Re: Functors and limits

[Aleks Kissinger <aleks0@gmail.com>]

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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