A classifying map or classifying morphism for a given object is a morphism into a classifying space that classified this object.

Where can I find a construction of the classifying morphism to a classifying space for a G-bundle with connection - using the connections as a 1-form?

Examples

For subobjects one typically speaks of characteristic maps or characteristic functions. The corresponding classifiyng space is a subobject classifier . See there for more.