Suppose we have the sequence of sets $\mathbb{R}$, $\mathbb{R}^2$, $\mathbb{R}^3$, … Is there a Kan simplicial structure on this sequence of sets, that is not $n$-coskeletal for some $n \in \mathbb{N}$?

To be more precise, is there a simplicial set (functor) $R$ with $R([n]) = \mathbb{R}^{n+1}$ that is not $n$-coskeletal for some $n \in \mathbb{N}$?

And very closely related: is there a simplicial set (functor) $R$ with $R([n]) = \mathbb{R}^{n}$ (with $R([0]))=\{0\}$), that is not $n$-coskeletal for some $n \in \mathbb{N}$ ?

]]>Suppose we have a simplicial set X and a m-truncated Kan simplicial set Y. Then how is it possible to construct $Hom_{Simpl}(X,Y)$ as a subset $H \subset Hom_{Set}(X_m,Y_m)$?

Since Severa used this in his work on the n-jet functor (for X the nerve of the pair groupoid over an arbitrary set), it should be possible. Nevertheless I can’t find an explicit construction including a proof that what he constructed is indeed $Hom_{Simpl}(X,Y)$.

By an explicit construction I mean something like: Let $f \in H$ be given, then the appropriate simplicial morphism $F$ is given by $F[n]= X_n \rightarrow Y_n$ as follows : ??? where the commutation with the face and degeneracy maps is seen as follows ??? … On the other side we that any simplicial morphism is given that way, because ???

….

So if someone could give me a proof (I think it will be an induction on something like $Hom_{Simpl}(Sk^n X,Y) \subset Hom_{Set}(X_n,Y_n)$ or $Hom_{Simpl}(Horn_j^n X,Y) \subset Hom_{Set}(X_n,Y_n)$ ) it would be great.

Likely this doesn’t work for arbitrary simplicial sets X, so another topic is to find the appropriate conditions on X .

Moreover this should be put into the nLab, too…

If nobody knows a proof it would be nice, if we could work it out together. At the end I will take the time to put in the nLab. Unfortunately my skills on simplicial sets are not good enough, to do it by myself.

]]>It is well known that a category can be defined as a certain simplicial set obtained by iterated fibred products which satisfies the internal horn filler condition; moreover, requiring horn filling for all horns (i.e., the Kan condition) one obtains the notion of groupoid. Then both the notions of category and groupoid should have an internalization in any category where one is able to arrange things in a way to have the required fibered products, and to state the horn filling condition.

This is what happens, e.g., when one defines Lie groupoids imposing that the source and target maps are submersions. Similarly one has a notion of Lie category, which by some reason seems to be less widely known of the more particular notion of Lie groupoid (maybe this is not surprising.. after all I suspect the notion of category is less known of that of group..). Another classical example are topological categories and groupoids.

Moving from categories and groupoids to oo-categories and oo-groupoids, one should have a similar internal simplicial object based notion of, e.g., Lie oo-groupoid. However, in the nLab the oo-sheaf point of view seems to be largely preferred to the internal Kan object point of view. Why is it so? is the oo-sheaf version just more general and powerful or there are problems with the internal version? I’m asking this since at internal infinity-groupoid it is said that a classical example of the internal Kan complex definition of oo-groupoid are Lie oo-groupoids, but then at Lie infinity-groupoid there is no trace of the internal Kan complex definition.

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