Molecular vibrations and vector bundle. A displacement of the molecule as a whole moves each of the atoms, and so is a function f such that f (C) ∈ EC and f (i) ∈ Ei, for i = 1, 2, 3, 4, which tells how each atom has been displaced from its equilibrium.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 22 / 42 Symmetries and groupoids.
Molecular vibrations and vector bundle. A displacement of the molecule as a whole moves each of the atoms, and so is a function f such that f (C) ∈ EC and f (i) ∈ Ei, for i = 1, 2, 3, 4, which tells how each atom has been displaced from its equilibrium.
Now, let us see how the group S 4 acts on the set of displacements. Consider, for example, the action of the element (123) ∈ S 4. On the molecule itself, at equilibrium, (123) leaves C fixed, rotates the chlorine atoms 1, 2 and 3 and leaves 4 fixed:
4 4
(123) C −→ C 3 3 1 1
2 2
Figura: The action of the element (123) ∈ S 4 on the displacements of Carbon Tetrachloride.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 22 / 42 Set M = {1, 2, 2, 3, 4, C} to be the U set of atoms, in the previous example. Then (E, π), where E = x ∈ M Ex and π : E → M is the obvious maps, is a S 4-equivariant vector bundle, or homogeneous vector bundle, whose associated module of global sections: n o Γ(E):= σ : M → E| π ◦ σ = identity
is the space of displacements of the molecule as a whole, and the action of S 4 on Γ(E) might be considered as the action of the symmetry group on the space of displacements. In general let us assume that a group G acts on set M and consider it associated action groupoid G := (G × M, M). Then any G-equivariant vector bundle over M leads to a linear representation on G. The converse also holds true, thus, any finite-dimensional (having the same dimension at each fibre) linear representation of G, gives rise to a G-equivariant vector bundle. There is in fact an equivalence of (symmetric monoidal) categories between the category of G-equivariant bundles over M and that of linear representations of G.
Homogeneous vector bundles.
Molecular vibrations and vector bundle.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 23 / 42 In general let us assume that a group G acts on set M and consider it associated action groupoid G := (G × M, M). Then any G-equivariant vector bundle over M leads to a linear representation on G. The converse also holds true, thus, any finite-dimensional (having the same dimension at each fibre) linear representation of G, gives rise to a G-equivariant vector bundle. There is in fact an equivalence of (symmetric monoidal) categories between the category of G-equivariant bundles over M and that of linear representations of G.
Homogeneous vector bundles.
Molecular vibrations and vector bundle. Set M = {1, 2, 2, 3, 4, C} to be the U set of atoms, in the previous example. Then (E, π), where E = x ∈ M Ex and π : E → M is the obvious maps, is a S 4-equivariant vector bundle, or homogeneous vector bundle, whose associated module of global sections: n o Γ(E):= σ : M → E| π ◦ σ = identity
is the space of displacements of the molecule as a whole, and the action of S 4 on Γ(E) might be considered as the action of the symmetry group on the space of displacements.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 23 / 42 There is in fact an equivalence of (symmetric monoidal) categories between the category of G-equivariant bundles over M and that of linear representations of G.
Homogeneous vector bundles.
Molecular vibrations and vector bundle. Set M = {1, 2, 2, 3, 4, C} to be the U set of atoms, in the previous example. Then (E, π), where E = x ∈ M Ex and π : E → M is the obvious maps, is a S 4-equivariant vector bundle, or homogeneous vector bundle, whose associated module of global sections: n o Γ(E):= σ : M → E| π ◦ σ = identity
is the space of displacements of the molecule as a whole, and the action of S 4 on Γ(E) might be considered as the action of the symmetry group on the space of displacements. In general let us assume that a group G acts on set M and consider it associated action groupoid G := (G × M, M). Then any G-equivariant vector bundle over M leads to a linear representation on G. The converse also holds true, thus, any finite-dimensional (having the same dimension at each fibre) linear representation of G, gives rise to a G-equivariant vector bundle.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 23 / 42 Homogeneous vector bundles.
Molecular vibrations and vector bundle. Set M = {1, 2, 2, 3, 4, C} to be the U set of atoms, in the previous example. Then (E, π), where E = x ∈ M Ex and π : E → M is the obvious maps, is a S 4-equivariant vector bundle, or homogeneous vector bundle, whose associated module of global sections: n o Γ(E):= σ : M → E| π ◦ σ = identity
is the space of displacements of the molecule as a whole, and the action of S 4 on Γ(E) might be considered as the action of the symmetry group on the space of displacements. In general let us assume that a group G acts on set M and consider it associated action groupoid G := (G × M, M). Then any G-equivariant vector bundle over M leads to a linear representation on G. The converse also holds true, thus, any finite-dimensional (having the same dimension at each fibre) linear representation of G, gives rise to a G-equivariant vector bundle. There is in fact an equivalence of (symmetric monoidal) categories between the category of G-equivariant bundles over M and that of linear representations of G.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 23 / 42 1 Rappels on categories and equivalences. Categories, functors and natural transformations. Adjunctions and equivalence between categories. The pull-back in general categories.
2 Groupoids and Symmetries. Abstract groupoids: Definition and examples. Left and right stars and the isotropy groups. Symmetries and groupoids. Homogeneous vector bundles
3 Linear representations of abstract groupoids. The context, motivations and overviews. Linear representations of groupoids. The representative functions functor. The contravariant adjunction.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 24 / 42 Not only groups and their representations are used in physics but also their algebra of representatives functions, that is, Hopf algebras (e.g., in the normalization process in QFT). The classical picture of groups is represented by the following diagram
Rk
duality ' Grp CHAlgk O g O χk continous k=R ( ? ? anti-equivalence ] CTGrp CHAlgR O h O
smooth ( ? CLGrp? anti-equivalence CHalg] h R
] where CHAlgk the subcategory of commutative real Hopf algebras with gauge (i.e., a Hopf integral coming from the Haar measure) and with dense character group in the linear dual.
The context, motivations and overviews.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 25 / 42 The classical picture of groups is represented by the following diagram
Rk
duality ' Grp CHAlgk O g O χk continous k=R ( ? ? anti-equivalence ] CTGrp CHAlgR O h O
smooth ( ? CLGrp? anti-equivalence CHalg] h R
] where CHAlgk the subcategory of commutative real Hopf algebras with gauge (i.e., a Hopf integral coming from the Haar measure) and with dense character group in the linear dual.
The context, motivations and overviews.
Not only groups and their representations are used in physics but also their algebra of representatives functions, that is, Hopf algebras (e.g., in the normalization process in QFT).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 25 / 42 ] where CHAlgk the subcategory of commutative real Hopf algebras with gauge (i.e., a Hopf integral coming from the Haar measure) and with dense character group in the linear dual.
The context, motivations and overviews.
Not only groups and their representations are used in physics but also their algebra of representatives functions, that is, Hopf algebras (e.g., in the normalization process in QFT). The classical picture of groups is represented by the following diagram
Rk
duality ' Grp CHAlgk O g O χk continous k=R ( ? ? anti-equivalence ] CTGrp CHAlgR O h O
smooth ( ? CLGrp? anti-equivalence CHalg] h R
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 25 / 42 The context, motivations and overviews.
Not only groups and their representations are used in physics but also their algebra of representatives functions, that is, Hopf algebras (e.g., in the normalization process in QFT). The classical picture of groups is represented by the following diagram
Rk
duality ' Grp CHAlgk O g O χk continous k=R ( ? ? anti-equivalence ] CTGrp CHAlgR O h O
smooth ( ? CLGrp? anti-equivalence CHalg] h R
] where CHAlgk the subcategory of commutative real Hopf algebras with gauge (i.e., a Hopf integral coming from the Haar measure) and with dense character group in the linear dual.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 25 / 42 The ultimate goal of our research on linear representations of groupoids is to give a new approach to Tannaka-Krein duality for compact topological groupoids, and try to recognize the theory of compact Lie groupoids as a part of the theory of real affine algebraic groupoids. In a very simplest way, we are attempted to complete the following diagram
Rk
duality ' Grpd CHAlgdk Grpd: abstract groupoids O h O χk k=R ? CTGrpd? anti-equivalence & ? CTGrpd: comapct topological groupoids O h O
? CLGrpd? anti-equivalence & ? CLGrpd: compact Lie groupoids h
In this talk, we will see how to construct the functor Rk and show the main steps in building up the duality between the category of transitive groupoids and the category of (geometrically) transitive commutative Hopf algebroids.
