The Stacks project

4.4 Products of pairs

Definition 4.4.1. Let $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. A product of $x$ and $y$ is an object $x \times y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ together with morphisms $p\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times y, x)$ and $q\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times y, y)$ such that the following universal property holds: for any $w\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and morphisms $\alpha \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(w, x)$ and $\beta \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(w, y)$ there is a unique $\gamma \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(w, x \times y)$ making the diagram

\[ \xymatrix{ w \ar[rrrd]^\beta \ar@{-->}[rrd]_\gamma \ar[rrdd]_\alpha & & \\ & & x \times y \ar[d]_ p \ar[r]_ q & y \\ & & x & } \]


If a product exists it is unique up to unique isomorphism. This follows from the Yoneda lemma as the definition requires $x \times y$ to be an object of $\mathcal{C}$ such that

\[ h_{x \times y}(w) = h_ x(w) \times h_ y(w) \]

functorially in $w$. In other words the product $x \times y$ is an object representing the functor $w \mapsto h_ x(w) \times h_ y(w)$.

Definition 4.4.2. We say the category $\mathcal{C}$ has products of pairs of objects if a product $x \times y$ exists for any $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

We use this terminology to distinguish this notion from the notion of “having products” or “having finite products” which usually means something else (in particular it always implies there exists a final object).

Comments (2)

Comment #7472 by Fawzy N. Hegab on

Sorry for being pediantic, but in the final sentence, after the definition of products of pairs of objects, a remark about terminology mentions initial objects. However, so far in the project, initial objects in a category are not defined. So, I think it is better to either add a hyperlink to where the definition is, or to recall the definition or something like that.

Comment #7621 by on

Dear Fawzy N. Hegab, this is indeed just a tad pedantic! Since the word "final" occurs in the text and not in a statement or proof of a mathematical result, I am not going to change it.

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