nLab
projective representation

A projective representation of a group $G$ is a group homomorphism $G\to \mathrm{PGL}(V)$, where $V$ is a $𝕂$-vector space. Via the projection $\mathrm{GL}(V)\to \mathrm{PGL}(V)=\mathrm{GL}(V)/{𝕂}^{*}$, every linear representation of $G$ induces a projective representation.

By the fibration sequence

$$\begin{array}{ccc}B{𝕂}^{*}& \to & B\mathrm{GL}(V)\\ \downarrow & & \downarrow \\ *& \to & B\mathrm{PGL}(V)\end{array}$$\array{ \mathbf{B}\mathbb{K}^*&\to & \mathbf{B}GL(V)\\ \downarrow&& \downarrow\\ *&\to&\mathbf{B}PGL(V) }

the obstruction to lift a projective representation of $G$ to a linear representation is represented by an element $c_{\rho }$ in $H^2(G,{𝕂}^{*})$.

A refinement of this idea consists in looking at the 2-groupoid $B\mathrm{GL}(V)//K^{*}$. Then a functor $(\rho ,\lambda ):𝔹G\to B\mathrm{GL}(V)//K^{*}$ consists of two maps $\rho :G\to \mathrm{GL}(V)$ and $\lambda :G\times G\to {𝕂}^{*}$ such that $\rho (g)\rho (h)=\lambda (g,h)\rho (gh)$, and $\lambda$ is a 2-cocycle on $G$ with values in ${𝕂}^{*}$ representing the cohomology class $c_{\rho }$.