The Stacks project

Example 92.6.1 (Representations of a topological group). Let $\Gamma $ be a topological group. Let $\mathcal{F}$ be the category defined as follows

  1. an object is a triple $(A, M, \rho )$ consisting of an object $A$ of $\mathcal{C}_\Lambda $, a finite projective $A$-module $M$, and a continuous homomorphism $\rho : \Gamma \to \text{GL}_ A(M)$ where $\text{GL}_ A(M)$ is given the discrete topology1, and

  2. a morphism $(f, g) : (B, N, \tau ) \to (A, M, \rho )$ consists of a morphism $f : B \to A$ in $\mathcal{C}_\Lambda $ together with a map $g : N \to M$ which is $f$-linear and $\Gamma $-equivariant and induces an isomorpism $N \otimes _{B, f} A \cong M$.

The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda $ sends $(A, M, \rho )$ to $A$ and $(f, g)$ to $f$. It is clear that $p$ is cofibred in groupoids. Given a finite dimensional $k$-vector space $V$ and a continuous representation $\rho _0 : \Gamma \to \text{GL}_ k(V)$, let $x_0 = (k, V, \rho _0)$ be the corresponding object of $\mathcal{F}(k)$. We set

\[ \mathcal{D}\! \mathit{ef}_{V, \rho _0} = \mathcal{F}_{x_0} \]
[1] An alternative would be to require the $A$-module $M$ with $G$-action given by $\rho $ is an $A\text{-}G$-module as defined in √Čtale Cohomology, Definition 59.57.1. However, since $M$ is a finite $A$-module, this is equivalent.

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