# LisMath Seminar

### Nonabelian Cohomology

For a given groupoid $G$ and $M$ a $G$-module, the $n$-th cohomology is defined as the set of homotopy classes $H^n(G,M)=[F_{\star}^{st} (G), K_n(M,G);\phi ]$, where $F_{\star}^{st} (G)$ is the free crossed resolution of $G$, and $\phi : F_1^{st} (G)\to G$ is the standard morphism.

In this talk we assign a free crossed complex to a cover $\mathcal{U}$ of the topological space $X$, so we get the notion of nonabelian cohomology.

We finish our talk by introducing a long exact sequence for nonabelian cohomology.

Bibliography:

[1] R. Brown , P. Higgins and R. Sivera. Nonabelian algebraic topology. European Mathematical Society, 2010

[2] T. Nikolaus and K. Waldorf. Lifting problems and transgression for non-abelian gerbes, Advances in Mathematics 242, pp. 50–79, 2013

[3] L. Breen. Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, pp. 401–476, 2007.