nLab
groupoid representation

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contentsrepre

Idea

A groupoid representation is a representation of a groupoid.

Definition

Definition

(groupoid representation)

Let 𝒢\mathcal{G} be a groupoid. Then:

A linear representation of 𝒢\mathcal{G} is a groupoid homomorphism (functor)

ρ:𝒢Core(Vect) \rho \;\colon\; \mathcal{G} \longrightarrow Core(Vect)

to the groupoid core of the category Vect of vector spaces (this example). Hence this is

  1. For each object xx of 𝒢\mathcal{G} a vector space V xV_x;

  2. for each morphism xfyx \overset{f}{\longrightarrow} y of 𝒢\mathcal{G} a linear map ρ(f):V xV y\rho(f) \;\colon\; V_x \to V_y

such that

  1. (respect for composition) for all composable morphisms xfygzx \overset{f}{\to}y \overset{g}{\to} z in the groupoid we have an equality

    ρ(g)ρ(f)=ρ(gf) \rho(g) \circ \rho(f) = \rho(g \circ f)
  2. (respect for identities) for each object xx of the groupoid we have an equality

    ρ(id x)=id V x. \rho(id_x) = id_{V_x} \,.

Similarly a permutation representation of 𝒢\mathcal{G} is a groupoid homomorphism (functor)

ρ:𝒢Core(Set) \rho \;\colon\; \mathcal{G} \longrightarrow Core(Set)

to the groupoid core of Set. Hence this is

  1. For each object xx of 𝒢\mathcal{G} a set S xS_x;

  2. for each morphism xfyx \overset{f}{\longrightarrow} y of 𝒢\mathcal{G} a function ρ(f):S xS y\rho(f) \;\colon\; S_x \to S_y

such that composition and identities are respected, as above.

For ρ 1\rho_1 and ρ 2\rho_2 two such representations, then a homomorphism of representations

ϕ:ρ 1ρ 2 \phi \;\colon\; \rho_1 \longrightarrow \rho_2

is a natural transformation between these functors, hence is

  • for each object xx of the groupoid a (linear) function

    (V 1) xϕ(x)(V 2) x (V_1)_x \overset{\phi(x)}{\longrightarrow} (V_2)_x
  • such that for all morphisms xfyx \overset{f}{\longrightarrow} y we have

    ϕ(y)ρ 1(f)=ρ 2(x)ϕ(x)AAAAAA(V 1) x ϕ(x) (V 2) x ρ 1(f) ϕ 2(f) (V 1) y ϕ(y) (V 2) y \phi(y) \circ \rho_1(f) = \rho_2(x) \circ \phi(x) \phantom{AAAAAA} \array{ (V_1)_x &\overset{\phi(x)}{\longrightarrow}& (V_2)_x \\ {}^{\mathllap{\rho_1(f)}}\downarrow && \downarrow^{\mathrlap{\phi_2(f)}} \\ (V_1)_y &\underset{\phi(y)}{\longrightarrow}& (V_2)_y }

A permutation representation of 𝒢\mathcal{G} is often called a “𝒢\mathcal{G}-set” (see at G-set) and the category of permutation representations is also often denoted

𝒢SetAAAAAorAAAAASet 𝒢 \mathcal{G}Set \phantom{AAAAA} \text{or} \phantom{AAAAA} Set^{\mathcal{G}}

Properties

Corollary

(groupoid representations are products of group representations)

Assuming the axiom of choice then the following holds:

Let 𝒢\mathcal{G} be a groupoid. Then its category of groupoid representations is equivalent to the product category indexed by the set of connected components π 0(𝒢)\pi_0(\mathcal{G}) (this def.) of group representations of the automorphism group G iAut 𝒢(x i)G_i \coloneqq Aut_{\mathcal{G}}(x_i) (this def.) for x ix_i any object in the iith connected component:

Rep Grpd(𝒢)iπ 0(𝒢)Rep(G i). Rep_{Grpd}(\mathcal{G}) \;\simeq\; \underset{i \in \pi_0(\mathcal{G})}{\prod} Rep(G_i) \,.
Proof

Let 𝒞\mathcal{C} be the category that the representation is on. Then by definition

Rep Grpd(𝒢)=Hom(𝒢,𝒞). Rep_{Grpd}(\mathcal{G}) = Hom( \mathcal{G} , \mathcal{C} ) \,.

Consider the injection functor of the skeleton (from this lemma)

inc:iπ 0(𝒢)BG i𝒢. inc \;\colon\; \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i \overset{}{\longrightarrow} \mathcal{G} \,.

By this lemma the pre-composition with this constitutes a functor

inc *:Hom(𝒢,𝒞)Hom(iπ 0(𝒢)BG i,𝒞) inc^\ast \;\colon\; Hom( \mathcal{G}, \mathcal{C} ) \longrightarrow Hom( \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i, \mathcal{C} )

and by combining this lemma with this lemma this is an equivalence of categories. Finally, by this example the category on the right is the product of group representation categories as claimed.

Examples

Example

(groupoid representation of delooping groupoid is group representation)

If BGB G is the delooping groupoid of a group GG (this example), then a groupoid representation of BGB G according to def. is equivalently a group representation of the group GG:

Rep Grpd(BG)Rep(G). Rep_{Grpd}(B G) \simeq Rep(G) \,.
Example

(fundamental theorem of covering spaces)

For XX a topological space then forming monodromy is a functor from the category of covering spaces over XX to that of permutation representations of the fundamental groupoid of XX:

Fib:Cov(C)Set Π 1(X). Fib \;\colon\; Cov(C) \longrightarrow Set^{\Pi_1(X)} \,.

If XX is locally path connected and semi-locally simply connected, then this is an equivalence of categories. See at fundamental theorem of covering spaces for details.

Last revised on July 11, 2017 at 05:14:30. See the history of this page for a list of all contributions to it.