The context, motivations and overviews.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 26 / 42 In a very simplest way, we are attempted to complete the following diagram
Rk
duality ' Grpd CHAlgdk Grpd: abstract groupoids O h O χk k=R ? CTGrpd? anti-equivalence & ? CTGrpd: comapct topological groupoids O h O
? CLGrpd? anti-equivalence & ? CLGrpd: compact Lie groupoids h
In this talk, we will see how to construct the functor Rk and show the main steps in building up the duality between the category of transitive groupoids and the category of (geometrically) transitive commutative Hopf algebroids.
The context, motivations and overviews.
The ultimate goal of our research on linear representations of groupoids is to give a new approach to Tannaka-Krein duality for compact topological groupoids, and try to recognize the theory of compact Lie groupoids as a part of the theory of real affine algebraic groupoids.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 26 / 42 In this talk, we will see how to construct the functor Rk and show the main steps in building up the duality between the category of transitive groupoids and the category of (geometrically) transitive commutative Hopf algebroids.
The context, motivations and overviews.
The ultimate goal of our research on linear representations of groupoids is to give a new approach to Tannaka-Krein duality for compact topological groupoids, and try to recognize the theory of compact Lie groupoids as a part of the theory of real affine algebraic groupoids. In a very simplest way, we are attempted to complete the following diagram
Rk
duality ' Grpd CHAlgdk Grpd: abstract groupoids O h O χk k=R ? CTGrpd? anti-equivalence & ? CTGrpd: comapct topological groupoids O h O
? CLGrpd? anti-equivalence & ? CLGrpd: compact Lie groupoids h
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 26 / 42 The context, motivations and overviews.
The ultimate goal of our research on linear representations of groupoids is to give a new approach to Tannaka-Krein duality for compact topological groupoids, and try to recognize the theory of compact Lie groupoids as a part of the theory of real affine algebraic groupoids. In a very simplest way, we are attempted to complete the following diagram
Rk
duality ' Grpd CHAlgdk Grpd: abstract groupoids O h O χk k=R ? CTGrpd? anti-equivalence & ? CTGrpd: comapct topological groupoids O h O
? CLGrpd? anti-equivalence & ? CLGrpd: compact Lie groupoids h
In this talk, we will see how to construct the functor Rk and show the main steps in building up the duality between the category of transitive groupoids and the category of (geometrically) transitive commutative Hopf algebroids.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 26 / 42 Let k denotes a ground base field, Vectk its category of vector spaces, and
vectk the full subcategory of finite dimensional ones.
For a given groupoid s / G : G1 o ι G0 , t / we consider the category of all G-representations as the symmetric monoidal k-linear abelian category of functors G, Vectk with identity object
I : G0 → Vectk, x → k, g → 1k.
For any G-representation V the image of an object x ∈ G0 is denoted by Vx, and referred to as the fibre of V at x.
The disjoint union of all the fibres of a G-representation V is denoted by S V = V and the canonical projection by πV : V → G . This called the x ∈ G0 x 0 associated vector G-bundle of the representation V.
Finite dimensional representations of groupoids.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 27 / 42 For a given groupoid s / G : G1 o ι G0 , t / we consider the category of all G-representations as the symmetric monoidal k-linear abelian category of functors G, Vectk with identity object
I : G0 → Vectk, x → k, g → 1k.
For any G-representation V the image of an object x ∈ G0 is denoted by Vx, and referred to as the fibre of V at x.
The disjoint union of all the fibres of a G-representation V is denoted by S V = V and the canonical projection by πV : V → G . This called the x ∈ G0 x 0 associated vector G-bundle of the representation V.
Finite dimensional representations of groupoids.
Let k denotes a ground base field, Vectk its category of vector spaces, and
vectk the full subcategory of finite dimensional ones.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 27 / 42 For any G-representation V the image of an object x ∈ G0 is denoted by Vx, and referred to as the fibre of V at x.
The disjoint union of all the fibres of a G-representation V is denoted by S V = V and the canonical projection by πV : V → G . This called the x ∈ G0 x 0 associated vector G-bundle of the representation V.
Finite dimensional representations of groupoids.
Let k denotes a ground base field, Vectk its category of vector spaces, and
vectk the full subcategory of finite dimensional ones.
For a given groupoid s / G : G1 o ι G0 , t / we consider the category of all G-representations as the symmetric monoidal k-linear abelian category of functors G, Vectk with identity object
I : G0 → Vectk, x → k, g → 1k.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 27 / 42 The disjoint union of all the fibres of a G-representation V is denoted by S V = V and the canonical projection by πV : V → G . This called the x ∈ G0 x 0 associated vector G-bundle of the representation V.
Finite dimensional representations of groupoids.
Let k denotes a ground base field, Vectk its category of vector spaces, and
vectk the full subcategory of finite dimensional ones.
For a given groupoid s / G : G1 o ι G0 , t / we consider the category of all G-representations as the symmetric monoidal k-linear abelian category of functors G, Vectk with identity object
I : G0 → Vectk, x → k, g → 1k.
For any G-representation V the image of an object x ∈ G0 is denoted by Vx, and referred to as the fibre of V at x.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 27 / 42 Finite dimensional representations of groupoids.
Let k denotes a ground base field, Vectk its category of vector spaces, and
vectk the full subcategory of finite dimensional ones.
For a given groupoid s / G : G1 o ι G0 , t / we consider the category of all G-representations as the symmetric monoidal k-linear abelian category of functors G, Vectk with identity object
I : G0 → Vectk, x → k, g → 1k.
For any G-representation V the image of an object x ∈ G0 is denoted by Vx, and referred to as the fibre of V at x.
The disjoint union of all the fibres of a G-representation V is denoted by S V = V and the canonical projection by πV : V → G . This called the x ∈ G0 x 0 associated vector G-bundle of the representation V.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 27 / 42 Let V be a G-representation in G, vectk , we define its dimension function as the map dV : G0 −→ N, x 7−→ dimk Vx ,
which clearly extends to a map dV : π0(G) → N. A G-representation V in G, vectk is called a finite dimensional representation,
provided that the dimension function dV has a finite image, that is, dV(G0) is a finite subset of the set of positive integers N.
We denote by repk(G) the category of finite dimensional representation over G. Clearly, we have that repk(G) = G, vectk , when π0(G) is a finite set.
Let V and W be two representations in repk(G). Then
dV ⊕ W = dV + dW, dDV = dV, and dV ⊗ W = dV dW.
Therefore, the category repk(G) is a symmetric rigid monoidal k-linear abelian category. But NOT locally finite, in general.
Finite dimensional representations of groupoids.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 28 / 42 A G-representation V in G, vectk is called a finite dimensional representation,
provided that the dimension function dV has a finite image, that is, dV(G0) is a finite subset of the set of positive integers N.
We denote by repk(G) the category of finite dimensional representation over G. Clearly, we have that repk(G) = G, vectk , when π0(G) is a finite set.
Let V and W be two representations in repk(G). Then
dV ⊕ W = dV + dW, dDV = dV, and dV ⊗ W = dV dW.
Therefore, the category repk(G) is a symmetric rigid monoidal k-linear abelian category. But NOT locally finite, in general.
Finite dimensional representations of groupoids.
Let V be a G-representation in G, vectk , we define its dimension function as the map dV : G0 −→ N, x 7−→ dimk Vx ,
which clearly extends to a map dV : π0(G) → N.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 28 / 42 We denote by repk(G) the category of finite dimensional representation over G. Clearly, we have that repk(G) = G, vectk , when π0(G) is a finite set.
