The Stacks project

14.8 Pushouts of simplicial objects

Of course we should define the pushout of simplicial objects as the pushout in the category of simplicial objects. This may lead to the potentially confusing situation where the pushouts exist but are not as described below. To avoid this we define the pushout directly as follows.

Definition 14.8.1. Let $\mathcal{C}$ be a category. Let $U, V, W$ be simplicial objects of $\mathcal{C}$. Let $a : U \to V$, $b : U \to W$ be morphisms. Assume the pushouts $V_ n \amalg _{U_ n} W_ n$ exist in $\mathcal{C}$. The pushout of $V$ and $W$ over $U$ is the simplicial object $V\amalg _ U W$ defined as follows:

  1. $(V \amalg _ U W)_ n = V_ n \amalg _{U_ n} W_ n$,

  2. $d^ n_ i = (d^ n_ i, d^ n_ i)$, and

  3. $s^ n_ i = (s^ n_ i, s^ n_ i)$.

In other words, $V\amalg _ U W$ is the pushout of the presheaves $V$ and $W$ over the presheaf $U$ on $\Delta $.

Lemma 14.8.2. If $U, V, W$ are simplicial objects in the category $\mathcal{C}$, and if $a : U \to V$, $b : U \to W$ are morphisms and if $V\amalg _ U W$ exists, then we have

\[ \mathop{\mathrm{Mor}}\nolimits (V\amalg _ U W, T) = \mathop{\mathrm{Mor}}\nolimits (V, T) \times _{\mathop{\mathrm{Mor}}\nolimits (U, T)} \mathop{\mathrm{Mor}}\nolimits (W, T) \]

for any fourth simplicial object $T$ of $\mathcal{C}$.

Proof. Omitted. $\square$

Comments (0)

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Your email address will not be published.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 016V. Beware of the difference between the letter 'O' and the digit '0'.