Re: Compatibility of functors with limits

Re: Compatibility of functors with limits

[Tom Leinster]

Mamuka Jibladze wrote:

> A related question: does anybody know any analogs of the
> Freyd's Adjoint Functor Theorems for functors between
> in(co)complete categories?

Borceux states the 'More General Adjoint Functor Theorem' in Vol 1, 6.6.1
of his Handbook.  This requires only that the codomain of the hoped-for
left adjoint is Cauchy-complete (and of course that the known functor has
some properties: it is 'absolutely flat' and satisfies some solution set
conditions).

Here's a representability theorem, presumably related.  Let C be a small,
Cauchy-complete category and let X: C ---> Set.  Then

   X is representable  <=>  X respects small limits.

The same goes for familial representability and connected limits.  Proofs
are at

   http://www.ihes.fr/~leinster/rr.ps

Tom

Re: Compatibility of functors with limits

[Tom LEINSTER]

Mamuka Jibladze wrote:

> It just occurred to me that there is something closely related
> in lattice theory; unfortunately I cannot give a reference, but
> I remember that one calls a subposet P' of a poset P
> relatively (co)complete if whenever a subset of P' has an upper
> bound in P, it has a least upper bound in P'.

This is quite similar, but not the same.  I'll take the dual concept (glbs
rather than lubs), since it was respect of limits that I wrote about
originally.  Take an inclusion P' into P of posets, and take a diagram D
in P', which might as well be just a subset of P'.  Then to say that the
inclusion of P' into P respects meets for D is to say that

  join {lower bounds of D in P'}
= meet D

where both join and meet are taken in P.  (I'm assuming that P is
complete; if not, respect of meets for D also asserts that the join and
the meet exist.)  In my first mail I described, vaguely, respect of limits
as meaning that the limit of the image is "no bigger than it needs to be".
Order theory is (unsurprisingly) the context in which this makes the most
sense: the greatest lower bound of D in P obviously needs to be greater
than all the lower bounds of D in P', but that understood, it's minimal.

The dual of Mamuka's statement is that, with D and P' and P as above, if D
has a lower bound in P then it has a greatest lower bound in P'.  Here's
an example where meets are respected but this condition (= relative
completeness?) fails.  Let 0 be the empty category.  For any category C,
the unique functor 0 ---> C respects limits if and only if C has an object
that is both initial and terminal.  So if 1 is the one-element lattice
then 0 ---> 1 respects meets, and the subset 0 of 0 has a lower bound in 1
but no lower bound in 0.

Tom

Re: Compatibility of functors with limits

[Mamuka Jibladze]

It just occurred to me that there is something closely related
in lattice theory; unfortunately I cannot give a reference, but
I remember that one calls a subposet P' of a poset P
relatively (co)complete if whenever a subset of P' has an upper
bound in P, it has a least upper bound in P'.

A related question: does anybody know any analogs of the
Freyd's Adjoint Functor Theorems for functors between
in(co)complete categories?

Mamuka

Re: Compatibility of functors with limits

[Tom LEINSTER]

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).

Compatibility of functors with limits

[leinster]

I have the sensation that I'm about to ask a question to which half the
readers of this list will be able to see an answer immediately.
Unfortunately, I'm one of the other half.

What should it mean for a functor to "respect limits"?  Consider the
following informal definition: a functor respects limits if given any
diagram in the domain category, the limit of the image of the diagram is no
bigger than it needs to be.  Formally, 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 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).

* Example: if D has a limit in A then the limit is a terminal object of
  Cone(D), so F respects limits for D if and only if it preserves the limit
  in the usual sense.

* Example: let B = Set and let A be the category consisting of a pair of
  parallel arrows; a functor F: A ---> B consists of sets and functions

      sigma, tau: F_0 ---> F_1.

  The condition that F respects pullbacks says that sigma and tau are monic
  and that the images of sigma and tau are disjoint.

The thought behind "no bigger than it needs to be" (a very approximate
description, I know) is that if we have a cone on D with vertex v then
there's an induced map from F(v) to lim(FD), which in some sense places a
"lower bound" on lim(FD): e.g. if B = Set and F(v) is nonempty then lim(FD)
is nonempty.  For F to respect limits for D means that lim(FD) is built up
freely from these F(v)s.

So the question is: is this notion of "respecting limits" well-known or
well-understood?  Is there, for instance, some way of rephrasing it that
brings it into more familiar territory?

Thanks very much,
Tom