Re: subgroupoids of V-categories

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

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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