Let V and W be two representations in repk(G). Then
dV ⊕ W = dV + dW, dDV = dV, and dV ⊗ W = dV dW.
Therefore, the category repk(G) is a symmetric rigid monoidal k-linear abelian category. But NOT locally finite, in general.
Finite dimensional representations of groupoids.
Let V be a G-representation in G, vectk , we define its dimension function as the map dV : G0 −→ N, x 7−→ dimk Vx ,
which clearly extends to a map dV : π0(G) → N. A G-representation V in G, vectk is called a finite dimensional representation,
provided that the dimension function dV has a finite image, that is, dV(G0) is a finite subset of the set of positive integers N.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 28 / 42 Let V and W be two representations in repk(G). Then
dV ⊕ W = dV + dW, dDV = dV, and dV ⊗ W = dV dW.
Therefore, the category repk(G) is a symmetric rigid monoidal k-linear abelian category. But NOT locally finite, in general.
Finite dimensional representations of groupoids.
Let V be a G-representation in G, vectk , we define its dimension function as the map dV : G0 −→ N, x 7−→ dimk Vx ,
which clearly extends to a map dV : π0(G) → N. A G-representation V in G, vectk is called a finite dimensional representation,
provided that the dimension function dV has a finite image, that is, dV(G0) is a finite subset of the set of positive integers N.
We denote by repk(G) the category of finite dimensional representation over G. Clearly, we have that repk(G) = G, vectk , when π0(G) is a finite set.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 28 / 42 Therefore, the category repk(G) is a symmetric rigid monoidal k-linear abelian category. But NOT locally finite, in general.
Finite dimensional representations of groupoids.
Let V be a G-representation in G, vectk , we define its dimension function as the map dV : G0 −→ N, x 7−→ dimk Vx ,
which clearly extends to a map dV : π0(G) → N. A G-representation V in G, vectk is called a finite dimensional representation,
provided that the dimension function dV has a finite image, that is, dV(G0) is a finite subset of the set of positive integers N.
We denote by repk(G) the category of finite dimensional representation over G. Clearly, we have that repk(G) = G, vectk , when π0(G) is a finite set.
Let V and W be two representations in repk(G). Then
dV ⊕ W = dV + dW, dDV = dV, and dV ⊗ W = dV dW.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 28 / 42 But NOT locally finite, in general.
Finite dimensional representations of groupoids.
Let V be a G-representation in G, vectk , we define its dimension function as the map dV : G0 −→ N, x 7−→ dimk Vx ,
which clearly extends to a map dV : π0(G) → N. A G-representation V in G, vectk is called a finite dimensional representation,
provided that the dimension function dV has a finite image, that is, dV(G0) is a finite subset of the set of positive integers N.
We denote by repk(G) the category of finite dimensional representation over G. Clearly, we have that repk(G) = G, vectk , when π0(G) is a finite set.
Let V and W be two representations in repk(G). Then
dV ⊕ W = dV + dW, dDV = dV, and dV ⊗ W = dV dW.
Therefore, the category repk(G) is a symmetric rigid monoidal k-linear abelian category.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 28 / 42 Finite dimensional representations of groupoids.
Let V be a G-representation in G, vectk , we define its dimension function as the map dV : G0 −→ N, x 7−→ dimk Vx ,
which clearly extends to a map dV : π0(G) → N. A G-representation V in G, vectk is called a finite dimensional representation,
provided that the dimension function dV has a finite image, that is, dV(G0) is a finite subset of the set of positive integers N.
We denote by repk(G) the category of finite dimensional representation over G. Clearly, we have that repk(G) = G, vectk , when π0(G) is a finite set.
Let V and W be two representations in repk(G). Then
dV ⊕ W = dV + dW, dDV = dV, and dV ⊗ W = dV dW.
Therefore, the category repk(G) is a symmetric rigid monoidal k-linear abelian category. But NOT locally finite, in general.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 28 / 42 Example of representations. Consider the set X = {1, 2} and denote by G{1,2} the associated groupoid of pairs. Thus G0 = {1, 2} and G1 = (1, 1), (1, 2), (2, 1), (2, 2) . {1,2} An object in repk(G ) is then a pair (n, N), where n is a positive integer, and N ∈ GLn(k). The vector spaces of homomorphisms are given by
{1,2} repk(G ) (n, N), (m, M) = Mm, n(k),
the k-vector space of m × n matrices with matrix multiplication.
{1,2} The other operations in repk(G ) are ! N 0 (n, N) ⊕ (m, M) = n + m, , D(n, N) = (n, Nt) 0 M (n, N) ⊗ (m, M) = nm, (N bi j)1≤i, j≤m , where M = (bi j), and I = (1, 1). Tr(n, N) = n.
Finite dimensional representations of groupoids.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 29 / 42 Consider the set X = {1, 2} and denote by G{1,2} the associated groupoid of pairs. Thus G0 = {1, 2} and G1 = (1, 1), (1, 2), (2, 1), (2, 2) . {1,2} An object in repk(G ) is then a pair (n, N), where n is a positive integer, and N ∈ GLn(k). The vector spaces of homomorphisms are given by
{1,2} repk(G ) (n, N), (m, M) = Mm, n(k),
the k-vector space of m × n matrices with matrix multiplication.
{1,2} The other operations in repk(G ) are ! N 0 (n, N) ⊕ (m, M) = n + m, , D(n, N) = (n, Nt) 0 M (n, N) ⊗ (m, M) = nm, (N bi j)1≤i, j≤m , where M = (bi j), and I = (1, 1). Tr(n, N) = n.
Finite dimensional representations of groupoids.
Example of representations.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 29 / 42 {1,2} An object in repk(G ) is then a pair (n, N), where n is a positive integer, and N ∈ GLn(k). The vector spaces of homomorphisms are given by
{1,2} repk(G ) (n, N), (m, M) = Mm, n(k),
the k-vector space of m × n matrices with matrix multiplication.
{1,2} The other operations in repk(G ) are ! N 0 (n, N) ⊕ (m, M) = n + m, , D(n, N) = (n, Nt) 0 M (n, N) ⊗ (m, M) = nm, (N bi j)1≤i, j≤m , where M = (bi j), and I = (1, 1). Tr(n, N) = n.
Finite dimensional representations of groupoids.
Example of representations. Consider the set X = {1, 2} and denote by G{1,2} the associated groupoid of pairs. Thus G0 = {1, 2} and G1 = (1, 1), (1, 2), (2, 1), (2, 2) .
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 29 / 42 The vector spaces of homomorphisms are given by
{1,2} repk(G ) (n, N), (m, M) = Mm, n(k),
the k-vector space of m × n matrices with matrix multiplication.
{1,2} The other operations in repk(G ) are ! N 0 (n, N) ⊕ (m, M) = n + m, , D(n, N) = (n, Nt) 0 M (n, N) ⊗ (m, M) = nm, (N bi j)1≤i, j≤m , where M = (bi j), and I = (1, 1). Tr(n, N) = n.
Finite dimensional representations of groupoids.
Example of representations. Consider the set X = {1, 2} and denote by G{1,2} the associated groupoid of pairs. Thus G0 = {1, 2} and G1 = (1, 1), (1, 2), (2, 1), (2, 2) . {1,2} An object in repk(G ) is then a pair (n, N), where n is a positive integer, and N ∈ GLn(k).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 29 / 42 {1,2} The other operations in repk(G ) are ! N 0 (n, N) ⊕ (m, M) = n + m, , D(n, N) = (n, Nt) 0 M (n, N) ⊗ (m, M) = nm, (N bi j)1≤i, j≤m , where M = (bi j), and I = (1, 1). Tr(n, N) = n.
Finite dimensional representations of groupoids.
