Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

Not to be confused with Product category (an operation on categories), this is an operation on objects of a category.

Definition

Product of two objects

Fix a category C. Let X1 and X2 be objects of C. A product of X1 and X2 is an object X, typically denoted X1 × X2, equipped with a pair of morphisms π1 : X → X1, π2 : X → X2 satisfying the following universal property:

  • For every object Y and every pair of morphisms f1 : Y → X1, f2 : Y → X2, there exists a unique morphism f : Y → X1 × X2 such that the following diagram commutes:
Universal property of the product

Whether a product exists may depend on C or on X1 and X2. If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product.

The morphisms π1 and π2 are called the canonical projections or projection morphisms. Given Y and f1, f2, the unique morphism f is called the product of morphisms f1 and f2 and is denoted ⟨f1, f2⟩.

Product of an arbitrary family

Instead of two objects, we can start with an arbitrary family of objects indexed by a set I.

Given a family (Xi)i∈I of objects, a product of the family is an object X equipped with morphisms πi : X → Xi satisfying the following universal property:

  • For every object Y and every I-indexed family of morphisms fi : Y → Xi, there exists a unique morphism f : Y → X such that the following diagrams commute for all i in I:
Universal product of the product

The product is denoted Πi∈I Xi. If I = {1, ..., n}, then it is denoted X1 × ... × Xn and the product of morphisms is denoted ⟨ f1, ..., fn ⟩.

Equational definition

Alternatively, the product may be defined through equations. So, for example, for the binary product:

  • Existence of f is guaranteed by existence of the operation ⟨ −, − ⟩.
  • Commutativity of the diagrams above is guaranteed by the equality ∀f1, ∀f2 ∀i ∈ {1, 2},  πi ∘ ⟨ f1, f2 ⟩ = fi.
  • Uniqueness of f is guaranteed by the equality ∀g : Y → X1 × X2,  ⟨ π1∘g, π2∘g ⟩ = g.[1]

As a limit

The product is a special case of a limit. This may be seen by using a discrete category (a family of objects without any morphisms, other than their identity morphisms) as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set I considered as a discrete category. The definition of the product then coincides with the definition of the limit, { f }i being a cone and projections being the limit (limiting cone).

Universal property

Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take J as the discrete category with two objects, so that CJ is simply the product category C × C. The diagonal functor Δ : C → C × C assigns to each object X the ordered pair (X, X) and to each morphism f the pair (f, f). The product X1 × X2 in C is given by a universal morphism from the functor Δ to the object (X1, X2) in C × C. This universal morphism consists of an object X of C and a morphism (X, X) → (X1, X2) which contains projections.

Examples

In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets Xi the product is defined as

Πi∈I Xi := { (xi)i∈I | ∀i∈I, xi∈Xi }

with the canonical projections

πj : Πi∈I Xi → Xj, πj((xi)i∈I) := xj.

Given any set Y with a family of functions fi : Y → Xi, the universal arrow f : Y → Πi∈I Xi is defined by f(y) := (fi(y))i∈I .

Other examples:

Discussion

An example in which the product does not exist: In the category of fields, the product Q × Fp does not exist, since there is no field with homomorphisms to both Q and Fp.

Another example: An empty product (i.e. I is the empty set) is the same as a terminal object, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group G there are infinitely many morphisms ℤ → G, so G cannot be terminal.

If I is a set such that all products for families indexed with I exist, then one can treat each product as a functor CI → C.[2] How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the binary product functor, which is a bifunctor. For f1 : X1 → Y1, f2 : X2 → Y2 we should find a morphism X1 × X2 → Y1 × Y2. We choose ⟨ f1 o π1, f2 o π2 ⟩. This operation on morphisms is called cartesian product of morphisms.[3] Second, consider the general product functor. For families {X}i,{Y}i, fi : Xi → Yi we should find a morphism Πi∈I Xi → Πi∈I Yi. We choose the product of morphisms {fi o πi}i.

A category where every finite set of objects has a product is sometimes called a cartesian category[3] (although some authors use this phrase to mean "a category with all finite limits").

The product is associative. Suppose C is a cartesian category, product functors have been chosen as above, and 1 denotes a terminal object of C. We then have natural isomorphisms

These properties are formally similar to those of a commutative monoid; a cartesian category with its finite products is an example of a symmetric monoidal category.

Distributivity
Main article: Distributive category

For any objects X, Y, and Z of a category with finite products and coproducts, there is a canonical morphism X × Y + X × Z → X × (Y + Z), where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct X × Y + X × Z guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):

The universal property of the product X × (Y + Z) then guarantees a unique morphism X × Y + X × Z → X × (Y + Z) induced by the dashed arrows in the above diagram. A distributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, one has the canonical isomorphism

.

See also

References
  1. Lambek J., Scott P. J. (1988). Introduction to Higher-Order Categorical Logic. Cambridge University Press. p. 304.
  2. Lane, S. Mac (1988). Categories for the working mathematician (1st ed.). New York: Springer-Verlag. p. 37. ISBN 0-387-90035-7.
  3. Michael Barr, Charles Wells (1999). Category Theory – Lecture Notes for ESSLLI. p. 62. Archived from the original on 2011-04-13.
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