Re: Compatibility of functors with limits

[Tom LEINSTER <leinster@ihes.fr>]

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