Example of representations. Consider the set X = {1, 2} and denote by G{1,2} the associated groupoid of pairs. Thus G0 = {1, 2} and G1 = (1, 1), (1, 2), (2, 1), (2, 2) . {1,2} An object in repk(G ) is then a pair (n, N), where n is a positive integer, and N ∈ GLn(k). The vector spaces of homomorphisms are given by
{1,2} repk(G ) (n, N), (m, M) = Mm, n(k),
the k-vector space of m × n matrices with matrix multiplication.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 29 / 42 Finite dimensional representations of groupoids.
Example of representations. Consider the set X = {1, 2} and denote by G{1,2} the associated groupoid of pairs. Thus G0 = {1, 2} and G1 = (1, 1), (1, 2), (2, 1), (2, 2) . {1,2} An object in repk(G ) is then a pair (n, N), where n is a positive integer, and N ∈ GLn(k). The vector spaces of homomorphisms are given by
{1,2} repk(G ) (n, N), (m, M) = Mm, n(k),
the k-vector space of m × n matrices with matrix multiplication.
{1,2} The other operations in repk(G ) are ! N 0 (n, N) ⊕ (m, M) = n + m, , D(n, N) = (n, Nt) 0 M (n, N) ⊗ (m, M) = nm, (N bi j)1≤i, j≤m , where M = (bi j), and I = (1, 1). Tr(n, N) = n.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 29 / 42 Recall that a groupoid G is said to be transitive if for any
two objects x, y ∈ G0, there is an arrow g ∈ G1 such that s(g) = x and t(g) = y, or equivalently, π0(G) is a singleton, i.e., a set with only one element.
Let G be a transitive groupoid. Then, the category repk(G) is a symmetric rigid monoidal locally finite k-linear abelian category.
Furthermore, repk(G) admits a non trivial fibre functor to the category of finite
dimensional vector spaces. Namely, fix an object x ∈ G0, and consider the functor ωx : repk(G) −→ vectk, V −→ Vx .
Then ωx is a non trivial fibre functor, and ωx ωy, for any x, y ∈ G0. I I On the other hand, we have that k Endrepk(G)( ), where is the identity G-representation.
Summarizing (repk(G), ωx) is a (neutral) Tannakian category in the sense of Saavedra-Rivano, Deligne and Milne.
Finite dimensional representations of groupoids.
The transitive case.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 30 / 42 Let G be a transitive groupoid. Then, the category repk(G) is a symmetric rigid monoidal locally finite k-linear abelian category.
Furthermore, repk(G) admits a non trivial fibre functor to the category of finite
dimensional vector spaces. Namely, fix an object x ∈ G0, and consider the functor ωx : repk(G) −→ vectk, V −→ Vx .
Then ωx is a non trivial fibre functor, and ωx ωy, for any x, y ∈ G0. I I On the other hand, we have that k Endrepk(G)( ), where is the identity G-representation.
Summarizing (repk(G), ωx) is a (neutral) Tannakian category in the sense of Saavedra-Rivano, Deligne and Milne.
Finite dimensional representations of groupoids.
The transitive case. Recall that a groupoid G is said to be transitive if for any
two objects x, y ∈ G0, there is an arrow g ∈ G1 such that s(g) = x and t(g) = y, or equivalently, π0(G) is a singleton, i.e., a set with only one element.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 30 / 42 Furthermore, repk(G) admits a non trivial fibre functor to the category of finite
dimensional vector spaces. Namely, fix an object x ∈ G0, and consider the functor ωx : repk(G) −→ vectk, V −→ Vx .
Then ωx is a non trivial fibre functor, and ωx ωy, for any x, y ∈ G0. I I On the other hand, we have that k Endrepk(G)( ), where is the identity G-representation.
Summarizing (repk(G), ωx) is a (neutral) Tannakian category in the sense of Saavedra-Rivano, Deligne and Milne.
Finite dimensional representations of groupoids.
The transitive case. Recall that a groupoid G is said to be transitive if for any
two objects x, y ∈ G0, there is an arrow g ∈ G1 such that s(g) = x and t(g) = y, or equivalently, π0(G) is a singleton, i.e., a set with only one element.
Let G be a transitive groupoid. Then, the category repk(G) is a symmetric rigid monoidal locally finite k-linear abelian category.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 30 / 42 I I On the other hand, we have that k Endrepk(G)( ), where is the identity G-representation.
Summarizing (repk(G), ωx) is a (neutral) Tannakian category in the sense of Saavedra-Rivano, Deligne and Milne.
Finite dimensional representations of groupoids.
The transitive case. Recall that a groupoid G is said to be transitive if for any
two objects x, y ∈ G0, there is an arrow g ∈ G1 such that s(g) = x and t(g) = y, or equivalently, π0(G) is a singleton, i.e., a set with only one element.
Let G be a transitive groupoid. Then, the category repk(G) is a symmetric rigid monoidal locally finite k-linear abelian category.
Furthermore, repk(G) admits a non trivial fibre functor to the category of finite
dimensional vector spaces. Namely, fix an object x ∈ G0, and consider the functor ωx : repk(G) −→ vectk, V −→ Vx .
Then ωx is a non trivial fibre functor, and ωx ωy, for any x, y ∈ G0.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 30 / 42 Summarizing (repk(G), ωx) is a (neutral) Tannakian category in the sense of Saavedra-Rivano, Deligne and Milne.
Finite dimensional representations of groupoids.
The transitive case. Recall that a groupoid G is said to be transitive if for any
two objects x, y ∈ G0, there is an arrow g ∈ G1 such that s(g) = x and t(g) = y, or equivalently, π0(G) is a singleton, i.e., a set with only one element.
Let G be a transitive groupoid. Then, the category repk(G) is a symmetric rigid monoidal locally finite k-linear abelian category.
Furthermore, repk(G) admits a non trivial fibre functor to the category of finite
dimensional vector spaces. Namely, fix an object x ∈ G0, and consider the functor ωx : repk(G) −→ vectk, V −→ Vx .
Then ωx is a non trivial fibre functor, and ωx ωy, for any x, y ∈ G0. I I On the other hand, we have that k Endrepk(G)( ), where is the identity G-representation.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 30 / 42 Finite dimensional representations of groupoids.
The transitive case. Recall that a groupoid G is said to be transitive if for any
two objects x, y ∈ G0, there is an arrow g ∈ G1 such that s(g) = x and t(g) = y, or equivalently, π0(G) is a singleton, i.e., a set with only one element.
Let G be a transitive groupoid. Then, the category repk(G) is a symmetric rigid monoidal locally finite k-linear abelian category.
Furthermore, repk(G) admits a non trivial fibre functor to the category of finite
dimensional vector spaces. Namely, fix an object x ∈ G0, and consider the functor ωx : repk(G) −→ vectk, V −→ Vx .
Then ωx is a non trivial fibre functor, and ωx ωy, for any x, y ∈ G0. I I On the other hand, we have that k Endrepk(G)( ), where is the identity G-representation.
Summarizing (repk(G), ωx) is a (neutral) Tannakian category in the sense of Saavedra-Rivano, Deligne and Milne.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 30 / 42 G0 Let G be a groupoid and denote by A0(G):= k its base algebra and by G1 A1(G):= k its total algebra. By reflecting the groupoid structure of G, we have a diagram of algebras: s∗ ∗ / A0(G) o ι A1(G). t∗ / Let V be a finite dimensional G-representation and denote by n o dV(G0):= n1, n2, ··· , nN ordered as n1 < n2 < ··· < nN (where obviously the maximal and minimal indices depend upon V).
SN i The set of objects G0 is then a disjoint union G0 = i=1 GV, where each of the i i −1 GV’s is the inverse image GV := dV {ni} , for any i = 1, ··· , N.
Finite dimensional representations of groupoids.
The fibre functor on repk(G).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 31 / 42 Let V be a finite dimensional G-representation and denote by n o dV(G0):= n1, n2, ··· , nN ordered as n1 < n2 < ··· < nN (where obviously the maximal and minimal indices depend upon V).
SN i The set of objects G0 is then a disjoint union G0 = i=1 GV, where each of the i i −1 GV’s is the inverse image GV := dV {ni} , for any i = 1, ··· , N.
