Lemma 14.15.2 (017M)—The Stacks project

Lemma 14.15.2. With $X$, $k$ and $U$ as above.

For any simplicial object $V$ of $\mathcal{C}$ we have the following canonical bijection

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}(\mathcal{C})}(V, U) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V_ k, X). \]

which maps $\gamma $ to the morphism $\gamma _ k$ composed with the projection onto the factor corresponding to $\text{id}_{[k]}$.
Similarly, if $W$ is an $k$-truncated simplicial object of $\mathcal{C}$, then we have

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ k(\mathcal{C})}(W, \text{sk}_ k U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W_ k, X). \]
The object $U$ constructed above is an incarnation of $\mathop{\mathrm{Hom}}\nolimits (C[k], X)$ where $C[k]$ is the cosimplicial set from Example 14.5.6.

Proof. We first prove (1). Suppose that $\gamma : V \to U$ is a morphism. This is given by a family of morphisms $\gamma _{\alpha } : V_ n \to X$ for $\alpha : [k] \to [n]$. The morphisms have to satisfy the rules that for all $\varphi : [m] \to [n]$ the diagrams

\[ \xymatrix{ X \ar[d]^{\text{id}_ X} & V_ n \ar[d]^{V(\varphi )} \ar[l]^{\gamma _{\varphi \circ \alpha '}} \\ X & V_ m \ar[l]_{\gamma _{\alpha '}} } \]

commute for all $\alpha ' : [k] \to [m]$. Taking $\alpha ' = \text{id}_{[k]}$, we see that for any $\varphi : [k] \to [n]$ we have $\gamma _\varphi = \gamma _{\text{id}_{[k]}} \circ V(\varphi )$. Thus the morphism $\gamma $ is determined by the component of $\gamma _ k$ corresponding to $\text{id}_{[k]}$. Conversely, given such a morphism $f : V_ k \to X$ we easily construct a morphism $\gamma $ by putting $\gamma _\alpha = f \circ V(\alpha )$.

The truncated case is similar, and left to the reader.

Part (3) is immediate from the construction of $U$ and the fact that $C[k]_ n = \mathop{\mathrm{Mor}}\nolimits ([k], [n])$ which are the index sets used in the construction of $U_ n$. $\square$

Comments (2)

Comment #1299 by JuanPablo on February 08, 2015 at 14:11

I do not understand the proof of part (3).

$\text{Mor}(V,\text{Hom}(\Delta[k],X))=\text{Mor}(V\times\Delta[k],X)=$

$\{(f_{n,\beta}:V_n\rightarrow X)\mid \beta:[n]\rightarrow [k] \text{ compatible}\}$ .

Whereas

$\text{Mor}(V,U)=\{(f_{n,\alpha}:V_n\rightarrow X)\mid \alpha:[k]\rightarrow [n] \text{ compatible}\}$ .

So that $\alpha$ and $\beta$ appear to be different.

More specifically, as $V\times \Delta[0]=V$ we have $\text{Hom}(\Delta[0],X)=X$ , but $U_n= \prod_{\alpha\in \text{Mor}([0],[n])} X=X^{n+1}$ .

Comment #1309 by Johan on February 11, 2015 at 00:56

Oops, yes indeed! And in this case there was a bit of fallout from this that had to be fixed. As well I moved a couple of lemmas to obsolete.tex. You can find the changes here. Thanks very much!

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