The Stacks project

Lemma 14.13.4. With $X$ and $k$ as above. For any simplicial object $V$ of $\mathcal{C}$ we have the following canonical bijection

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

which maps $\gamma $ to the restriction of the morphism $\gamma _ k$ to the component corresponding to $\text{id}_{[k]}$. Similarly, for any $n \geq k$, if $W$ is an $n$-truncated simplicial object of $\mathcal{C}$, then we have

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ n(\mathcal{C})}(\text{sk}_ n(X \times \Delta [k]), W) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W_ k). \]

Proof. A morphism $\gamma : X \times \Delta [k] \to V$ is given by a family of morphisms $\gamma _\alpha : X \to V_ n$ where $\alpha : [n] \to [k]$. The morphisms have to satisfy the rules that for all $\varphi : [m] \to [n]$ the diagrams

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

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

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

Comments (4)

Comment #1015 by correction_bot on

About the second part of the lemma, the skeleton functors haven't been defined yet (they aren't until Section 14.17).

Comment #1028 by on

OK, I moved the introduction of skelet functors earlier. See here. Thanks!

Comment #2054 by shom on

A minor typo: In Lemma 14.13.4 after the first equation "wich" should be "which".

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