For $C$ and $D$ two categories, the product category $C \times D$ is the category whose
objects are ordered pairs $(c,d)$ with $c$ an object of $C$ and $d$ an object of $D$: $Obj(C \times D) = Obj(C) \times Obj(D)$.
morphisms are ordered pairs $((c \stackrel{f}{\to} c'),(d \stackrel{g}{\to} d'))$: $Mor(C \times D) = Mor(C) \times Mor(D)$
composition of morphisms is defined componentwise by composition in $C$ and $D$.
This operation is the cartesian product in the 1-category Cat.
We discuss cartesian products for categories in homotopy type theory.
Note: UFP 2013 calls a category a “precategory” and a univalent category a “category”, but here we shall refer to the standard terminology of “category” and “univalent category” respectively.
For categories $A$ and $B$, their product $A \times B$ is a category with $(A\times B)_0\equiv A_0 \times B_0$ and
Identities are defined by $1_{(a,b)}\equiv (1_a,1_b)$ and composition by
(Lemma 9.5.3 in UFP13)
For categories $A,B,C$, the following types are equivalent:
Given $F : A \times B \to C$, for any $a:A$ we obviously have a functor $F_a : B \to C$. This gives a function $A_0 \to (C^B)_0$. Next, for any $f: hom_A(a,a')$, we have for any $b:B$ the morphisms $F_{(a,b),(a',b')}(f,1_b):F_a(b) \to F_{a'}(b)$.
These are the components of a natural transformation $F_a \to F_{a'}$. Functoriality in $a$ is easy to check, so we have a functor $\hat{F} : A \to C^B$.
Conversely, suppose given $G:A \to C^B$. Then for any $a:A$ and $b:B$ we have the object $G(a)(b):C$, giving a function $A_0 \times B_0 \to C_0$. And for $f:hom_A(a,a')$ and $g : hom_B(b,b')$, we have the morphism
in $hom_C(G(a)(b),G(a')(b'))$. Functoriality is again easy to check, so we have a functor $\v{G} : A \times B \to C.$ Finally, it is also clear that these operations are inverses.
Textbook accounts:
Discussion in homotopy type theory:
Last revised on June 3, 2023 at 08:31:51. See the history of this page for a list of all contributions to it.