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I understood that the old terminology was ’projective system’, and ’projective limit’ refereed to the limit of a projective system. Can anyone confirm that? if I am right the present entry is slightly incorrect, but this needs checking first before changing it.
The meaning of words projective system and inverse system, projective limit and inverse limit really depends on author. Some limit such limits to (co)filtered case, some to directed case, some just mean small limits. I do not think there was ever a uniform terminology here. So nowdays we should not really distinguish projective from inverse limits in the generality. Who needs directed or (co)filtered case can specify.
That was not quite my point. The current entry derives pro-object from projective limit rather than projective system. Of course, projective was used as a synonym for inverse in this context…. and has far too many other uses as well!
Shakespeare’s Stratford upon Avon…Edit: comment in a wrong discussion. Sorry.
The treatment in pro-object anyway looks good and standard. I mean one can talk ladders and equivalences among ladders for morphisms but it is equivalent as talking about lim colim homs.
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