The Stacks project

Lemma 14.19.6. Let $n$ be an integer $\geq 1$. Let $U$ be a $n$-truncated simplicial object of $\mathcal{C}$. Consider the contravariant functor from $\mathcal{C}$ to $\textit{Sets}$ which associates to an object $T$ the set

\[ \{ (f_0, \ldots , f_{n + 1}) \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(T, U_ n) \mid d^ n_{j - 1} \circ f_ i = d^ n_ i \circ f_ j \ \forall \ 0\leq i < j\leq n + 1\} \]

If this functor is representable by some object $U_{n + 1}$ of $\mathcal{C}$, then there exists an $(n + 1)$-truncated simplicial object $\tilde U$, with $\text{sk}_ n \tilde U = U$ and $\tilde U_{n + 1} = U_{n + 1}$ such that the following adjointness holds

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_{n + 1}(\mathcal{C})}(V, \tilde U) = \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ n(\mathcal{C})}(\text{sk}_ nV, U) \]

Proof. By Lemma 14.19.3 there are identifications

\[ U_ i = \mathop{\mathrm{lim}}\nolimits _{(\Delta /[i])_{\leq n}^{opp}} U(i) \]

for $0 \leq i \leq n$. By Lemma 14.19.5 we have

\[ U_{n + 1} = \mathop{\mathrm{lim}}\nolimits _{(\Delta /[n + 1])_{\leq n}^{opp}} U(n). \]

Thus we may define for any $\varphi : [i] \to [j]$ with $i, j \leq n + 1$ the corresponding map $\tilde U(\varphi ) : \tilde U_ j \to \tilde U_ i$ exactly as in Lemma 14.19.2. This defines an $(n + 1)$-truncated simplicial object $\tilde U$ with $\text{sk}_ n \tilde U = U$.

To see the adjointness we argue as follows. Given any element $\gamma : \text{sk}_ n V \to U$ of the right hand side of the formula consider the morphisms $f_ i = \gamma _ n \circ d^{n + 1}_ i : V_{n + 1} \to V_ n \to U_ n$. These clearly satisfy the relations $d^ n_{j - 1} \circ f_ i = d^ n_ i \circ f_ j$ and hence define a unique morphism $V_{n + 1} \to U_{n + 1}$ by our choice of $U_{n + 1}$. Conversely, given a morphism $\gamma ' : V \to \tilde U$ of the left hand side we can simply restrict to $\Delta _{\leq n}$ to get an element of the right hand side. We leave it to the reader to show these are mutually inverse constructions. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 14.19: Coskeleton functors

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0187. Beware of the difference between the letter 'O' and the digit '0'.