I know about Ehresmann connections on fiber bundles and covariant derivatives as an (equivalent) way to define linear Ehresmann connections on vector bundles. My question is:
Is there any notion of covariant derivative equivalent to Ehresmann connection in the most general setting concerning fiber bundles?
When I say "the most general setting", I am emphasizing that the fiber bundle do not have any further structure than being just a fiber bundle (it may not be a vector bundle nor a principal bundle).
Thanks in advance,
PS: I'm concerning the case when the fiber bundles are smooth. I don't worry about the non smooth case.