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• CommentRowNumber1.
• CommentAuthorGuest
• CommentTimeOct 14th 2009
Comment at (n,r)-category about the equivalence of fundamental categories. This is a coarser notion than equivalence of categories in the usual sense.

-David Roberts
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 14th 2009

Thanks. Do you have a reference for that notion? Is that in Grandis' work? We should have an entry on this.

Recently, after I carried this question to the CatTheory mailing list I received a reply by Peter Bubenik who wrote that together with David Spivak they are in the process of proving the "directed homotopy hypothesis" relating (oo,1)-categories and some flavor of directed topological spaces.

But even though I tried, I couldn't make him tell me what exactly it is they are proving and precisely which notions of equivalence etc they are using.

• CommentRowNumber3.
• CommentAuthorEric
• CommentTimeNov 16th 2009

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 16th 2009

I have replied.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 3rd 2011
• (edited Jun 3rd 2011)

I have added to the Definition-section at (n,r)-category a precise definition:

In terms of the standard notion of (∞,n)-categories we can make this precise as follows:

For $-2 \leq n \leq \infty$, an (n,0)-category is an ∞-groupoid that is n-truncated: an n-groupoid.

For $0 \leq r \lt \infty$, an (n,r)-category is an (∞,r)-category $C$ such that for all objects $X,Y \in C$ the $(\infty,r-1)$-categorical hom-object $C(X,Y)$ is an $(n-1,r-1)$-category.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeSep 1st 2018

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeSep 1st 2018

Is there anything in those query boxes worth keeping?

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeSep 1st 2018

I think their conclusions should be incorporated into the page.