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added to exact functor the characterization of left exact functors as those preserving terminal object and pullbacks. This was previously stated only at finitely complete category.
I find the discussion at exact functor a bit strange. Usually a functor is DEFINED to be left exact if it preserves finite limits. That is a clear precise definition… but is given as being just the ’idea’ whilst the ’definition’ is in terms of the characterisation (already essentially in Grothendieck’s Bourbaki seminar(195) in 1960 by the way) that this corresponded to a certain comma category being (co)filtering. This property is fine fro generalisation, but it seems very odd to give it as the definition. Does anyone else agree with this or am I being a traditionalist!!!!
I agree. I would only use the term “left exact functor” for a finite-limit preserving functor between categories that have finite limits. The more general notion I would call flat.
I would agree with your extra point on flat, Mike. Mine was a bit more pickie! I think we should avoid having the deep plunge into abstraction too soon in entries. (This could be seen as a criticism on nLab by some in the recent discussion.) The question is in any given context / entry should a pedagogic or perhaps historical perspective come fairly early on (even if there is a historical paragraph later on perhaps). I have to write up something on Grothendieck’s Galois theory for another piece of work and have been looking back at his Techniques de descent. (Very clearly written) I will try to restructure exact functor later when I have finished what I have to do. (I also think that Grothendieck’s Galois theory needs a bit a work, but cannot see yet what needs adding.)
I have made some readjustments to exact functor.
Thanks! I tried to clarify further.
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