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A definition that always works and does agree with that one in the finite-dimensional case is the following: put $$ C^k(\mathfrak{g})=({\Lambda}^k\mathfrak{g})^*=\operatorname{Hom}({\Lambda}^k\mathfra …

The two definitions are the same. The thesis of Lefèvre-Hasegawa does not require the differential to be zero, it requires the component $m_1$ of the differential to be equal to zero: minimality trans …

To add a bit to what Damien says, addressing your question on how to generalise the gauge approach (which is equivalent to the approach outlined by Damien, as proved by several people): You can view …

If you want to find another $\sigma'_{V,W}\colon V\otimes W\to W\otimes V$ so that $\sigma'_{U,V\otimes W}=\sigma'_{U,V}\sigma'_{U,W}$ and assume that on homogeneous elements $\sigma'_{V,W}(v\otimes w …

OK let me try a naive answer, and then maybe you will elaborate a bit on what it is that worries you? The key idea is very simple - the bar differential has a part coming from the differential on $A …

Maybe http://arxiv.org/abs/0707.0889 could be of any help? It's general enough - representations of properads cover a huge variety of cases, from algebraic structures to formal differential geometric …