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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?

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  • $\begingroup$ 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 $\endgroup$
    – Fernando
    Commented Feb 13, 2014 at 2:49
  • $\begingroup$ Btw, the adjoints you're looking for are quite well known: ncatlab.org/nlab/show/simplicial+skeleton $\endgroup$
    – roman
    Commented Mar 5, 2014 at 13:03

1 Answer 1

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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 $>n$.

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  • $\begingroup$ 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 $\endgroup$
    – FShrike
    Commented Oct 23, 2022 at 21:19

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