The Stacks project

4.10 Equalizers

Definition 4.10.1. Suppose that $X$, $Y$ are objects of a category $\mathcal{C}$ and that $a, b : X \to Y$ are morphisms. We say a morphism $e : Z \to X$ is an equalizer for the pair $(a, b)$ if $a \circ e = b \circ e$ and if $(Z, e)$ satisfies the following universal property: For every morphism $t : W \to X$ in $\mathcal{C}$ such that $a \circ t = b \circ t$ there exists a unique morphism $s : W \to Z$ such that $t = e \circ s$.

As in the case of the fibre products above, equalizers when they exist are unique up to unique isomorphism. There is a straightforward generalization of this definition to the case where we have more than $2$ morphisms.

Comments (2)

Comment #8139 by Ahmed on

I think a proof of uniqueness is a good idea since in the next section, uniqueness of coequalizers was argued using that of equalizers.

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Your email address will not be published.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 0027. Beware of the difference between the letter 'O' and the digit '0'.