Most of our computational tools in homological algebra break pretty badly if you don't assume surjections in $\operatorname{Set}$ split. Here's an illustration: given a group $G$ and a $G$-module $A$, we can identify (without using choice) $H^2(G; A)$ with extensions of $G$ by $A$. But $A$-valued 2-cocycles on $G$ only describe extensions $0 \to A \to E \to G \to 1$ where the surjection splits as a map of sets! In essence, removing choice adds another level to the problem of classification of extensions: first, we must classify extensions of $G$ by $A$ as sets, and then we must classify the compatible group structure to put on top of each such set extension. The same issue arises if you want to compute cohomology of Lie algebras over $\mathbb{Z}$: there are two extension problems that must be solved concurrently.

definitionof trivial Kan fibration that should be changed when AC fails.) $\endgroup$ – Zhen Lin Nov 4 '13 at 23:02