Re: Compatibility of functors with limits

[Tom LEINSTER <leinster@ihes.fr>]

A week ago I asked this list a question, thinking that lots of people
would know the answer.  Either I was wrong or those who know are keeping
it to themselves, as I didn't get any answers at all.  But I now
understand the issue better than before, so I'd like to try re-asking my
question in a different way and see if that elicits a response.

I'm trying to understand a certain notion of compatibility of a functor
with limits.  There are of course several well-known such notions:
preserves, reflects, creates.  I called mine (provisionally) "respecting"
limits.  It is close to preservation, but not quite the same; it seems
more constructive and perhaps (dare I say it?) more natural.
Preservation is all very well when the domain category has all limits, or
all limits of whatever type we're concerned with.  But otherwise, it seems
a bit suspect.  For let F: A ---> B be a functor; preservation says that
given a diagram D: I ---> A in A,

- if D *does* admit a limit cone, then the image of that cone under F is
  also a limit cone,

- if D *doesn't* admit a limit cone, then... well, nothing.

"Respect", on the other hand, says the same as preservation in the first
case, but also says something in the second case.  (So respect is in
general stronger than preservation.)

Last time I gave a definition of respect in terms of categories of cones;
that definition is appended to this mail.  Here's a different way to put
it: F "respects limits for D" if the canonical map

  \int^a  (\int_i A(a, Di)) \times Fa   ---->  \int_i FDi

is an isomorphism.  Here \int^a denotes coend over a in A, \int_i is limit
over i in I, and I'm working under the assumption that these (co)limits
exist in the codomain category B.  Now, if D does have a limit in A then
the left-hand side is

  \int^a  A(a, \int_i Di) \times Fa

which by density is just F \int_i Di; so, as claimed, respect is the same
as preservation in the case where the limit exists.

My question was whether anyone understood "respect of limits" well, or
could shed any light on it.  It seems to me that, as well as being just
the right thing in certain examples I've been considering, it's a very
natural concept.

Tom


The definition of respect from last time:

> Let F: A ---> B be a functor, where B is a category with (for
> sake of argument) all small limits and colimits.  Let I be a small
> category and D: I ---> A a diagram in A; write Cone(D) for the category
> of cones on/into D in A, write Cone(FD) for the category of cones on FD
> in B, and write
>
>   F_*: Cone(D) ---> Cone(FD)
>
> for the induced functor.  Then F can be said to "respect limits for D"
> if the colimit of F_* is the terminal object of Cone(FD) (that is, the
> limit cone on FD).