subgroupoids of V-categories

subgroupoids of V-categories

[Emily Riehl]

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


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Re: subgroupoids of V-categories

[Zhen Lin Low]

Dear Emily,

I'm sure someone else will have pointed this out by now, but it seems
quite unlikely that the maximal subgroupoid has a V-enrichment in
general! For instance, suppose V is the category of pointed sets: then
a V-category is simply a category with a system of distinguished zero
morphisms, but a zero morphism is hardly ever an isomorphism. (Of
course, one might argue that the maximal V-subgroupoid _does_ in fact
exist and is the full subcategory of zero objects, but that seems
quite uninteresting.)

Perhaps things are better if we focus on those categories V which are
sufficiently Set-like. Suppose V has pullbacks and the "forgetful"
functor U : V -> Set is an isofibration (= transportable functor) and
has a fully faithful right adjoint. Then, for any injective map f :  X
-> U B in Set, there exists a monomorphism g : A -> B in V such that f
= U g and for any morphism h : C -> B in V such that U h = f k in Set,
there is a unique morphism m : C -> A with k = U m and h = g m. For
example, if V = Top, this is just the construction of the subspace
topology on a subset, and if V = sSet, this constructs the maximal
simplicial subset containing a fixed subset of vertices. It is clear
how to use this to make the set of isomorphisms between two objects in
a V-category into a V-object. Composition is inherited if U is
strongly monoidal: the universal property of these "initial lifts"
gives the required factorisations. And there is nothing special about
the maximal subgroupoid: this construction works for any subcategory
whatsoever.

I suppose one could also work "internally" in the cartesian monoidal
case and try to define the V-object of isomorphisms by pulling back
the "name" of the identity morphism along composition and then
projecting down to one of the factors. But I don't see how to
generalise this to the non-cartesian case.

Best wishes,
--
Zhen Lin


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Re: subgroupoids of V-categories

[Ronnie Brown]

Dear Emily,

The following seems relevant to the question and Zhen's answer.

A related question is how to obtain a maximal subgroup object of a
monoid object wrt tensor in a monoidal category.  A particular case
concerned is in

59.  (with N.D. GILBERT), ``Algebraic models of 3-types and
automorphism  structures for crossed modules'', {\em Proc. London
Math. Soc.} (3) 59 (1989)  51-73.

where the monoidal closed category is XMod, that of crossed modules over
groupoids, with  internal hom written XMOD. For an object  C one can
define END(C)= XMOD(C,C), but what should be AUT(C)?

   In this example  there is a functor say U: XMod \to Set which takes
the zero dimensional part.   We then define AUT(C) to be the part of
END(C) over the maximal subgroup of END(C)_0.  This is a "group-like
object".  One argument for this is that the nerve N(C) of an object C is
a simplicial set, so N(END(C)) is a simplicial monoid;  it turns out
that N(AUT(C)) is a simplicial group, which gives a kind of justification.

The above ideas extend to crossed complexes, using Andy Tonks extension
of the Eilenberg-Zilber Theorem from chain complexes to crossed complexes.

All this was part of the following scheme for "higher order symmetry"
(with applications to .....?!),  which clearly should be part of "higher
order group theory":

sets give models of homotopy 0-types;
the automorphisms of a set  form a group which is a model of (pointed,
connected) homotopy 1-types (ancient history);
the automorphisms  of a group form (part of the structure of) a crossed
module which is a model of (pointed, connected) homotopy 2-types (Mac
Lane-Whitehead);
and in the above paper: the automorphisms of a crossed module form (part
of the structure of) a crossed square, (see also a paper of Norrie),
which is a model of (pointed, connected) 3-types (Loday);

the automorphisms of a crossed square form ......... (???)

Best wishes

Ronnie








On 19/11/2013 18:51, Emily Riehl wrote:
> 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
>


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Re: subgroupoids of V-categories

[Jeff Egger]

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



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