The Stacks project

Lemma 14.20.3. Let $\mathcal{C}$ be a category with fibred products. Let $f : Y\to X$ be a morphism of $\mathcal{C}$. Let $U$ be the simplicial object of $\mathcal{C}$ whose $n$th term is the $(n + 1)$fold fibred product $Y \times _ X Y \times _ X \ldots \times _ X Y$. See Example 14.3.5. For any simplicial object $V$ of $\mathcal{C}$ we have

\begin{align*} \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(V, U) & = \mathop{Mor}\nolimits _{\text{Simp}_1(\mathcal{C})}(\text{sk}_1 V, \text{sk}_1 U) \\ & = \{ g_0 : V_0 \to Y \mid f \circ g_0 \circ d^1_0 = f \circ g_0 \circ d^1_1\} \end{align*}

In particular we have $U = \text{cosk}_1 \text{sk}_1 U$.

Proof. Suppose that $g : \text{sk}_1V \to \text{sk}_1U$ is a morphism of $1$-truncated simplicial objects. Then the diagram

\[ \xymatrix{ V_1 \ar@<1ex>[r]^{d^1_0} \ar@<-1ex>[r]_{d^1_1} \ar[d]_{g_1} & V_0 \ar[d]^{g_0} \\ Y \times _ X Y \ar@<1ex>[r]^{pr_1} \ar@<-1ex>[r]_{pr_0} & Y \ar[r] & X } \]

is commutative, which proves that the relation shown in the lemma holds. We have to show that, conversely, given a morphism $g_0$ satisfying the relation $f \circ g_0 \circ d^1_0 = f \circ g_0 \circ d^1_1$ we get a unique morphism of simplicial objects $g : V \to U$. This is done as follows. For any $n \geq 1$ let $g_{n, i} = g_0 \circ V([0] \to [n], 0 \mapsto i) : V_ n \to Y$. The equality above implies that $f \circ g_{n, i} = f \circ g_{n, i + 1}$ because of the commutative diagram

\[ \xymatrix{ [0] \ar[rd]_{0 \mapsto 0} \ar[rrrrrd]^{0 \mapsto i} \\ & [1] \ar[rrrr]^{0 \mapsto i, 1\mapsto i + 1} & & & & [n] \\ [0] \ar[ru]^{0 \mapsto 1} \ar[rrrrru]_{0 \mapsto i + 1} } \]

Hence we get $(g_{n, 0}, \ldots , g_{n, n}) : V_ n \to Y \times _ X\ldots \times _ X Y = U_ n$. We leave it to the reader to see that this is a morphism of simplicial objects. The last assertion of the lemma is equivalent to the first equality in the displayed formula of the lemma. $\square$


Comments (1)

Comment #1019 by correction_bot on

Being a robot, I noticed that in the statement of the lemma, appears but should be to be consistent with the notation introduced earlier.


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Your email address will not be published.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 018I. Beware of the difference between the letter 'O' and the digit '0'.