Re terminology:

[Urs Schreiber <urs.schreiber@googlemail.com>]

Dear Ronnie Brown,

you write::

> My own problem is with term `infinity groupoid' which
>  is used to describe something which is not even a groupoid,

It seems to follow the well established terminology in higher category
theory, which proceeds: category, 2-category, 3-category, ....
infinity-category and  groupoid, 2-groupoid, 3-groupoid, ...
infinity-groupoid.

> It seems to be an example of these confusions is the way the simplicial
> singular complex of a space is called an infinity-groupoid, even the
> `fundamental infinity groupoid', when what seems to be referred to is that
> it is a Kan complex, i.e. satisfies the Kan extension condition, studied
> since 1955.

The notion of Kan complex is one model for the notion of
infinity-groupoid. There are other, equivalent models. And there are
models that model stricter subclasses of infinity-groupoids, such as
those you are famous for having studied. Part of the point of saying
"infinity-groupoid" instead of "Kan complex" or else is to amplify the
general notion over its concrete implementation.

> this distracts from studying the not so simple proofs that strict
>  higher homotopical structures exist,

I don't quite see why the term should distract from or otherwise
diminish accomplishments made in the study of strict
infinity-groupoids. On the contrary, to my mind the general theory of
infinity-groupoids puts many of these constructions into the full
perspective of higher category theory and thereby amplifies their
relevance.


> I know the term (\infty,n)-groupoid has been well used recently


Not quite: the term (infinity,n)-category is used to denote an
infinity-category in which all k-morphisms for k greater than n are
equivalences. So an infinity-groupoid is an (infinity,0)-category.

This terminology was not invented in order to hide anybody's previous
accomplishments. On the contrary, I think, this terminology follows
established use in higher category theory and serves to nicely
organize past, present and future insights into higher category theory
in a coherent picture.

I am hoping that eventually we find a constructive way of thinking
about these things, where past insights are seen as fitting into the
beautiful picture of higher category theory that has recently been
emerging, and are not seen to be in conflict with them.

Here is an example I quite like: in the context of
(infinity,1)-category theory Jacob Lurie gave an entirely intrinsic
category-theoretic definition of (infinity,1)-sheaves, also known as
infinity-stacks: these are infinity-groupoid valued presheaves
satisfying a suitable descent condition.

Working backwards from the abstract higher category-theoretic
definition of these, one can work out how this notion matches related
models that were previously investigated.

Among these are two main strands:

1) the homotopical structures/model category structures on categories
of ordinary presheaves with values in simplicial sets, as developed by
Brown, Joyal, Jardine, Dugger and others.

2) The notion of presheaves with values in strict infinity-groupoids /
strict omega-groupoids, as originally conceived by John Roberts and
then formalized by Ross Street and others.

One might worry that both these decade-old developments might not
harmonize with the intrinsic higher-categorical notion of
(infinity,1)-sheaf. But the opposite is true: one finds that they are
neatly subsumed in the abstract definition and conversely provide
concrete workable models for the abstract notion.

For point 1) this is proven in Jacob Lurie's book on higher topos
theory: the Joyal/Jardine model structure models precisely those
(infinity,1)-sheaves which are "hypercomplete". More generally, the
left Bousfiled localization of the model structure on simplicial
presheaves at Cech nerves models (infinity,1)-sheaves.

For point 2) this has been proven by Dominic Verity, following a
conjecture I made: one can show that under mild conditons, under the
inclusion of strict omega-groupoids / strict infinity-groupoids into
all infinity-groupoids, the Roberts-Street notion of descent for such
strict oo-groupoid sheaves does model the abstractly found
(infinity,1)-sheaf condition.

http://ncatlab.org/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves

So this means now not a diminishing but a considerable increase in
value of the old work on presheaves with values in strict
infinity-groupoids / omega-groupoids: since it embeds these
constructions into a powerful abstract theory, we now conversely have
all the abstract tools and insights available to study and use the
former.

I have been using this embeddingg of strict oo-groupoid valued
oo-stacks into all oo-stacks quite a lot in my research, originally
starting with the observation that the BCSS-model of the string
2-group realizes it as a strict 2-groupoid valued stack on Diff, which
is quite useful for some applications. All along I have greatly
benfitted by having your book and nonabelian algebraic topology next
to me, together with Ross Street's articles on descent, and at the
same time having Higher Topos Theory on the table. I find that that
these two aspects interact very nicely, and I was therefore a bit
saddened by hearing what sounded like accusations that new
developments in higher category theory try to intentionally diminish
previous development.

I think math is a win-win game, not a competition: one person's
insight does not dimish the other person's insight, but both increase
each other's value. I am dearly hoping that those who practiced
aspects of higher category theory for so long see the new developments
not as in conflictt with their work, but as a beautiful blossoming of
the theory that they started developing.

Because it is true.


Best regards,
Urs


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