Product category

In the mathematical field of category theory, the product of two categories C and D, denoted C × D and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors.

Definition

The product category C × D has:

as objects:
pairs of objects (A, B), where A is an object of C and B of D;
as arrows from (A₁, B₁) to (A₂, B₂):
pairs of arrows (f, g), where f : A₁ → A₂ is an arrow of C and g : B₁ → B₂ is an arrow of D;
as composition, component-wise composition from the contributing categories:
(f₂, g₂) o (f₁, g₁) = (f₂ o f₁, g₂ o g₁);
as identities, pairs of identities from the contributing categories:
1_{(A, B)} = (1_A, 1_B).

A product of a family of categories is defined exactly the same way.

Universal property

Just like for sets, a product of a family of categories is characterized by the following universal property. Given categories $C_{i}$ indexed by a set $I$ , $P = \prod C_{i}, p_{j} : P \to C_{j}, j \in I$ satisfy:

given a family of functors

f_{i} : D \to C_{i}

, there exists a unique functor

f : D \to P

such that

f_{j} = p_{j} \circ f

for each

j \in I

.

Put in another way, a product of a family of small categories is exactly the categorical product of them in the category of small categories $𝖢 𝖺 𝗍$ . Thus, for example, $𝖥 𝖼 𝗍 (A, \prod_{i} B_{i}) ≃ \prod_{i} 𝖥 𝖼 𝗍 (A, B_{i})$

where $𝖥 𝖼 𝗍$ denotes a functor category.

Functoriality

Given two functors $f : C \to D, g : C^{'} \to D^{'}$ , the product $f \times g : C \times C^{'} \to D \times D^{'}$ is defined component-wise; that is, $(f \times g) (x, x^{'}) = (f (x), g (x^{'}))$ for a pair of objects or morphisms $x, x^{'}$ . (This product may also be characterized by the universal property similar to that for categories.) This way, we get the functor

\times : 𝖢 𝖺 𝗍 \times 𝖢 𝖺 𝗍 \to 𝖢 𝖺 𝗍 .

It satisfies the tensor-hom adjunction in the sense

{Hom}_{𝖢 𝖺 𝗍} (A \times B, C) ≃ {Hom}_{𝖢 𝖺 𝗍} (A, 𝖥 𝖼 𝗍 (B, C))

where $𝖥 𝖼 𝗍$ denotes a functor category.

Example: C × 2

Let $f, g : C \to D$ be functors. Suppose there is a natural transformation $φ : f \to g$ . Then $φ$ determines the functor

h : C \times \underline{2} \to D

such that

h (\cdot, 0) = f, h (\cdot, 1) = g

,

where $\underline{2} = {0, 1}$ is the category with two objects and the non-identity morphism $⇝ : 0 \to 1$ . Intuitively, h is a non-invertible homotopy from $f$ to $g$ . Indeed, define $h$ by, for $x : a \to b$ in $C$ ,

h (x, {id}_{0}) = f (x), h (x, {id}_{1}) = g (x), h (x, ⇝) = g (x) \circ φ_{a} = φ_{b} \circ f (x) .

Conversely, given $h : C \times \underline{2} \to D$ , we get $f, g, φ$ by $f = h (\cdot, 0), g = h (\cdot, 1)$ and $φ_{a} = h ({id}_{a}, ⇝)$ .

Bifunctor

A functor whose domain is a product category is called a bifunctor. A bifunctor can be defined in each variable separately in the following sense:

Proposition— Each bifunctor

F : A \times B \to C

determines the families of the functors, for objects $a$ in $A$ and $b$ in $B$ ,

F_{b} : A \to C, F_{a} : B \to C

given by

F_{b} a = F (a, b)

and

F_{b} f = F (f, {id}_{b})

for $f : a \to a^{'}$ and similarly for $F_{a}$ . They commute in the sense:

F_{a^{'}} g \circ F_{b} f = F_{b^{'}} f \circ F_{a} g

.

Conversely, given families of functors $F_{b}, F_{a}$ as above, if they commute, they define the bifunctor $F : A \times B \to C$ by

F (f, g) = F_{b^{'}} f \circ F_{a} g

.

For example, consider $(a, b) \mapsto Hom (a, b) : C^{o p} \times C \to 𝖲 𝖾 𝗍$ . For each fixed $b$ in $B$ , we have the functor

Hom (-, b) : C^{o p} \to 𝖲 𝖾 𝗍

by pullback; i.e., $f : a \to a^{'}$ goes to the function

f^{*} : Hom (a^{'}, b) \to Hom (a, b)

defined by $f^{*} g = g \circ f$ . On the other hand, $Hom (a, -) : C \to 𝖲 𝖾 𝗍$ is defined by pushforward; i.e., $f \mapsto f_{*} = f \circ -$ . Clearly, these two functors commute (the associativity of composition) and so, by the proposition, we get the functor called the Hom functor

Hom (-, -) : C^{o p} \times C \to 𝖲 𝖾 𝗍,

which is explicitly given as: $(f, g) \mapsto (h \mapsto g \circ h \circ f) .$

There is a similar result for natural transformations between bifunctors:

Proposition— Let $F, G : A, B \to C$ be bifunctors and

α = {α_{a, b} : F (a, b) \to G (a, b) ∣ a \in Ob (A), b \in Ob (B)}

a family of morphisms. Then $α : F \to G$ is a natural transformation if and only if it is natural in the first variable and the second variable separately; i.e., for each object $b$ in $B$ ,

α_{-, b} : F (-, b) \to G (-, b)

is a natural transformation and similarly in the second variable.

References

↑ Mac Lane 1978, p. 37.
↑ Mac Lane 1978, Ch. II., § 5., Exercise 2.
1 2 Mac Lane 1978, Ch. II., § 3.
↑ Mac Lane 1978, Ch. II., § 5., Exercise 1.
↑ Mac Lane 1978, Ch. II., § 4., Exercise 8.
↑ Mac Lane 1978, Ch. II., § 3., Proposition 1.
↑ Mac Lane 1978, Ch. II., § 3., Proposition 2.

Definition 1.6.5 in Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]. Vol. 1. Cambridge University Press. p. 22. ISBN 0-521-44178-1.
Product category at the nLab
Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. pp. 36–40. ISBN 1441931236. OCLC 851741862.

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[FOOTNOTEMac_Lane197837-1] Mac Lane 1978, p. 37.

[2] Mac Lane 1978, Ch. II., § 5., Exercise 2.

[products-3] 1 2 Mac Lane 1978, Ch. II., § 3.

[4] Mac Lane 1978, Ch. II., § 5., Exercise 1.

[5] Mac Lane 1978, Ch. II., § 4., Exercise 8.

[6] Mac Lane 1978, Ch. II., § 3., Proposition 1.

[7] Mac Lane 1978, Ch. II., § 3., Proposition 2.