Finite dimensional representations of groupoids.
The fibre functor on repk(G).
G0 Let G be a groupoid and denote by A0(G):= k its base algebra and by G1 A1(G):= k its total algebra. By reflecting the groupoid structure of G, we have a diagram of algebras: s∗ ∗ / A0(G) o ι A1(G). t∗ /
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 31 / 42 SN i The set of objects G0 is then a disjoint union G0 = i=1 GV, where each of the i i −1 GV’s is the inverse image GV := dV {ni} , for any i = 1, ··· , N.
Finite dimensional representations of groupoids.
The fibre functor on repk(G).
G0 Let G be a groupoid and denote by A0(G):= k its base algebra and by G1 A1(G):= k its total algebra. By reflecting the groupoid structure of G, we have a diagram of algebras: s∗ ∗ / A0(G) o ι A1(G). t∗ / Let V be a finite dimensional G-representation and denote by n o dV(G0):= n1, n2, ··· , nN ordered as n1 < n2 < ··· < nN (where obviously the maximal and minimal indices depend upon V).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 31 / 42 Finite dimensional representations of groupoids.
The fibre functor on repk(G).
G0 Let G be a groupoid and denote by A0(G):= k its base algebra and by G1 A1(G):= k its total algebra. By reflecting the groupoid structure of G, we have a diagram of algebras: s∗ ∗ / A0(G) o ι A1(G). t∗ / Let V be a finite dimensional G-representation and denote by n o dV(G0):= n1, n2, ··· , nN ordered as n1 < n2 < ··· < nN (where obviously the maximal and minimal indices depend upon V).
SN i The set of objects G0 is then a disjoint union G0 = i=1 GV, where each of the i i −1 GV’s is the inverse image GV := dV {ni} , for any i = 1, ··· , N.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 31 / 42 We can then define the functor which acts on objects by:
n1 nN ω : repk(G) −→ proj(A0(G)), V −→ PV = B1 × · · · × BN
an A0(G)-module which corresponds to the above decomposition.
By identifying a G-representation in repk(G) with its associated vector G-bundle, we can consider the k-vector space of ”global sections": n o V G → V | ◦ Γ( ):= s : 0 πV s = idG0 .
Both functors ω and Γ are symmetric monoidal faithful functors. Moreover, there is a tensorial natural isomorphism ω Γ.
Finite dimensional representations of groupoids.
This leads to a decomposition of the base algebra A0(G):
A0(G) = B1 × · · · · · · × BN ,
i where each of Bi’s is the algebra of functions on GV.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 32 / 42 By identifying a G-representation in repk(G) with its associated vector G-bundle, we can consider the k-vector space of ”global sections": n o V G → V | ◦ Γ( ):= s : 0 πV s = idG0 .
Both functors ω and Γ are symmetric monoidal faithful functors. Moreover, there is a tensorial natural isomorphism ω Γ.
Finite dimensional representations of groupoids.
This leads to a decomposition of the base algebra A0(G):
A0(G) = B1 × · · · · · · × BN ,
i where each of Bi’s is the algebra of functions on GV.
We can then define the functor which acts on objects by:
n1 nN ω : repk(G) −→ proj(A0(G)), V −→ PV = B1 × · · · × BN
an A0(G)-module which corresponds to the above decomposition.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 32 / 42 Both functors ω and Γ are symmetric monoidal faithful functors. Moreover, there is a tensorial natural isomorphism ω Γ.
Finite dimensional representations of groupoids.
This leads to a decomposition of the base algebra A0(G):
A0(G) = B1 × · · · · · · × BN ,
i where each of Bi’s is the algebra of functions on GV.
We can then define the functor which acts on objects by:
n1 nN ω : repk(G) −→ proj(A0(G)), V −→ PV = B1 × · · · × BN
an A0(G)-module which corresponds to the above decomposition.
By identifying a G-representation in repk(G) with its associated vector G-bundle, we can consider the k-vector space of ”global sections": n o V G → V | ◦ Γ( ):= s : 0 πV s = idG0 .
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 32 / 42 Finite dimensional representations of groupoids.
This leads to a decomposition of the base algebra A0(G):
A0(G) = B1 × · · · · · · × BN ,
i where each of Bi’s is the algebra of functions on GV.
We can then define the functor which acts on objects by:
n1 nN ω : repk(G) −→ proj(A0(G)), V −→ PV = B1 × · · · × BN
an A0(G)-module which corresponds to the above decomposition.
By identifying a G-representation in repk(G) with its associated vector G-bundle, we can consider the k-vector space of ”global sections": n o V G → V | ◦ Γ( ):= s : 0 πV s = idG0 .
Both functors ω and Γ are symmetric monoidal faithful functors. Moreover, there is a tensorial natural isomorphism ω Γ.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 32 / 42 Let us recall first some general facts. Assume we are given a pair (T , ω) consisting of a symmetric monoidal rigid k-linear (essentially small) category and a non trivial symmetric monoidal faithful functor ω : T → proj(A), where A is a commutative k-algebra.
Associated to these data, there are at least two universal problems, which have a common solution: PR-1 The functor: A-Corings −→ Sets, n o C −→ the set of funtorial right C-comodules structure on ω
is representable.
PR-2 The functor: (A ⊗ A)-CAlgk −→ Sets, s / −→ ⊗ ∗ ∗ A t / C Iso t ω, s ω is representable.
The representative functions functor.
Tannakian reconstruction process and the universal Hopf algebroid.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 33 / 42 Assume we are given a pair (T , ω) consisting of a symmetric monoidal rigid k-linear (essentially small) category and a non trivial symmetric monoidal faithful functor ω : T → proj(A), where A is a commutative k-algebra.
Associated to these data, there are at least two universal problems, which have a common solution: PR-1 The functor: A-Corings −→ Sets, n o C −→ the set of funtorial right C-comodules structure on ω
is representable.
PR-2 The functor: (A ⊗ A)-CAlgk −→ Sets, s / −→ ⊗ ∗ ∗ A t / C Iso t ω, s ω is representable.
The representative functions functor.
Tannakian reconstruction process and the universal Hopf algebroid. Let us recall first some general facts.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 33 / 42 Associated to these data, there are at least two universal problems, which have a common solution: PR-1 The functor: A-Corings −→ Sets, n o C −→ the set of funtorial right C-comodules structure on ω
is representable.
PR-2 The functor: (A ⊗ A)-CAlgk −→ Sets, s / −→ ⊗ ∗ ∗ A t / C Iso t ω, s ω is representable.
The representative functions functor.
Tannakian reconstruction process and the universal Hopf algebroid. Let us recall first some general facts. Assume we are given a pair (T , ω) consisting of a symmetric monoidal rigid k-linear (essentially small) category and a non trivial symmetric monoidal faithful functor ω : T → proj(A), where A is a commutative k-algebra.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 33 / 42 PR-1 The functor: A-Corings −→ Sets, n o C −→ the set of funtorial right C-comodules structure on ω
is representable.
PR-2 The functor: (A ⊗ A)-CAlgk −→ Sets, s / −→ ⊗ ∗ ∗ A t / C Iso t ω, s ω is representable.
The representative functions functor.
Tannakian reconstruction process and the universal Hopf algebroid. Let us recall first some general facts. Assume we are given a pair (T , ω) consisting of a symmetric monoidal rigid k-linear (essentially small) category and a non trivial symmetric monoidal faithful functor ω : T → proj(A), where A is a commutative k-algebra.
Associated to these data, there are at least two universal problems, which have a common solution:
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 33 / 42 PR-2 The functor: (A ⊗ A)-CAlgk −→ Sets, s / −→ ⊗ ∗ ∗ A t / C Iso t ω, s ω is representable.
The representative functions functor.
Tannakian reconstruction process and the universal Hopf algebroid. Let us recall first some general facts. Assume we are given a pair (T , ω) consisting of a symmetric monoidal rigid k-linear (essentially small) category and a non trivial symmetric monoidal faithful functor ω : T → proj(A), where A is a commutative k-algebra.
