The Stacks project

Lemma 4.31.12. Let

\[ \xymatrix{ \mathcal{C}' \ar[r] \ar[d] & \mathcal{S} \ar[d]^\Delta \\ \mathcal{C} \ar[r]^{G_1 \times G_2} & \mathcal{S} \times \mathcal{S} } \]

be a $2$-fibre product of categories. Then there is a canonical isomorphism

\[ \mathcal{C}' \cong (\mathcal{C} \times _{G_1, \mathcal{S}, G_2} \mathcal{C}) \times _{(p, q), \mathcal{C} \times \mathcal{C}, \Delta } \mathcal{C}. \]

Proof. An object of the right hand side is given by $((C_1, C_2, \phi ), C_3, \psi )$ where $\phi : G_1(C_1) \to G_2(C_2)$ is an isomorphism and $\psi = (\psi _1, \psi _2) : (C_1, C_2) \to (C_3, C_3)$ is an isomorphism. Hence we can associate to this the triple $(C_3, G_1(C_1), (G_1(\psi _1^{-1}), \phi ^{-1} \circ G_2(\psi _2^{-1})))$ which is an object of $\mathcal{C}'$. Details omitted. $\square$

Comments (0)

There are also:

  • 6 comment(s) on Section 4.31: 2-fibre products

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Your email address will not be published.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 02XF. Beware of the difference between the letter 'O' and the digit '0'.