Product and co-product

In category theory , product and co- product are mutually dual concepts for assigning an object to families of objects in a category. Duality of two terms means, as is usual in category theory, that one term arises from the other by reversing the morphism arrows, as can be easily recognized from the definition given below. Both can only be clearly defined except for natural isomorphism . The product arises from a generalization of the Cartesian product and the coproduct from a generalization of the (outer) disjoint union of sets . The product and coproduct cover the Cartesian product and the disjoint union as special cases on the category of sets.

If the product coincides with the co-product, it is called a biproduct .

Definitions

Let there be any category, any index set, and any family of objects in . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle I}$ ${\displaystyle \left(\,A_{i}\mid i\in I\,\right)}$ ${\displaystyle {\mathcal {C}}}$

An object of together with morphisms , the projections onto the respective -th component, is called the product which , if the universal property holds: ${\displaystyle \Pi }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathrm {pr} _{i}\colon \Pi \to A_{i}}$ ${\displaystyle i}$ ${\displaystyle A_{i}}$

For every object of with morphisms there is exactly one morphism that satisfies for all . ${\displaystyle X}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f_{i}\colon X\to A_{i}}$ ${\displaystyle f\colon X\to \Pi }$ ${\displaystyle f_{i}=\mathrm {pr} _{i}\circ f}$ ${\displaystyle i\in I}$

One then writes for one . ${\textstyle \prod _{i\in I}A_{i}}$ ${\displaystyle \Pi }$

An object of together with morphisms , the embedding in the respective -th component, is called a coproduct which , if the universal property applies: ${\displaystyle \amalg }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathrm {ins} _{i}\colon A_{i}\to \amalg }$ ${\displaystyle i}$ ${\displaystyle A_{i}}$

For every object of with morphisms there is exactly one morphism that satisfies for all . ${\displaystyle Y}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle g_{i}\colon A_{i}\to Y}$ ${\displaystyle g\colon \amalg \to Y}$ ${\displaystyle g_{i}=g\circ \mathrm {ins} _{i}}$ ${\displaystyle i\in I}$

One then writes for one . ${\textstyle \coprod _{i\in I}A_{i}}$ ${\displaystyle \amalg }$

Examples

Some common categories with their products and co-products are given.

category	product	Coproduct
amounts	Cartesian product	(outer) disjoint union
groups	direct product	free product
abelian groups		direct sum
Vector spaces
Modules over a ring
Commutative rings with one		Tensor product of rings (considered as -algebras) ${\displaystyle \mathbb {Z} }$
(quasi-) projective varieties	associated Segre variety	(no special term)
topological spaces	Product topology	disjoint union with the obvious topology
compact Hausdorff rooms		(no special term)
dotted topological spaces		Wedge product
Banach rooms	Countable linear combinations with , i.e. absolutely limited coefficients, with the weighted supremum of the norms as the norm ${\displaystyle \ell ^{\infty }}$	Countable linear combinations with , i.e. absolutely summable coefficients, with the weighted sum of the norms as the norm ${\displaystyle \ell ^{1}}$
partial orders	Infimum	Supremum

For Abelian groups, modules, vector spaces and Banach spaces, the finite products agree with the finite coproducts, so they deliver a biproduct. Their existence is required in the definition of Abelian categories , in particular Abelian groups, vector spaces and modules form Abelian categories over a ring.

In the category of topological spaces the product is exactly the Cartesian product provided with the coarsest topology where the projections are continuous, and the coproduct is the disjoint union with the same open sets on each of the spaces as before and their unions. ${\displaystyle \mathrm {pr} _{i}}$

In the category of Abelian groups, modules and vector spaces, the product is exactly the Cartesian product with component-wise connection; the co-product consists of the elements of the product, the components of which are almost everywhere (i.e. everywhere except for a finite number of places) zero.

If one interprets a quasi-order as the category of its elements with morphisms for , then the products give the infima and the coproducts the suprema of the corresponding elements. ${\displaystyle ({\mathcal {A}},\lesssim )}$ ${\displaystyle f\colon a\to b}$ ${\displaystyle a\lesssim b}$

literature

• Kurt Meyberg: Algebra . Part 2. Hanser Verlag, Munich 1976, ISBN 3-446-12172-2 ( Mathematical foundations for mathematicians, physicists and engineers ), see Chapter 10: Categories .

Individual evidence

1. ↑ ^{a b c d e f} Saunders Mac Lane : Categories for the Working Mathematician (=  Graduate Texts in Mathematics . No. 5 ). Springer, Berlin 1971, ISBN 3-540-90036-5 , p. 63 .