Correction: 1. reflects limits if F*(k) contains only limiting cones On 26 April 2011 16:15, Aleks Kissinger <aleks0@gmail.com> wrote:
Dear categorists,
The "standard" definitions for functors doing nice things with limits have always seemed a bit clumsy to me. Here's what I think is quite a natural way to unroll the quantifiers:
For a functor F : C --> D and a cone k, let F*(k) be the class of all cones k' in C s.t. F(k') = k.
For all limiting cones k in D, F.... 1. reflects limits if F*(k) != {} implies F*(k) contains a limiting cone 2. lifts limits if F*(k) contains a limiting cone 3. lifts limits uniquely if F*(k) contains exactly 1 limiting cone, but possibly other cones 4. creates limits if F*(k) = {k'}, for k' a limiting cone
This seems to read much more cleanly than the usual, quantifier-laden version that seems to be in most standard texts. Of course, they're all still there in the def, but there is no ambiguity in how they nest. For example, the difference in 3 in 4 ranges from subtle to all-but-invisible in most of the places I've seen them defined. Does this definition, or some close relative exist somewhere? If not, is it problematic somehow? For example, do you get into trouble when F*(k) is a proper class?
Thanks! Aleks
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