I was giving a talk in a seminar, and I mistakenly said that the coskeleton tower of a quasi-category was its Postnikov tower. Someone corrected me, but a discussion then ensued about what, precisely, this tower is. It appears to be homotopy-invariant, and each $k$-coskeleton looks like it is somehow related to something along the lines of a weak $(k,1)$-homotopy-category for $k\geq 2$, but the other members of the seminar said that the $(k,1)$-homotopy-category is constructed differently and doesn't seem to be equivalent. What's interesting is this tower seems to converge to $X$ when $X$ is a quasicategory.


Has this tower been studied before? Does it have a name? 



What is the coskeleton tower of a quasi-category?
It turns out the answer is yes: $k$-coskeletalization of a quasicategory models truncation of an $(\infty,1)$-category to a $(k-1,1)$-category.


Let's collect some easy observations.


1. We have an adjunction $sk\_k \dashv cosk\_k : sSet \to sSet$.
2. $cosk\_k$ preserves the property of being a quasicategory, i.e. descends to a functor $cosk\_k : qCat \to qCat$.


To see this, note that $sk\_k(\Lambda^i[n] \to \Delta[n])$ is an isomorphism for $k \leq n-2$, a horn inclusion for $i \geq n$, and $\Lambda^i[n] \to \partial \Delta[n]$ for $k=n-1$, which can be extended to a horn inclusion by postcomposing $\partial \Delta[n] \to \Delta[n]$. So if $X$ is a quasicategory, we can transpose any horn lifting problem for $cosk\_k X$ to one for $X$, and use this analysis to solve the lifting problem there.


	* Thinking about this a bit more reveals that $cosk\_k$ also preserves inner fibrations between quasicategories -- although not between arbitrary simplicial sets.
3. $cosk\_k$ preserves equivalences between quasicategories. Equivalently, $cosk\_k$ preserves homotopies between functors between quasicategories.


This follows from the fact that the walking isomorphism is 0-coskeletal and $cosk\_k$ preserves binary products (in fact, all limits).
4. If $X$ is an $k$-coskeletal quasicategory, then its hom-spaces are $(k-1)$-coskeletal Kan complexes, and in particular they are $(k-1)$-truncated spaces.


To see this, use the model $Hom^R\_X(x,y)$. For a simplicial set $A$, maps $A \to Hom^R\_X(x,y)$ are represented by certain maps $A \ast \Delta[0] \to X$. In particular, a map $\partial \Delta[n-1] \to Hom^R\_X(x,y)$ is a map $\partial \Delta[n-1] \ast \Delta[0] \to X$. But there is a degeneracy condition which allows us to fill the last remaining face to obtain a map $\partial \Delta[n] \to X$. In turn this can be uniquely filled (for $n \geq k$ if $X$ is $k$-coskeletal) to obtain a map $\Delta[n] = \Delta[n-1] \ast \Delta [0] \to X$, i.e. a map $\Delta[n-1] \to Hom^R\_X(x,y)$. Because $Hom^R\_X(x,y)$ is a Kan complex, the fact that it is $(k-1)$-coskeletal implies that it is $(k-1)$-truncated.
5. The map $Hom^R\_X(x,y) \to Hom\_{cosk\_k X}^R(x,y)$ is an isomorphism on $(k-1)$-skeleta. So it is the $(k-1)$-coskeletalization map, i.e. the $(k-1)$-truncation map.


An $n$-simplex of $Hom^R\_X(x,y)$ is an $(n+1)$-simplex of $X$ satisfying some condition involving certain faces, so it depends only on the $(n+1)$-skeleton of $X$. Since $k$-coskeletalization doesn't change the $(n+1)$-skeleton for $n \leq k-1$, this is clear.


