Lemma 4.18.3. Let $\mathcal{C}$ be a category. The following are equivalent:

Nonempty finite limits exist in $\mathcal{C}$.

Products of pairs and equalizers exist in $\mathcal{C}$.

Products of pairs and fibre products exist in $\mathcal{C}$.

Lemma 4.18.3. Let $\mathcal{C}$ be a category. The following are equivalent:

Nonempty finite limits exist in $\mathcal{C}$.

Products of pairs and equalizers exist in $\mathcal{C}$.

Products of pairs and fibre products exist in $\mathcal{C}$.

**Proof.**
Since products of pairs, fibre products, and equalizers are limits with nonempty index categories we see that (1) implies both (2) and (3). Assume (2). Then finite nonempty products and equalizers exist. Hence by Lemma 4.14.11 we see that finite nonempty limits exist, i.e., (1) holds. Assume (3). If $a, b : A \to B$ are morphisms of $\mathcal{C}$, then the equalizer of $a, b$ is

\[ (A \times _{a, B, b} A)\times _{(pr_1, pr_2), A \times A, \Delta } A. \]

Thus (3) implies (2), and the lemma is proved. $\square$

Your email address will not be published.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.

## Comments (0)

There are also: