In the context of homological algebra its pretty common to talk about lifts, I think. For example, if you have a short exact sequence 0->A->B->C->0 then any c in C lifts to some b in B. For example, see Weibel's book on page 316 (http://books.google.ie/books?id=flm-dBXfZ_gC&lpg=PA14&dq=short%20exact%20sequence%20lift%20weibel&pg=PA316#v=onepage&q=lift&f=false). I didn't have any problems finding the .lift() method.
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