In category theory, the product of two 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.
Notable quotes
“But if capitalism had built up science as a productive force, the very character of the new mode of production was serving to make capitalism itself unnecessary.”
— John Desmond Bernal (1959) Marx and Science. p. 39.
“It makes unavoidably necessary an entirely new organization of society in which production is no longer directed by mutually competing individual industrialists but rather by the whole society operating according to a definite plan and taking account of the needs of all.”
— Friedrich Engels, (1847)
“Capitalism [is] a system of wage-labour and commodity production for sale, exchange, and profit, rather than for the immediate need of the producers.”
— Gordon Marshall ed. The Oxford Dictionary of Sociology, 2nd edition. Lemma "Capitalism".
“The product of mental labor — science — always stands far below its value, because the labor-time necessary to reproduce it has no relation at all to the labor-time required for its original production.”
— Karl Marx Addenda, "Relative and Absolute Surplus Value" in Economic Manuscripts (1861–63).
“Corporations are necessary to the effective use of the forces of production and commerce under modern conditions.”
— Theodore Roosevelt, The Progressives, Past and Present (1910)
“Production for sale in a market in which the object is to realize the maximum profit is the essential feature of a capitalist world-economy. In such a system production is constantly expanded as long as further production is profitable, and men constantly innovate new ways of producing things that will expand the profit margin.”
— Immanuel Wallerstein (1979) The Capitalist World-Economy. p. 15.
Quotes via Wikiquote (CC BY-SA), each with its original source.
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.
Contents
Definition
Product of two objects
Fix a category
C
.
{\displaystyle C.}
Let
X
1
{\displaystyle X_{1}}
and
X
2
{\displaystyle X_{2}}
be objects of
C
.
{\displaystyle C.}
A product of
X
1
{\displaystyle X_{1}}
and
X
2
{\displaystyle X_{2}}
is an object
X
,
{\displaystyle X,}
typically denoted
X
1
×
X
2
Product of an arbitrary family
Instead of two objects, we can start with an arbitrary family of objects indexed by a set
I
.
{\displaystyle I.}
Given a family
(
X
i
)
i
∈
I
{\displaystyle \left(X_{i}\right)_{i\in I}}
of objects, a product of the family is an object
X
{\displaystyle X}
equipped with morphisms
π
i
:
X
→
X
i
,
{\displaystyle \pi _{i}:X\to X_{i},}
satisfying the following universal property:
For every object
Y
{\displaystyle Y}
Equational definition
Alternatively, the product may be defined through equations. So, for example, for the binary product:
Existence of
f
{\displaystyle f}
is guaranteed by existence of the operation
⟨
⋅
,
⋅
⟩
.
{\displaystyle \langle \cdot ,\cdot \rangle .}
Commutativity of the diagrams above is guaranteed by the equality: for all
f
1
,
f
2
{\displaystyle f_{1},f_{2}}
and all
i
∈
{
1
,
2
}
,
{\displaystyle i\in \{1,2\},}
π
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
{\displaystyle I}
considered as a discrete category. The definition of the product then coincides with the definition of the limit,
{
f
}
i
{\displaystyle \{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
{\displaystyle \mathbf {J} }
as the discrete category with two objects, so that
In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets
X
i
{\displaystyle X_{i}}
the product is defined as
∏
i
∈
I
X
i
:=
{
(
x
i
)
i
∈
I
:
x
i
∈
X
i
for all
i
∈
I
}
{\displaystyle \prod _{i\in I}X_{i}:=\left\{\left(x_{i}\right)_{i\in I}:x_{i}\in X_{i}{\text{ for all }}i\in I\right\}}
with the canonical projections
Discussion
An example in which the product does not exist: In the category of fields, the product
Q
×
F
p
{\displaystyle \mathbb {Q} \times F_{p}}
does not exist, since there is no field with homomorphisms to both
Q
{\displaystyle \mathbb {Q} }
and
F
p
.
{\displaystyle F_{p}.}
Another example: An empty product (that is,
I
{\displaystyle 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
{\displaystyle G}
there are infinitely many morphisms
Z
→
G
,
{\displaystyle \mathbb {Z} \to G,}
Distributivity
For any objects
X
,
Y
,
and
Z
{\displaystyle X,Y,{\text{ and }}Z}
of a category with finite products and coproducts, there is a canonical morphism
where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct
X
×
Y
+
X
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,
{\displaystyle X_{1}\times X_{2},}
equipped with a pair of morphisms
π
1
:
X
→
X
1
,
{\displaystyle \pi _{1}:X\to X_{1},}
π
2
:
X
→
X
2
{\displaystyle \pi _{2}:X\to X_{2}}
satisfying the following universal property:
For every object
Y
{\displaystyle Y}
and every pair of morphisms
f
1
:
Y
→
X
1
,
{\displaystyle f_{1}:Y\to X_{1},}
f
2
:
Y
→
X
2
,
{\displaystyle f_{2}:Y\to X_{2},}
there exists a unique morphism
f
:
Y
→
X
1
×
X
2
{\displaystyle f:Y\to X_{1}\times X_{2}}
such that the following diagram commutes:
Whether a product exists may depend on
C
{\displaystyle C}
or on
X
1
{\displaystyle X_{1}}
and
X
2
.
{\displaystyle X_{2}.}
If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product. This has the following meaning: if
X
′
,
π
1
′
,
π
2
′
{\displaystyle X',\pi _{1}',\pi _{2}'}
is another product, there exists a unique isomorphism
h
:
X
′
→
X
1
×
X
2
{\displaystyle h:X'\to X_{1}\times X_{2}}
such that
π
1
′
=
π
1
∘
h
{\displaystyle \pi _{1}'=\pi _{1}\circ h}
and
π
2
′
=
π
2
∘
h
{\displaystyle \pi _{2}'=\pi _{2}\circ h}
.
The morphisms
π
1
{\displaystyle \pi _{1}}
and
π
2
{\displaystyle \pi _{2}}
are called the canonical projections or projection morphisms; the letter
In the category of topological spaces, the product is the space whose underlying set is the Cartesian product and which carries the product topology. The product topology is the coarsest topology for which all the projections are continuous.
In the category of modules over some ring
R
,
{\displaystyle R,}
the product is the Cartesian product with addition defined componentwise and distributive multiplication.
In the category of groups, the product is the direct product of groups given by the Cartesian product with multiplication defined componentwise.
In the category of graphs, the product is the tensor product of graphs.
In the category of relations, the product is given by the disjoint union. (This may come as a bit of a surprise given that the category of sets is a subcategory of the category of relations.)
In the category of algebraic varieties, the product is given by the Segre embedding.
In the category of semi-abelian monoids, the product is given by the history monoid.
In the category of Banach spaces and short maps, the product carries the l∞ norm.
A partially ordered set can be treated as a category, using the order relation as the morphisms. In this case the products and coproducts correspond to greatest lower bounds (meets) and least upper bounds (joins).
so
G
{\displaystyle G}
cannot be terminal.
If
I
{\displaystyle I}
is a set such that all products for families indexed with
I
{\displaystyle I}
exist, then one can treat each product as a functor
C
I
→
C
.
{\displaystyle \mathbf {C} ^{I}\to \mathbf {C} .}
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
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.
×
Z
{\displaystyle X\times Y+X\times Z}
guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):
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, there is the canonical isomorphism
