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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.
This article also has query boxes, which is against the recommendation of the “Anything I shouldn’t do?” section of the writing in the nLab article. Should the query boxes be removed?
Yes!
first query box posted here for posterity
+– {: .query} Eric: What is the category of all (small) $(n,r)$-categories? An $(n+1,r+1)$-category?
Urs Schreiber: yes, that should be right. Roughly the argument is that a $(k+1)$-morphism of $(n,r)Cat$ is a (n,k)-transformation:
a 1-morphism in $(n,r)Cat$ is an $n$-functor $C \stackrel{F}{\to} D$ , hence an “$(n,0)$-transformation”
a 2-morphism is a transformation between $n$-functors, hence a “(n,1)-transformation”.
and so on
finally an $(n+1)$-morphism is an $(n,n)$-transformation.
So $(n,r)Cat$ is an $(n+1)$-category.
The invertibiliy of the $(n,k)$-transformations is that of their components which are $(\ell \geq k)$-morphisms in the target $n$-category $D$. So if all $(\ell \gt r)$-morphisms in $D$ are invertible, then so are all $(n,\ell \gt r)$-transformations between $C$ and $D$ hence all $(\ell \gt r+1)$-morphisms in $(n,r)Cat$. So $(n,r)Cat$ is an $(n+1,r+1)$-category. =–
second query box
+–{: .query} Mike Shulman: I am not convinced that the homotopy hypothesis applies to anything directed. I’ll believe that maybe an $r$-directed $n$-type (whatever that means) should have a fundamental $(n,r)$-category, and that this operation has a left adjoint that geometrically realizes an $(n,r)$-category as an $r$-directed $n$-type. But I don’t see why to expect this adjunction to be an equivalence in the directed world, unless all of your $r$-directed $n$-types come equipped with a chosen CW-complex-like $n$-skeleton which you restrict your fundamental categories to.
More concretely: take the interval category. Realize it as a directed space; presumably you get a directed topological interval $[0,1]$. Now take the fundamental category of this space: you get the ordered set $[0,1]$ considered as a category—quite different from the interval category! In order to get back the interval category, you need to do something like remember the endpoints of the directed topological interval, and only use these chosen points as the objects of your fundamental category. Perhaps everyone talking about identifying directed homotopy types with higher categories has some fix like this in mind, but if so I think it should be stressed. (Alternately, maybe someone can tell me why I’m completely wrong.)
David Roberts: Perhaps one could take a leaf out of Ronnie Brown’s book and consider filtered/stratified directed spaces. The relative fundamental category is, as you point out, the 'correct' answer.
Urs Schreiber: right. I didn’t mean to imply that there is an established theory of directed spaces that yields a directed homotopy hypothesis-theorem yet. Instead the idea was that “in view of the homotopy hypothesis” we should be entitled to think of an $(n,r)$-category as an $r$-directed $n$-type. Over at directed space I say more explicitly that one option is to defined what a (nice) $r$-directed $n$-type is this way. I have very little online time today, otherwise I would now add a paragraph along these lines to the above. Maybe one of you feels like doing it. I still think that th slogan “An $(n,r)$-category is an $r$-directed $n$-type.” is a very useful guiding principle, and be it for the right definition of directed space. My impression is that the theory of directed spaces is at the time still tentative and not set in stonee. But if that’s wrong, then I’d still keep the above slogan but put an explicit caveat that this uses the notion “diected space” differently to that established in the literature.
David Corfield: During a discussion on fundamental categories with duals of statified spaces, we had this description of a project to provide a geometric picture of directed homtopy. Speaking of categories with duals, couldn’t nLab do with some more pages on them?
Mike Shulman: Your definition at directed space (“a directed space is a topological space in which not every cell is traversable in all directions”) doesn’t say anything about a stratification, so I think it’s misleading to then say that they could be defined as $(n,r)$-categories without making a point that this would change the notion. My impression from the very little I’ve read about directed spaces is that they don’t necessarily come with any sort of stratification. Do we have any reason to want to define “$r$-directed $n$-type” to mean “$(n,r)$-category”, other than that it would be cute if the homotopy hypothesis could be generalized? We like $(n,r)$-categories for lots of reasons—but would calling them $r$-directed $n$-types really be useful to us or anyone else?
Toby: I don't think that it helps our understanding of $(n,r)$-categories, at least not yet, which is why I moved this section down here. But I think that it may help us to understand directed spaces, particularly to suggest the idea that spaces might be $r$-directed.
Urs Schreiber: I agree with Mike that the statements may currently be too misleading, and with Toby about what they should still achieve for us. Will try to improve on the state of the two entries a bit tomorrow – unless someone beats me to it.
David Roberts: Going back to Mike’s original comment, having read a little about fundamental categories (fingers automatically started typing ’groupoid’ there :), the concept of equivalence of categories has to be expanded so as to capture ’directed homotopy equivalence’. In particular, there is the notions of past retract and future retract - these should be considered as equivalences, but are not equivalences of categories in the usual sense. From memory they are more like (co)relexive subcategories. =–
$(n, r)$-categories don’t reflect the actual combinatorial explosion of category-like objects in category theory.
Already in the case where $n = 1$, one has groupoids, categories, Pos-enriched groupoids, and 2-posets, depending upon which level of $k$-morphism is reversible for $0 \leq k \leq n$. When $n = 2$ there are 8 such structures:
groupoids,
Cat-enriched groupoids
(Pos-enriched groupoid)-enriched groupoids
2Pos-enriched groupoid
(2,1)-categories
2-categories
(Pos-enriched groupoid)-enriched (2,1)-categories
3-posets
In general, for any $n$ there should be $2^{n + 1}$ different structures one could create to generalize sets and posets.
by ’groupoid’ in the $n = 2$ list I meant ’2-groupoid’
@14 There’s a similar combinatorial explosion when talking about truncatedness and connectedness, where one could say that at each level $n$ the collection of $n$-morphisms are contractible, resulting in $2^n$ such combinations of truncations and connections.
the relation between groupoids and Pos-enriched groupoids is basically whatever the directed version of $n$-connectedness is, perhaps ’irreversibility’.
There is a third dimension neglected in this discussion of $(n, r)$-categories: Rezk-completeness. One could and should be talking about various flavors of $(n, r)$-precategories, and really $(n + 1, r)$-preorders if one wants to also remove Rezk-completeness from the type of $n$-morphisms.
Perhaps it’s time to create an article on $(n, r, k)$-precategories where one talks about the hom-$j$-morphism types being Rezk-complete for $j \gt k$.
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