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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.
Asked a question at (n,r)-category.
I have replied.
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.
Is there anything in those query boxes worth keeping?
I think their conclusions should be incorporated into the page.
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