On the homology of partial group representations

We study how the partial group (co)homology of a group $G$ with coefficient in a partial representation $M$ can be described using the usual group (co)homology. To address this, we introduce the concept of the $\mathit{\text{universal}}\mathit{\text{globalization}}$ $\mathrm{\Lambda }(M)$ of a partial group representation $M$ of $G$. Our main result shows that the partial group homology $H_{\bullet }^{\text{par}}(G,M)$ is naturally isomorphic to the classical group homology $H_{\bullet }(G,\mathrm{\Lambda }(M))$. We extend this result to the cohomological framework, obtaining a spectral sequence involving the classical group cohomology that converges to the partial group cohomology. Notably, when $G$ is countable, the spectral sequence collapses, resulting in a natural isomorphism $H_{\text{par}}^{\bullet }(G,M)\cong H^{\bullet }(G,{\mathrm{Hom}}_{K_{\text{par}}G}(\mathrm{\Lambda }(K_{par}G),M))$, where $K_{par}G$ stands for the partial group algebra of $G$.