Re: subgroupoids of V-categories

[Jeff Egger <jeffegger@yahoo.ca>]

Hi Emily,

I'm a bit worried about the notion of "local monomorphism".
If V is, say, a poset, then every arrow in it is (tautologously)
monic, and therefore every V-functor is a local monomorphism.

Let us consider, for instance, generalised metric spaces.

Suppose C has two vertices, p and q, with C(p,q)=0 and
C(q,p)=oo (infinity). Then, for any e>0, we can choose G_e
with G_e(p,q)=e and G(q,p)=oo, and the identity-on-objects
functor G_e --> C has the property you desire.

[I'm sure you realise that the definition of "isomorphism"
you gave is equivalent to that of "isomorphism in the
underlying category"; the underlying category of C is "the
arrow", whereas that of G_e is discrete.]

So "the maximal subgroupoid" of C is not defined uniquely,
except up to "equivalence of underlying category". Moreover,
whenever e>d>0, the i-o-o functor G_e --> C factors through
G_d, so there is not even an "optimal" choice of e.

More precisely, if one were to define a V-groupoid to be a
V-category whose underlying category is a groupoid (which
is, implicitly, what you seem to be doing), then I've just
shown that the forgetful functor from [0,oo]-groupoids to
[0,oo]-categories does not have a right adjoint.

Of course, I think it would be better to define a V-groupoid
in a more Hopf-y kind of way---e.g., in terms of what one
might call a co-Maltsev operation---but that might not suit
your purposes, and would seem to make things much
more difficult, at least in general.

Cheers,
Jeff.

--------------------------------------------
On Tue, 11/19/13, Emily Riehl <eriehl@math.harvard.edu> wrote:

  Subject: categories: subgroupoids of V-categories
  To: "Categories" <categories@mta.ca>
  Received: Tuesday, November 19, 2013, 1:51 PM

  Hi all,

  For general V (closed symmetric monoidal, bicomplete), is
  there a general
  way to construct the maximal subgroupoid of a V-category C?

  I think I know how to *detect* the maximal subgroupoid. A
  map in C is an
  isomorphism iff it is representably so: Writing 1 for the
  monoidal unit,
  we say f : 1 -> C(x,y) is an iso iff the induced map f_*
  : C(z,x) ->
  C(z,y) is an iso in V for all z. So we might say that a
  V-category G with
  the same objects and C and an identity-on-objects local
  monomorphism G ->
  C is the maximal subgroupoid provided that a morphism f
  factors through
  G(x,y) -> C(x,y) just when f is an isomorphism.

  In examples, this is probably good enough, but I still would
  feel better
  if I had a general construction of the maximal subgroupoid.
  I feel like
  this should be some sort of weighted limit, perhaps with
  some additional
  structure on V?

  Thanks,
  Emily



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]