# Question on the coskeleton functor $\mathbf{coskel}'$ for pointed simplicial sets

There is an abstract construction of the coskeleton functor for simplicial set as follows:

Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq n}\hookrightarrow \Delta$. The induced functor $$i_{n*}\colon sSet\to sSet_{\leq n}$$ given by $X\mapsto X\circ i_n$ has a right-adjoint $i_n^!\colon sSet_{\leq n}\to sSet$ and the composition $\mathbf{coskel}_n=i_n^!\circ i_{n*}$ is the $n$-coskeleton functor and it comes with a natural map $X\to \mathbf{coskel}_n(X)$.

I want to do the same for the pointed situation: Let $n\geq 0$ be an integer and cosinder the functor $$i'_{n*}\colon sSet_*\to sSet_{*\leq n}$$ given by $X\mapsto X\circ i_n$ for a pointed simplicial set $X\in sSet_*=*\downarrow sSet\cong\operatorname{Fun}(\Delta^{op},Set_*)$. This has a right-adjoint ${i_n'}^{!}\colon sSet_{*\leq n}\to sSet_*$. Let the composition $\mathbf{coskel'}_n={i_n'}^{!}\circ i_{n*}'$ be the pointed $n$-coskeleton functor. It comes with a natural map $X\to \mathbf{coskel}'_n(X)$.

Is $X\to \mathbf{coskel}'_n(X)$ for a pointed simplicial set $X\in sSet_*$ an isomorphism on the simplicial homotopy group $\pi_i(-):=[\Delta^i/\partial\Delta^i, -]_{pointed}$ for $0\leq i<n$ and is $\pi_i(\mathbf{coskel}'_n(X))=0$ for $i\geq n$?

For the unpointed situation this seem to be true as stated here where an isomorphism of the simplicial homotopy groups mean an isomorphism with respect to all basepoints. The question is thus probably easy to answer but I am unsure if there is a picky basepoint-detail going wrong in the pointed case. Thank you.

• Your notations are inverted: the functor $- \circ i_n$ is typically denoted by $i_n^*$, not $(i_n)_*$ (because postcomposition is contravariant), whereas its right adjoint is often denoted $(i_n)_!$ (because now it's covariant). Jun 20, 2016 at 15:18

The pointed question reduces to the unpointed one in the obvious way.

I will change notation to better align with what I'm more familiar with. I will also suppress the $n$ subscripts. In particular:

• $i^*$ denotes the pullback functor $\mathrm{sSet} \to \mathrm{sSet}_{\leq n} : X \mapsto X \circ i$
• $i_*$ denotes the right adjoint of $i^*$
• $i_!$ denotes the left adjoint of $i^*$

and I will use primes to indicate the corresponding functors on pointed objects.

A pointed object in a category $\mathcal{C}$ with a terminal object can expressed as a functor $\mathbf{2} \to \mathcal{C}$ (where $\mathbf{2}$ is the category $0 \to 1$ with one arrow) with the additional property that $0$ is mapped to a terminal object. Write the category of such functors as $\mathrm{Fun}^L(\mathbf{2}, \mathcal{C})$.

Since right adjoints preserve all limits, and in particular preserve terminal objects, the adjunction $i^* \dashv i_*$ induces on the functor categories by precomposition restricts to an adjunction $i^{*'} \dashv i_*'$ between $\mathrm{Fun}^L(\mathbf{2}, \mathrm{sSet})$ and $\mathrm{Fun}^L(\mathbf{2}, \mathrm{sSet}_{\leq n})$.

($i_!$ also sends the terminal object to the terminal object, so you get $i_!' \vdash i^{*'}$ too)

In particular this means $\mathbf{coskel}'$ is computed by applying $\mathbf{coskel}$; in particular, $\mathbf{coskel}'(X,x) = (\mathbf{coskel}(X), \mathbf{coskel}(x))$, and thus

$$\pi_i (\mathbf{coskel}'(X,x)) \cong \pi_i(\mathbf{coskel}(X), \mathbf{coskel}(x))$$