added perspective of higher categories as combinatorial models of ’abstract higher structures’ (prompted by a comment by David Corfield in the thread on ’geometric n-category’)

please feel free to rewrite

]]>One of them used to be ω-categories, so I’ve changed back to that.

]]>Might have been a side-wide programmatic edit via a script that messed things up.

]]>"These combinatorial or algebraic models are known as n-categories or ... as ∞-categories or ∞-categories, or, ..."

What is the difference between an ∞-category and an ∞-category?

Cordially,

Joaquin Miller ]]>

Put in “one of the driving forces” etc.

]]>I’m ok with that.

]]>Agreed with “one of the driving forces”.

]]>The wording that Tim removed seemed inoffensive to me, but if the problem was with “to a large extent”, then I think “one of the driving forces” would be a reasonable replacement (and incontestable I believe).

Other driving forces could be added in as one sees fit.

]]>Many people developed their theory with regard to purely categorical aspects or to links with topology or algebraic geometry. There were several breakthroughs, one being from earlier with Boardman and Vogt, followed on by Cordier with weak Kan complexes and, of course, Street’s orientals were important and owed inspiration to John Roberts. The page you link to, Urs, makes the point about Ronnie’s work, but all the work by researchers such as Batanin, Makkai, Lenster, Cheng, Verity, Joyal, is independent of TQFT ideas and even John Baez and Jim Dolan is only distantly linked to extended QFTs. The homotopy hypothesis etc dates from interpretations of Grothendieck’s pursuing stacks which was a very important input to the development. I am not suggesting there was no link with TQFTs merely that very few of those people I mention and who were very important in the development of the theory of higher categories, would have mentioned extended TQFTs as a dominant inspiration. John Roberts work was very important as motivation for Ross Street, but mostly because of the link with non-Abelian cohomology. Much later, of course, Lurie worked both on quasi-categories and on cobordisms but that was not an early motivation.

]]>Seems to me equally contentious to remove it. It’s easier to forget than to remember, way back, John Roberts introduced $\omega$-categories for purposes of AQFT (see here).

]]>I removed a somewhat contentious wording. The main driving force for the development of higher category theory is not possible applications in extended QFT although that has fed in some interesting conjectures that have shaped the way the theory grew.

]]>Removed an incorrect statement about the relation between $(\infty,\infty)$-categories of cobordisms and cobordism spectra.

Rune Haugseng

]]>started to (re)structure the entry higher category theory roughly along the lines of the new structure at category theory. But for the moment many sections just contain link lists.

]]>