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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?
For subobjects one typically speaks of characteristic maps or characteristic functions. The corresponding classifiyng space is a subobject classifier .
More generally, in an (infinity,1)-topos every “small” (see there) object in a slice (infinity,1)-topos is given by a classifying morphism into the object classifier;
In dependent/homotopy type theory these classifying morphisms are the categorical semantics for functions into a type of types that classify dependent types. See at categorical model of dependent types for more on this.