Product category

Key takeaways

Product category belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Product category to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Product category from memory before moving on to harder problems.

Reference excerpt

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) = (1A, 1B). 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 {\displaystyle C_{i}} indexed by a set I {\displaystyle I} , P = ∏ C i , p j : P → C j , j ∈ I {\displaystyle P=\prod C_{i},p_{j}:P\to C_{j},j\in I} satisfy:

given a family of functors f i : D → C i {\displaystyle f_{i}:D\to C_{i}} , there exists a unique functor f : D → P {\displaystyle f:D\to P} such that f j = p j ∘ f {\displaystyle f_{j}=p_{j}\circ f} for each j ∈ I {\displaystyle 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 {\displaystyle {\mathsf {Cat}}} . Thus, for example,

F c t ( A , ∏ i B i ) ≃ ∏ i F c t ( A , B i ) {\displaystyle \textstyle {\mathsf {Fct}}(A,\prod _{i}B_{i})\simeq \prod _{i}{\mathsf {Fct}}(A,B_{i})}

where F c t {\displaystyle {\mathsf {Fct}}} denotes a functor category.

Functoriality Given two functors f : C → D , g : C ′ → D ′ {\displaystyle f:C\to D,g:C'\to D'} , the product f × g : C × C ′ → D × D ′ {\displaystyle f\times g:C\times C'\to D\times D'} is defined component-wise; that is,

( f × g ) ( x , x ′ ) = ( f ( x ) , g ( x ′ ) ) {\displaystyle (f\times g)(x,x')=(f(x),g(x'))}

for a pair of objects or morphisms x , x ′ {\displaystyle x,x'} . (This product may also be characterized by the universal property similar to that for categories.) This way, we get the functor

× : C a t × C a t → C a t . {\displaystyle \times :{\mathsf {Cat}}\times {\mathsf {Cat}}\to {\mathsf {Cat}}.}

It satisfies the tensor-hom adjunction in the sense

Hom C a t ⁡ ( A × B , C ) ≃ Hom C a t ⁡ ( A , F c t ( B , C ) ) {\displaystyle \operatorname {Hom} _{\mathsf {Cat}}(A\times B,C)\simeq \operatorname {Hom} _{\mathsf {Cat}}(A,{\mathsf {Fct}}(B,C))}

where F c t {\displaystyle {\mathsf {Fct}}} denotes a functor category.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Product category

Start with the simplest possible case. Write down what Product category claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Product category before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Product category ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Product category

In research: Product category appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Product category in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Product category is common in secondary-school and first-year university syllabi. It links to neighbouring topics Category theory, Category theory stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Product category outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Product category in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Product category means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Product category out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Product category in simple terms?

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.

Why does Product category matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Product category?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Product category.