Associated to these data, there are at least two universal problems, which have a common solution: PR-1 The functor: A-Corings −→ Sets, n o C −→ the set of funtorial right C-comodules structure on ω
is representable.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 33 / 42 The representative functions functor.
Tannakian reconstruction process and the universal Hopf algebroid. Let us recall first some general facts. Assume we are given a pair (T , ω) consisting of a symmetric monoidal rigid k-linear (essentially small) category and a non trivial symmetric monoidal faithful functor ω : T → proj(A), where A is a commutative k-algebra.
Associated to these data, there are at least two universal problems, which have a common solution: PR-1 The functor: A-Corings −→ Sets, n o C −→ the set of funtorial right C-comodules structure on ω
is representable.
PR-2 The functor: (A ⊗ A)-CAlgk −→ Sets, s / −→ ⊗ ∗ ∗ A t / C Iso t ω, s ω is representable.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 33 / 42 where TP is the endomorphism algebra of an object P ∈ T and J is the A-sub-bimodule generated by D E J := ψ λ ⊗T p − ψ ⊗T λp P Q ψ ∈ ω(Q)∗, p ∈ ω(P), λ:P→Q ∈ T It turns out that A, Lk(T , ω) is a commutative Hopf algebroid, such that there is a commutative diagram:
T / comodLk(T ,ω)
ω ( u O proj(A)
where comodLk(T ,ω) is the full subcategory of comodules with finitely generated and projective underlying A-modules.
The representative functions functor.
The universal solution for both PR-1-2 is given by the following A-bimodule L ∗ ⊗ ω(P) TP ω(P) ∈ T L (T , ω):= P , k J
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 34 / 42 It turns out that A, Lk(T , ω) is a commutative Hopf algebroid, such that there is a commutative diagram:
T / comodLk(T ,ω)
ω ( u O proj(A)
where comodLk(T ,ω) is the full subcategory of comodules with finitely generated and projective underlying A-modules.
The representative functions functor.
The universal solution for both PR-1-2 is given by the following A-bimodule L ∗ ⊗ ω(P) TP ω(P) ∈ T L (T , ω):= P , k J
where TP is the endomorphism algebra of an object P ∈ T and J is the A-sub-bimodule generated by D E J := ψ λ ⊗T p − ψ ⊗T λp P Q ψ ∈ ω(Q)∗, p ∈ ω(P), λ:P→Q ∈ T
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 34 / 42 The representative functions functor.
The universal solution for both PR-1-2 is given by the following A-bimodule L ∗ ⊗ ω(P) TP ω(P) ∈ T L (T , ω):= P , k J
where TP is the endomorphism algebra of an object P ∈ T and J is the A-sub-bimodule generated by D E J := ψ λ ⊗T p − ψ ⊗T λp P Q ψ ∈ ω(Q)∗, p ∈ ω(P), λ:P→Q ∈ T It turns out that A, Lk(T , ω) is a commutative Hopf algebroid, such that there is a commutative diagram:
T / comodLk(T ,ω)
ω ( u O proj(A)
where comodLk(T ,ω) is the full subcategory of comodules with finitely generated and projective underlying A-modules.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 34 / 42 The commutative Hopf algebroid A0(G), Rk(G) is called the representative functions algebra (or algebroid) of the groupoid G.
The terminology ”functions” is justified by the following
(A0(G) ⊗k A0(G))-algebra map:
ξ : Rk(G) −→ A1(G),
where A1(G) is an A0(G) ⊗k A0(G)-algebra, as before by reflecting the groupoid structure of G. The representative functions establish a contravariant functor:
Rk : Grpd −→ CHAlgk from the category of abstract groupoids to the category of commutative Hopf algebroids.
The representative functions functor. Let G be a groupoid and consider the pair repk(G), ω . Applying the previous general constructions, we obtain a commutative Hopf algebroid A0(G), Lk repk(G), ω , which we denote by A0(G), Rk(G) .
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 35 / 42 The terminology ”functions” is justified by the following
(A0(G) ⊗k A0(G))-algebra map:
ξ : Rk(G) −→ A1(G),
where A1(G) is an A0(G) ⊗k A0(G)-algebra, as before by reflecting the groupoid structure of G. The representative functions establish a contravariant functor:
Rk : Grpd −→ CHAlgk from the category of abstract groupoids to the category of commutative Hopf algebroids.
The representative functions functor. Let G be a groupoid and consider the pair repk(G), ω . Applying the previous general constructions, we obtain a commutative Hopf algebroid A0(G), Lk repk(G), ω , which we denote by A0(G), Rk(G) . The commutative Hopf algebroid A0(G), Rk(G) is called the representative functions algebra (or algebroid) of the groupoid G.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 35 / 42 The representative functions establish a contravariant functor:
Rk : Grpd −→ CHAlgk from the category of abstract groupoids to the category of commutative Hopf algebroids.
The representative functions functor. Let G be a groupoid and consider the pair repk(G), ω . Applying the previous general constructions, we obtain a commutative Hopf algebroid A0(G), Lk repk(G), ω , which we denote by A0(G), Rk(G) . The commutative Hopf algebroid A0(G), Rk(G) is called the representative functions algebra (or algebroid) of the groupoid G.
The terminology ”functions” is justified by the following
(A0(G) ⊗k A0(G))-algebra map:
ξ : Rk(G) −→ A1(G),
where A1(G) is an A0(G) ⊗k A0(G)-algebra, as before by reflecting the groupoid structure of G.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 35 / 42 The representative functions functor. Let G be a groupoid and consider the pair repk(G), ω . Applying the previous general constructions, we obtain a commutative Hopf algebroid A0(G), Lk repk(G), ω , which we denote by A0(G), Rk(G) . The commutative Hopf algebroid A0(G), Rk(G) is called the representative functions algebra (or algebroid) of the groupoid G.
The terminology ”functions” is justified by the following
(A0(G) ⊗k A0(G))-algebra map:
ξ : Rk(G) −→ A1(G),
where A1(G) is an A0(G) ⊗k A0(G)-algebra, as before by reflecting the groupoid structure of G. The representative functions establish a contravariant functor:
Rk : Grpd −→ CHAlgk from the category of abstract groupoids to the category of commutative Hopf algebroids.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 35 / 42 (•) Let X = {1, 2} be a set of two elements and consider as before the groupoid G{1,2} of pairs and denote by A := k × k its base algebra. Then L n n A ⊗Mn(k) A n ∈ N Rk(G) = . t n m hv ⊗Mn(k) λw − λ v ⊗Mm(k) wiv ∈ A , w ∈ A , λ ∈ Mm×n(k) % / (•) Let G: G × X o ι X be an action groupoid. Then there is a morphism pr1 / of Hopf algebroids: X X X X (k , k ⊗ Rk(G) ⊗ k ) −→ (k , Rk(G)). Furthermore, if the action is transitive, then any isotropy Hopf algebra x (kx, Rk(G) ), for x ∈ X, is isomorphic to (k, Rk(G)). In general, Rk(G) is not a split Hopf algebroid (i.e., not isomorphic to X k ⊗ Rk(G))
Representative functions algebroid. Examples.
(•) If G is a groupoids with only one object, that is, a group, then Rk(G) is the usual Hopf algebra of representative funcions on the group G. This is isomorphic to the finite dual k[G]o of the group algebra k[G].
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 36 / 42 % / (•) Let G: G × X o ι X be an action groupoid. Then there is a morphism pr1 / of Hopf algebroids: X X X X (k , k ⊗ Rk(G) ⊗ k ) −→ (k , Rk(G)). Furthermore, if the action is transitive, then any isotropy Hopf algebra x (kx, Rk(G) ), for x ∈ X, is isomorphic to (k, Rk(G)). In general, Rk(G) is not a split Hopf algebroid (i.e., not isomorphic to X k ⊗ Rk(G))
Representative functions algebroid. Examples.
