# Status of pairs/tuples in category theory

A product of two categories $\mathbb{C}, \mathbb{D}$ is defined by a new category $\mathbb{C} \times \mathbb{D}$ such that :

• Objects are ordered pairs (C, D) of objects from $\mathbb{C}$ and $\mathbb{D}$
• Morphisms are ordered pairs (f, g) of morphisms with :
• $f : C \rightarrow C'$
• $g : D \rightarrow D'$
• $C'$ and $D'$ are objects from $\mathbb{C}$ and $\mathbb{D}$

But considering that category theory gives a foundation of mathematics with primitive constructions such as objects and arrows, what is a pair ? It doesn't seem to be a primitive construction.

A naive idea that comes to my mind when thinking about a product of categories $\mathbb{C}, \mathbb{D}$ is to construct a new category $\mathbb{C} \times \mathbb{D}$ where objects are products of objects $C \times D$ from $\mathbb{C}$ and $\mathbb{D}$.

• @Mr.Chip No, $\operatorname{Hom}(A, B)$ is not required to be a set in a category. That would be a locally small category. – Arthur Oct 9 '17 at 10:27
• Yeah, not sure why I misremembered that. Still, the point is they have to form a class. And you can make sense of a product of classes. Ergo, "primitivity" is not violated by the notion described by the OP. – Mr. Chip Oct 9 '17 at 10:52
• What is the difference between a pair $(C, D)$ and the formal product $C \times D$? – Joppy Oct 9 '17 at 11:43

Nevertheless, there have been attempts to directly axiomatize a "category" of categories or a 2-category of categories. The way to handle products in these cases is simple: you just take the universal property of products as an axiom. (Indeed, this is also exactly how products of sets are formalized in ETCS.) That is, we axiomatically assert that given two categories $\mathbb{C}_1$ and $\mathbb{C}_2$, there exists another category which we'll call $\mathbb{C}_1\times\mathbb{C}_2$ such that there are two functors $\pi_i : \mathbb{C}_1\times\mathbb{C}_2\to\mathbb{C}_i$, and for any category $\mathbb{D}$ and pair of functors $f_1 : \mathbb{D}\to\mathbb{C}_1$ and $f_2 : \mathbb{D}\to\mathbb{C}_2$ there is a functor $\langle f_1, f_2 \rangle : \mathbb{D}\to\mathbb{C}_1\times\mathbb{C}_2$ such that $\langle \pi_1, \pi_2 \rangle = id$ and $\pi_i \circ \langle f_1, f_2 \rangle = f_i$. "Category" and "functor" would presumably be primitive concepts in such an axiomatization. If the axiomatization otherwise did an adequate job of formalizing the notion of "category" and "functor", we'd be able to prove that the usual set-theoretic definition of product category is a model of the axiomatically characterized product category. In an axiomatization like the above, objects of the categories are likely to be identified with functors $\mathbf{1}\to\mathbb{C}$, from which we can already start to see that an "object" (in this sense) of $\mathbb{C}_1\times\mathbb{C}_2$, i.e. a functor $X : \mathbf{1}\to\mathbb{C}_1\times\mathbb{C}_2$, gives rise to a pair of "objects", $\pi_i \circ X : \mathbf{1}\to\mathbb{C}_i$ and vice versa.