Re: Category without objects

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

The notion of groupoids having structure in (at least, pace Bill!) 2
dimensions, namely 0 and 1,  was important for me to explain how one
could get a van Kampen type theorem for the fundamental groupoid on a
set of base points which would then determine completely individual
fundamental groups, hidden in the middle. This seemed to be against the
experience in algebraic topology where invariants in adjacent dimensions
are determined by exact sequences which do not determine the result you
want completely. I was well aware of the latter feature since
determining extensions in some track exact sequences by fibration
methods  was the topic of my 1961 DPhil thesis, under Michael Barratt.

It seemed logical that for a theorem that determined completely
information on 1-type you needed an invariant, in this case a groupoid,
with information in dimensions 0 and 1, to take account of all the
1-type information in the pieces glued together.

This result on groupoids  led to the idea of having invariants with
structure in dimensions 0, ..., n. Though it took 9 years to understand,
it is perhaps not surprisingly that the invariants had to be not of a
bare space but of something which also had structure in dimensions
0,...,n; and so one can use filtered spaces, (with Philip Higgins) and
later, with Loday, n-cubes of spaces.

I mention here that some confusion arises in standard algebraic topology
where people often talk of "the fundamental group of a space";  of
course you can talk only of "the fundamental group of a space with base
point", which is a special case of a space with structure.

This leads me to share with you  some comments of Alexander Grothendieck
on base points, and I hope you enjoy the elegance of the language.   Of
course I entirely agree with his criticism of the limitations of the
concepts considered, but it is also true that even 2-fold groupoids in
general are not well understood, whereas cat^n-groups, i.e. (n+1)-fold
groupoids in which one structure is a group) have this amazing
equivalent format, discovered later by Ellis and Steiner, of crossed
n-cubes of groups.  The more general concept has not, I think, been used
for any specific gluing theorems, but has been studied as a model of
homotopy types Blanc and Paoli).

Comments from Alexander Grothendieck, 12 April, 1983

What you write about Loday's n-Cat-groups makes sense for me and is
quite interesting indeed. When you say they capture truncated homotopy
types, I guess you mean "pointed 0-connected (truncated) homotopy
types". This qualification seems to me an important one - while they are
presumably quite adequate for dealing with a number of situations, it is
kind of clear to me they are not for a "passe partout" description of
homotopy types - both the choice of a base point, and the
0-connectedness assumption, however innocuous they may seem at first
sight, seem to me of a very essential nature. To make an analogy, it
would be just impossible to work at ease with algebraic varieties, say,
if sticking from the outset (as had been customary for a long time) to
varieties which are supposed to be connected. Fixing one point, in this
respect (which wouldn't have occurred in the context of algebraic
geometry) looks still worse, as far as limiting elbow-freedom goes!
Also, expressing a pointed 0-connected homotopy type in terms of a group
object mimicking the loop space (which isn't a group object strictly
speaking), or conversely, interpreting the group object in terms of a
pointed "classifying space", is a very inspiring magic indeed - what
makes it so inspiring it that it relates objects which are definitively
of a very different nature - let's say, "spaces" and "spaces with group
law". The magic shouldn't make us forget though in the end that the
objects thus related are of different nature, and cannot be confused
without causing serious trouble.

(This is taken with thanks from the full edited correspondence available
from
http://webusers.imj-prg.fr/~georges.maltsiniotis/ps.html)

Ronnie









On 08/03/2015 19:53, F. William Lawvere wrote:
> It is difficult to understand "without objects"  without any definition
> of "object". Remember that , already before the 21st century, modern
> mathematics had begun to overcome medieval metaphysics. In fact ,
> in the late 1950s, Alexander Grothendieck had made explicit the definition
>  of "subobject", which seems relevant here, as does his powerful
> legacy of
> relativization in several senses. Now we understand that a category C
> in a category U is a truncated simplicial object C0->...->C3
> satisfying certain
> limit conditions. We are free to call C0 'objects" and C1 "maps" and
> since
> C0->C1 is a subobject of C1, we could also say that objects "are" maps,
> but "mimicked by" seems \x10unnecessary (as well as undefined).
>
> (Recall that it is actions of such a C in a topos U that form the topos
> enveloping, as a full subtopos of sheaves, the typical U-topos E->U).
>
> To give a category "with objects" i\x10n a serious sense would seem to be
> giving MORE than ju\x10st a category, for example an interpretation as
> structures
> C-> B^A, the (functor category also emphasized by Grothendieck)
> of structures of shape A in background B. (Where perhaps B is equipped
> with
> an internal embedding in U itself)
>
> The case of no structure and featureless background ( which seems to
> be the
>  default setting of modern mathematics despite the preference of MacLane's
> dear teacher for a vonNeuman-like setting) means in particular that
> the C0
> in a category there consists of "lauter Einsen" in the sense of Cantor.
>
> Those featureless elements X of C0 do obtain a structure by virtue
> of C1,C2
> because taking the latter into account we can see the inside of  X as
> the "comma" category
> C/X involving (not only the subobjects of X and their inclusions, but
> also singular
> figures and reparameterizations) as very extensively utilized by
> Grothendieck .
>
> Bill
>

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