Sentence:  Accordingly, a "diffeological principal S1-bundle" over a diffeological space "X" is a <WSD>"subduction</WSD>" π : "P" → "X" and a smooth map τ satisfying the same conditions; see [30] for a thorough discussion. 
Meaning of the word marked with <WSD> tags:  A surjection between diffeological spaces such that the target is identified as the pushforward of the source.