(•) If G is a groupoids with only one object, that is, a group, then Rk(G) is the usual Hopf algebra of representative funcions on the group G. This is isomorphic to the finite dual k[G]o of the group algebra k[G]. (•) Let X = {1, 2} be a set of two elements and consider as before the groupoid G{1,2} of pairs and denote by A := k × k its base algebra. Then L n n A ⊗Mn(k) A n ∈ N Rk(G) = . t n m hv ⊗Mn(k) λw − λ v ⊗Mm(k) wiv ∈ A , w ∈ A , λ ∈ Mm×n(k)
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 36 / 42 Representative functions algebroid. Examples.
(•) If G is a groupoids with only one object, that is, a group, then Rk(G) is the usual Hopf algebra of representative funcions on the group G. This is isomorphic to the finite dual k[G]o of the group algebra k[G]. (•) Let X = {1, 2} be a set of two elements and consider as before the groupoid G{1,2} of pairs and denote by A := k × k its base algebra. Then L n n A ⊗Mn(k) A n ∈ N Rk(G) = . t n m hv ⊗Mn(k) λw − λ v ⊗Mm(k) wiv ∈ A , w ∈ A , λ ∈ Mm×n(k) % / (•) Let G: G × X o ι X be an action groupoid. Then there is a morphism pr1 / of Hopf algebroids: X X X X (k , k ⊗ Rk(G) ⊗ k ) −→ (k , Rk(G)). Furthermore, if the action is transitive, then any isotropy Hopf algebra x (kx, Rk(G) ), for x ∈ X, is isomorphic to (k, Rk(G)). In general, Rk(G) is not a split Hopf algebroid (i.e., not isomorphic to X k ⊗ Rk(G))
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 36 / 42 The properties of Rk(G) when G is transitive. Let G be a transitive groupoid, then its algebra of representative functions enjoys the following properties:
(A0(G), Rk(G)) is a transitive Hopf algebroid, in the sense that each of the fibers of its associated presheaf of groupoids is actually a transitive groupoids (i.e., each of the groupoids Rk(G)(C), A0(G)(C) is transitive, for any commutative algebra C). The notation is R(C):= Algk(R, C).
The fibre functor ω : repk(G) → proj(A0(G)) induces a monoidal
equivalence between the category repk(G) and the category comodRk(G) of comodules with finitely generated underlying module.
Any comodule M in comodRk(G) is a locally free A0(G)-module with
constant rank, in the sense that, if for some x ∈ G0, we have dimk(Mx) = n, then so is the dimension of any other fibre.
The contravariant adjunction.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 37 / 42 Let G be a transitive groupoid, then its algebra of representative functions enjoys the following properties:
(A0(G), Rk(G)) is a transitive Hopf algebroid, in the sense that each of the fibers of its associated presheaf of groupoids is actually a transitive groupoids (i.e., each of the groupoids Rk(G)(C), A0(G)(C) is transitive, for any commutative algebra C). The notation is R(C):= Algk(R, C).
The fibre functor ω : repk(G) → proj(A0(G)) induces a monoidal
equivalence between the category repk(G) and the category comodRk(G) of comodules with finitely generated underlying module.
Any comodule M in comodRk(G) is a locally free A0(G)-module with
constant rank, in the sense that, if for some x ∈ G0, we have dimk(Mx) = n, then so is the dimension of any other fibre.
The contravariant adjunction.
The properties of Rk(G) when G is transitive.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 37 / 42 (A0(G), Rk(G)) is a transitive Hopf algebroid, in the sense that each of the fibers of its associated presheaf of groupoids is actually a transitive groupoids (i.e., each of the groupoids Rk(G)(C), A0(G)(C) is transitive, for any commutative algebra C). The notation is R(C):= Algk(R, C).
The fibre functor ω : repk(G) → proj(A0(G)) induces a monoidal
equivalence between the category repk(G) and the category comodRk(G) of comodules with finitely generated underlying module.
Any comodule M in comodRk(G) is a locally free A0(G)-module with
constant rank, in the sense that, if for some x ∈ G0, we have dimk(Mx) = n, then so is the dimension of any other fibre.
The contravariant adjunction.
The properties of Rk(G) when G is transitive. Let G be a transitive groupoid, then its algebra of representative functions enjoys the following properties:
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 37 / 42 The fibre functor ω : repk(G) → proj(A0(G)) induces a monoidal
equivalence between the category repk(G) and the category comodRk(G) of comodules with finitely generated underlying module.
Any comodule M in comodRk(G) is a locally free A0(G)-module with
constant rank, in the sense that, if for some x ∈ G0, we have dimk(Mx) = n, then so is the dimension of any other fibre.
The contravariant adjunction.
The properties of Rk(G) when G is transitive. Let G be a transitive groupoid, then its algebra of representative functions enjoys the following properties:
(A0(G), Rk(G)) is a transitive Hopf algebroid, in the sense that each of the fibers of its associated presheaf of groupoids is actually a transitive groupoids (i.e., each of the groupoids Rk(G)(C), A0(G)(C) is transitive, for any commutative algebra C). The notation is R(C):= Algk(R, C).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 37 / 42 Any comodule M in comodRk(G) is a locally free A0(G)-module with
constant rank, in the sense that, if for some x ∈ G0, we have dimk(Mx) = n, then so is the dimension of any other fibre.
The contravariant adjunction.
The properties of Rk(G) when G is transitive. Let G be a transitive groupoid, then its algebra of representative functions enjoys the following properties:
(A0(G), Rk(G)) is a transitive Hopf algebroid, in the sense that each of the fibers of its associated presheaf of groupoids is actually a transitive groupoids (i.e., each of the groupoids Rk(G)(C), A0(G)(C) is transitive, for any commutative algebra C). The notation is R(C):= Algk(R, C).
The fibre functor ω : repk(G) → proj(A0(G)) induces a monoidal
equivalence between the category repk(G) and the category comodRk(G) of comodules with finitely generated underlying module.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 37 / 42 The contravariant adjunction.
The properties of Rk(G) when G is transitive. Let G be a transitive groupoid, then its algebra of representative functions enjoys the following properties:
(A0(G), Rk(G)) is a transitive Hopf algebroid, in the sense that each of the fibers of its associated presheaf of groupoids is actually a transitive groupoids (i.e., each of the groupoids Rk(G)(C), A0(G)(C) is transitive, for any commutative algebra C). The notation is R(C):= Algk(R, C).
The fibre functor ω : repk(G) → proj(A0(G)) induces a monoidal
equivalence between the category repk(G) and the category comodRk(G) of comodules with finitely generated underlying module.
Any comodule M in comodRk(G) is a locally free A0(G)-module with
constant rank, in the sense that, if for some x ∈ G0, we have dimk(Mx) = n, then so is the dimension of any other fibre.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 37 / 42 This establishes a contravariant functor
χk : CHAlgdk −→ Grpd
from the category of commutative Hopf algebroids to the category of groupoids. Furthermore, for any groupoid G, we have a natural transformation:
G −→ χk ◦ Rk(G),
which is not in general a monomorphism. When it is, the groupoid χk(Rk(G)) is sometimes called the algebraic cover of G.
The contravariant adjunction
The character functor and the counit. Let (A, H) be a commutative Hopf algebroid such that A , 0 and Algk(A, k) , ∅. The groupoid
s∗ ∗ / H(k): Algk(H, k) o ι Algk(A, k), t∗ / is referred to as the character groupoid of (A, H).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 38 / 42 Furthermore, for any groupoid G, we have a natural transformation:
G −→ χk ◦ Rk(G),
which is not in general a monomorphism. When it is, the groupoid χk(Rk(G)) is sometimes called the algebraic cover of G.
