What is the coskeleton tower of a quasi-category?

Question

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?

Answer 1

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.