If one allows (and it makes good sense to allow this) curvature characteristic forms to be represented not necessarily by globally defined forms, but by cocycles in the abelian sheaf cohomology of the truncated de Rham complex$\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \to \cdots \to \Omega^n_{cl}(X)$ then a weaker notion than that of a connection serves the same purpose: that of a pseudo-connection .

Effectively a pseudo-connection on a principal bundle is locally a collection of Lie-algebra valued 1-forms, that do not have to satisfy the cocycle condition familiar from connections.

That makes pseudo-connections rather empty structures. And in a precise sense this is actually the point of them: there is a (non-concrete) Lie groupoid$\mathbf{B}G_{diff}$ such that Cech cocycles with values in it are those for $G$-principal bundles with pseudo-connections. This groupoid is in fact weakly equivalent to the delooping groupoid $\mathbf{B}G$ that serves to classify just bare smooth $G$-principal bundles

The purpose of pseudo-connections is precisely to form this resolution of $\mathbf{B}G$, which allows to represent an intrinsic curvature characteristic class to be modeled by an anafunctor with tip $\mathbf{B}G_{diff}$.

While for ordinary $G$-principal bundles the notion of pseudo-connection can be (and traditionally is) circumvented, it does become crucial for gerbes and principal 2-bundles and more generally in infinity-Chern-Weil theory. For instance an equivariantgerbewith connection is not an equivariant object in the 2-category of gerbes with ordinary connections, but a certain constrained equivariant gerbe with pseudo-connection.