My personal opinion is that this process is very much influenced by the pressure of "bibliometry", "impact factors" and other "modern trends" - people often not very scrupulously invent and reinvent terminology to be better cited, and, conscious or not, it often very much smells of imposture. Sergei Soloviev

Peter Sellinger writes recently:

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I think this is a very apt illustration of what happens if a term with an existing meaning is redefined to mean something else. Henceforth it is impossible for anybody to use the term (with either meaning) without first giving a definition.

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I completely agree. My own problem is with term `infinity groupoid' which is used to describe something which is not even a groupoid, and whose use seems to me to militate against the understanding of what has been achieved with the original and much earlier definition. I once asked Gian-Carl Rota about such change of terminology, in connection with a refereeing job, and he agreed that mathematicians are used to creating confusion in this way.

There are two easy tendencies: one is to use an old name in a quite different way, and the other is to use a new name for an old idea, so that the use of the old term looks old fashioned, and a lot of work may be consigned to the dustbin of history, becoming not easy of access for new students.

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 new term sounds like `dressing up' an old idea to look new. My personal objection to this change of terminology (i.e. axe to grind!) is that this distracts from studying the not so simple proofs that strict higher homotopical structures exist, which mainly are for structured spaces (in particular filtered spaces (Brown/Higgins, Ashley), n-cubes of spaces (Loday), and more recently smooth spaces (Faria Martins/Picken)). The analysis and comparison of these uses should be made. It was certainly a relief to Philip and I that we could do something with filtered spaces which we could not do for the absolute case; the significance of the fact that these constructions work and lead to specific calculations should be thought about.

The term `higher dimensional group theory' which was published in a paper with that title in 1982 was intended to suggest developing higher groupoid theory and its relations to homotopy theory in the spirit of group theory, which meant specific constructions relevant to geometry and calculations, even computer calculations, of many examples, in which actual numbers arise as a test of and examples of the general theory, and in which some aspects of group theory are sensibly seen as better represented in the higher dimensional theory; and example of this is the nonabelian tensor product of groups, where group theorists have found lots of pickings.

I am not sure how these terminological problems will be resolved, and I know the term (\infty,n)-groupoid has been well used recently but the problem of relation to the older ideas, which have had a certain success, should be recognised.

Ronnie Brown

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