TITLE: Unifying Geometry for Characteristic Classes
QUESTION [31 upvotes]: When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches:

Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, page 78, th. 3.2.
Characteristic Classes as Pullbacks of the Cohomology of Grassmannians. See, for instance, Vector Bundles and K-Theory, page 84, th. 3.9.
Locus of Linear Dependency of General Sections (and translating into cohomology by intersection). See, for instance, 3264 & All That: Intersection Theory in Algebraic Geometry, page 59, prop-def. 1.36.
Characteristic Polynomial of the Curvature Matrix (at least when working with smooth manifolds). See, for instance, Lecture Notes on Seiberg-Witten Invariants, page 19, sec. 1.5.

The approaches 2, 3 and 4 are proven to be equivalent by comparing the obtained objects with the axiomatic definition. But I wonder
How can be seen geometrically, in an intuitive way, that the approaches 2, 3 and 4, however different may seem, describe the same phenomena in the non-triviality of vector bundles?
Any intuitive explanation is welcome.

REPLY [5 votes]: (A lot of people here can write a better comment about connection b/w 2 and 4 — but per Jjm's request here is a rough sketch.)
Algebraically the universal $G$-bundle is modelled by Weil (DG-)algebra $W(g)=\Lambda(g^\ast)\otimes S(g)$.
For a principle $G$-bundle $E\to M$ a connection on $E$ gives a map $g^\ast\to\Omega^1(E)$ which together with curvature define a map $W(g)\to C_{de Rham}(E)$. This map agrees with filtrations so it induces a map of $E_2$-terms of corresponding spectral sequences, $H^q(g;S^p(g^\ast))\to H^{2p}(B;H^q(G))$. For $q=0$ we get a map from $S(g^*)^G=H(BG)$ to $H(M)$ — namely, an invariant polynomial $P$ goes to $P(\Omega)$. For $gl_n$ this gives the usual definition of Chern class via curvature.