The contravariant adjunction
The character functor and the counit. Let (A, H) be a commutative Hopf algebroid such that A , 0 and Algk(A, k) , ∅. The groupoid
s∗ ∗ / H(k): Algk(H, k) o ι Algk(A, k), t∗ / is referred to as the character groupoid of (A, H). This establishes a contravariant functor
χk : CHAlgdk −→ Grpd from the category of commutative Hopf algebroids to the category of groupoids.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 38 / 42 The contravariant adjunction
The character functor and the counit. Let (A, H) be a commutative Hopf algebroid such that A , 0 and Algk(A, k) , ∅. The groupoid
s∗ ∗ / H(k): Algk(H, k) o ι Algk(A, k), t∗ / is referred to as the character groupoid of (A, H). This establishes a contravariant functor
χk : CHAlgdk −→ Grpd from the category of commutative Hopf algebroids to the category of groupoids. Furthermore, for any groupoid G, we have a natural transformation:
G −→ χk ◦ Rk(G), which is not in general a monomorphism. When it is, the groupoid χk(Rk(G)) is sometimes called the algebraic cover of G.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 38 / 42 (A, H) is a flat Hopf algebroid. The category comodH is a symmetric rigid monoidal k-linear abelian category, and any object in this category is finitely generated and projective A-module; the canonical map
Lk(comodH , O) −→ H, where O : comodH → proj(A), is an isomorphism of Hopf algebroids; there is a symmetric monoidal functor F : comodH −→ rep χk(H) ,
which is a morphism of fibre functors. there is a natural morphism of Hopf transitive Hopf algebroids: (A, H) −→ A0(χk(H)), Rk(χk(H))
The contravariant adjunction
The unit. Let (A, H) be a transitive Hopf algebroid. Then
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 39 / 42 the canonical map
Lk(comodH , O) −→ H, where O : comodH → proj(A), is an isomorphism of Hopf algebroids; there is a symmetric monoidal functor F : comodH −→ rep χk(H) ,
which is a morphism of fibre functors. there is a natural morphism of Hopf transitive Hopf algebroids: (A, H) −→ A0(χk(H)), Rk(χk(H))
The contravariant adjunction
The unit. Let (A, H) be a transitive Hopf algebroid. Then
(A, H) is a flat Hopf algebroid. The category comodH is a symmetric rigid monoidal k-linear abelian category, and any object in this category is finitely generated and projective A-module;
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 39 / 42 there is a natural morphism of Hopf transitive Hopf algebroids: (A, H) −→ A0(χk(H)), Rk(χk(H))
The contravariant adjunction
The unit. Let (A, H) be a transitive Hopf algebroid. Then
(A, H) is a flat Hopf algebroid. The category comodH is a symmetric rigid monoidal k-linear abelian category, and any object in this category is finitely generated and projective A-module; the canonical map
Lk(comodH , O) −→ H, where O : comodH → proj(A), is an isomorphism of Hopf algebroids; there is a symmetric monoidal functor F : comodH −→ rep χk(H) ,
which is a morphism of fibre functors.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 39 / 42 The contravariant adjunction
The unit. Let (A, H) be a transitive Hopf algebroid. Then
(A, H) is a flat Hopf algebroid. The category comodH is a symmetric rigid monoidal k-linear abelian category, and any object in this category is finitely generated and projective A-module; the canonical map
Lk(comodH , O) −→ H, where O : comodH → proj(A), is an isomorphism of Hopf algebroids; there is a symmetric monoidal functor F : comodH −→ rep χk(H) ,
which is a morphism of fibre functors. there is a natural morphism of Hopf transitive Hopf algebroids: (A, H) −→ A0(χk(H)), Rk(χk(H))
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 39 / 42 As we have seen before, the represenative functions functor define a contravariant functor
Rk : TGrpd −→ TCHAlgdk, G −→ Rk(G).
On the other hand, the character functor obviously define a contravariant functor χk : TCHAlgdk −→ TGrpd, (A, H) −→ χk(H).
There is a natural isomorphism TGrpd − , χk(+) TCHAlgdk + , Rk(−) .
The contravariant adjunction.
Notations: Denotes by TGrpd the full subcategory of transitive groupoids, and by TCHAlgdk the full subcategory of transitive commutative Hopf algebroids.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 40 / 42 On the other hand, the character functor obviously define a contravariant functor χk : TCHAlgdk −→ TGrpd, (A, H) −→ χk(H).
There is a natural isomorphism TGrpd − , χk(+) TCHAlgdk + , Rk(−) .
The contravariant adjunction.
Notations: Denotes by TGrpd the full subcategory of transitive groupoids, and by TCHAlgdk the full subcategory of transitive commutative Hopf algebroids.
As we have seen before, the represenative functions functor define a contravariant functor
Rk : TGrpd −→ TCHAlgdk, G −→ Rk(G).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 40 / 42 There is a natural isomorphism TGrpd − , χk(+) TCHAlgdk + , Rk(−) .
The contravariant adjunction.
Notations: Denotes by TGrpd the full subcategory of transitive groupoids, and by TCHAlgdk the full subcategory of transitive commutative Hopf algebroids.
As we have seen before, the represenative functions functor define a contravariant functor
Rk : TGrpd −→ TCHAlgdk, G −→ Rk(G).
On the other hand, the character functor obviously define a contravariant functor χk : TCHAlgdk −→ TGrpd, (A, H) −→ χk(H).
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 40 / 42 The contravariant adjunction.
Notations: Denotes by TGrpd the full subcategory of transitive groupoids, and by TCHAlgdk the full subcategory of transitive commutative Hopf algebroids.
As we have seen before, the represenative functions functor define a contravariant functor
Rk : TGrpd −→ TCHAlgdk, G −→ Rk(G).
On the other hand, the character functor obviously define a contravariant functor χk : TCHAlgdk −→ TGrpd, (A, H) −→ χk(H).
There is a natural isomorphism TGrpd − , χk(+) TCHAlgdk + , Rk(−) .
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 40 / 42 Alan Weinstein:
“I hope to have convinced the reader that groupoids are worth knowing about and worth looking out for.” “Spero di aver convinto il lettore che i gruppoidi sono qualcosa che valga la pena conoscere e investigare.” “Espero haber convencido al lector de que merece la pena conocer los groupoids y quedarse a la expectativa.” “J’espère avoir convaincu le lecteur que les groupoïdes valent la peine d’être connus et méritent d’être recherchés.”
Alain, Connes:
“It is fashionable among mathematicians to despise groupoids and to consider that only groups have an authentic mathematical status, probably because of the pejorative suffix ’oid’. ”
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 41 / 42 Alain, Connes:
“It is fashionable among mathematicians to despise groupoids and to consider that only groups have an authentic mathematical status, probably because of the pejorative suffix ’oid’. ”
Alan Weinstein:
“I hope to have convinced the reader that groupoids are worth knowing about and worth looking out for.” “Spero di aver convinto il lettore che i gruppoidi sono qualcosa che valga la pena conoscere e investigare.” “Espero haber convencido al lector de que merece la pena conocer los groupoids y quedarse a la expectativa.” “J’espère avoir convaincu le lecteur que les groupoïdes valent la peine d’être connus et méritent d’être recherchés.”
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 41 / 42 References
Katherine, Brading and Elena, Castellani, eds. (2003), Symmetries in Physics: Philosophical Reflections. Cambridge: Cambridge University Press. R.Brown, From groups to groupoids: A brief survey. Bull. London Math. Soc. 19 (2), (1987), 113–134. Alain, Connes, Noncommutative Geometry. (1994), San Diego: Academic Press. L. El Kaoutit, On geometrically transitive Hopf algebroids, arXiv:1508.05761v2 [math:AC] (2015). L. El Kaoutit and N. Kowalzig, Morita theory for Hopf algebroids, principal bibundles, and weak equivalences. Documenta Math., 22, 551–609 (2017). Alexandre Guay and Brian Hepburn, Symmetry and Its Formalisms: Mathematical Aspects, Philosophy of Science, 76 (April 2009), 160?178. Alan, Weinstein, Groupoids: Unifying Internal and External Symmetry, Notices of the American Mathematical Society 43: 744–752. Hermann, Weyl, Symmetry. (1952) Princeton, N.J: Princeton University Press.
Departamento de Álgebra (UGR) Basic Theory of Abstract Groupoids. TETUÁN, December 2018. 42 / 42