# n-truncated simplicial set

It might be a trivial question. So, I apologise in advance. Let $\Delta ^{op}_n$ be the full subcategory of $\Delta ^{op}$ such that the set of objects of $\Delta ^{op}_n$ is $\left\{ 0, \ldots , n\right\}$. The inclusion $i: \Delta ^{op} _ n \to \Delta ^{op}$ induces a functor $i ^*: sSet\to Fun (\Delta _n ^{op}, Set )$. I wonder if this functor has a right adjoint.

On the other hand, I guess that this functor is right adjoint and the left adjoint is the functor which associates each truncated simplicial set to the "degenerated extension" (by degenerated extension, I mean that the higher simplices are the degenerated k-simplices for $k\leq n$). Am I right?

• My guess is that the functor $i^*$ is left adjoint to the functor $P$, which associates each truncated simplicial set to an extension, which is constructed by "pulling back" the low simplices. For example, if $X$ is a truncated simplicial set, then $P(X)$ is such that $P(X) _ {n+1}$ is the pullback of $X(d_n)$ and $X(d_0 )$ in which $d_n, d_0$ are faces. And so on. Am I right? Thank you in advance Commented Feb 13, 2014 at 2:49
• Btw, the adjoints you're looking for are quite well known: ncatlab.org/nlab/show/simplicial+skeleton Commented Mar 5, 2014 at 13:03

By general abstract nonsense, given any functor $F:\mathcal C \to \mathcal D$, the induced functor $F^*:\hat{\mathcal D} \to \hat{\mathcal C}$ between the associated presheaf categories has both a left adjoint $F_!$ and a right adjoint $F_*$ (the left and right Kan extensions).
For your case above, the left/right adjoint will map an $n$-truncated simplicial set $X$ to a simplicial set whose $\le n$ simplices agree with those of $X$, and the other simplices are all degenerate. There are two extreme choices for the new generate simplices: add as many as possible, or add as few as possible. The left adjoint adds as many as possible, and (as you claim, I think) is obtained by "pulling back" the lower degree simplicies (in fact, it suffices to only pullback the degree $n$ simplices) along the structure maps. The right adjoint adds just one new simplex at dimensions $>n$.
• What is meant by “the other simplicies are all degenerate”? This term ‘degenerate’ is confusing me, I can’t get a clear answer from looking it up. It should mean: “singleton” or “empty”, but I can’t tell if that’s the case. This is related to my confusion here. But also, how can we pull back along the structure maps which don’t yet exist? An $n$ truncated simplicial set has no notion of a map to $(n+1)$-simplices Commented Oct 23, 2022 at 21:19