nLab groupoid representation

Idea

A groupoid representation is a representation of a groupoid.

Definition

Definition

(groupoid representation)

Let $\mathcal{G}$ be a groupoid. Then:

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

$$\rho \;\colon\; \mathcal{G} \longrightarrow Core(Vect)$$

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

1. For each object $x$ of $\mathcal{G}$ a vector space $V_x$;

2. for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a linear map $\rho(f) \;\colon\; V_x \to V_y$

such that

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

   $$\rho(g) \circ \rho(f) = \rho(g \circ f)$$

2. (respect for identities) for each object $x$ of the groupoid we have an equality

   $$\rho(id_x) = id_{V_x} \,.$$

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

$$\rho \;\colon\; \mathcal{G} \longrightarrow Core(Set)$$

to the groupoid core of Set. Hence this is

1. For each object $x$ of $\mathcal{G}$ a set $S_x$;

2. for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a function $\rho(f) \;\colon\; S_x \to S_y$

such that composition and identities are respected, as above.

For $\rho_1$ and $\rho_2$ two such representations, then a homomorphism of representations

$$\phi \;\colon\; \rho_1 \longrightarrow \rho_2$$

is a natural transformation between these functors, hence is

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

  $$(V_1)_x \overset{\phi(x)}{\longrightarrow} (V_2)_x$$

• such that for all morphisms $x \overset{f}{\longrightarrow} y$ we have

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

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

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

Properties

Corollary

(groupoid representations are products of group representations)

Assuming the axiom of choice then the following holds:

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

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

Proof

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

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

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

$$inc \;\colon\; \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i \overset{}{\longrightarrow} \mathcal{G} \,.$$

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

$$inc^\ast \;\colon\; Hom( \mathcal{G}, \mathcal{C} ) \longrightarrow Hom( \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i, \mathcal{C} )$$

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

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

Example

(fundamental theorem of covering spaces)

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

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

If $X$ 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.