Functors and limits

Functors and limits

[Aleks Kissinger]

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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Re: Functors and limits

[Aleks Kissinger]

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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