List of Universal Properties

Introduction

This article lists the objects of the corpus that are defined by a universal property, that is, by the maps into or out of them rather than by a construction. The pattern is the same in every case: an object is required, together with a map satisfying a condition, such that every other candidate factors through it uniquely. The corpus meets the pattern in the free objects, in the quotients, in the tensor products, in the localizations, in the completions and in the products and coproducts.

Every entry points to the article that introduces the object and states the property that fixes it. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being a candidate for which the required factorisation fails, with the failure named and the article that records it.

Free Objects

A free object is the solution of a universal arrow from a set to a forgetful functor: the free monoid on a set is the universal monoid the set maps into, the free group and the free module are its analogues for groups and for modules, and the free algebra is the universal algebra containing a given module. In each case the universal map is the insertion of the generators, and a homomorphism out of the free object is determined by its values on them.

Object	The property it has	Introduced in
Free monoid $X^*$	the universal monoid on a set, with words as elements	Universal Properties and Categories
Free group $F(X)$	the universal group on a set, with reduced words as elements	Generators, Presentations and Free Products
Free abelian group	the universal abelian group on a set, with rank as its invariant	Infinite Abelian Groups
Free module $R^{(I)}$	the universal $R$-module on a set, with basis the inserted elements	Direct Sums, Free Modules and Rank
Free algebra $T(V)$	the universal associative algebra on a module, the tensor algebra	Tensor Powers and the Free Algebra
Free commutative algebra $\operatorname{Sym}(V)$	the universal commutative algebra on a module, the symmetric algebra	The Symmetric Algebra
Polynomial algebra $R[x_1,\dots,x_n]$	the free commutative $R$-algebra on $n$ generators	Polynomial Rings and Rational Functions
Free product $G_1 * G_2$	the coproduct of groups, the universal group receiving both factors	Generators, Presentations and Free Products
Free product of algebras $A \sqcup B$	the coproduct of associative algebras	Tensor Products of Algebras
Free operad	the free monoid in symmetric sequences, with trees as components	Operads
Non-example: $\mathbb{Z}/n\mathbb{Z}$ as a $\mathbb{Z}$-module	a module that fails to be free: it has no basis	Direct Sums, Free Modules and Rank
Non-example: a group with a presentation $\langle X \mid R \rangle$, $R \neq 1$	a quotient of the free group that fails freeness	Generators, Presentations and Free Products

Products, Coproducts and Limits

The product and the coproduct are the universal objects receiving the projections and carrying the injections; the initial and terminal objects are their empty cases, the equaliser and the coequaliser their diagrammatic forms, and the limit and the colimit the general construction. In the module-theoretic articles the coproduct is the direct sum and the product the direct product, and the two differ for infinite families. The pushout is the coproduct under a common base, which for commutative algebras is the tensor product.

Object	The property it has	Introduced in
Initial and terminal objects	the universal objects mapping out of and into every object	Universal Properties and Categories
Product $A \times B$	the universal object with two projections	Universal Properties and Categories
Coproduct $A + B$	the universal object with two injections	Universal Properties and Categories
Equaliser and coequaliser	the universal object equalising or coequalising a parallel pair	Universal Properties and Categories
Limit and colimit	the universal cone and cocone over a diagram	Universal Properties and Categories
Direct sum $\bigoplus_i M_i$	the coproduct of modules, the finitely supported submodule of the product	Direct Sums, Free Modules and Rank
Direct product $\prod_i M_i$	the product of modules, with its coordinate projections	Direct Sums, Free Modules and Rank
Pushout of commutative algebras	the coproduct under a common base, given by $\otimes_C$	Tensor Products of Algebras
Non-example: $\prod_i M_i$ for infinite $I$	the product is not the coproduct: its elements need not be finitely supported	Direct Sums, Free Modules and Rank
Non-example: a set with two elements	the coproduct in $\mathbf{Set}$ is the disjoint union, not the union	Sets, Functions and Relations

Quotients

A quotient is a coequaliser: the quotient of a set by an equivalence relation, of a group by a normal subgroup, of a ring by a two-sided ideal, of a module by a submodule and of an algebra by a two-sided ideal are the universal objects in which the collapsed elements become equal. The normal closure of a set of relations is the kernel of the corresponding quotient of the free group, and a presentation records the quotient in the language of generators and relations.

Object	The property it has	Introduced in
Quotient set $X/{\sim}$	the universal set in which equivalent elements coincide	Sets, Functions and Relations
Quotient group $G/N$	the universal group killing a normal subgroup	Groups
Abelianisation $G^{\mathrm{ab}}$	the universal abelian quotient of a group	Groups
Quotient ring $R/I$	the universal ring killing a two-sided ideal	Rings
Quotient module $M/N$	the universal module killing a submodule	Modules
Quotient algebra $A/I$	the universal algebra killing a two-sided ideal	Ideals and Quotients of Algebras
Normal closure $\langle\langle R \rangle\rangle$	the kernel of the quotient of $F(X)$ by the relations $R$	Generators, Presentations and Free Products
Non-example: the quotient of a group by a non-normal subgroup	the candidate fails: only normal subgroups carry a group structure on the cosets	Groups
Non-example: the quotient of a ring by a one-sided ideal	the candidate fails: the multiplication is not well defined on the cosets	Rings

Tensor Products

The tensor product is the universal object representing the bilinear maps; over a commutative ring it is the balanced product, and it is the underlying construction of the tensor, symmetric, exterior and Clifford algebras alike. The tensor power, the symmetric power and the exterior power are its quotients or its graded pieces, and each of them has the universal property of the multilinear maps it represents.

Object	The property it has	Introduced in
Balanced product $M \otimes_R N$	the universal object representing the $R$-bilinear maps	The Balanced Product
Tensor product of algebras $A \otimes_R B$	the universal algebra receiving two commuting images	Tensor Products of Algebras
Bimodule tensor $M_A \otimes_A {}_A N$	the balanced product over a non-commutative ring, forcing centrality	The Balanced Product over an Algebra
Tensor power $V^{\otimes n}$	the universal object for the $n$-multilinear maps	Tensor Powers and the Free Algebra
Symmetric power $\operatorname{Sym}^n M$	the universal object for the symmetric $n$-multilinear maps	Symmetric Powers
Symmetric algebra $\operatorname{Sym}(M)$	the quotient of the tensor algebra by $x \otimes y - y \otimes x$	The Symmetric Algebra
Exterior power $\Lambda^n M$	the universal object for the alternating $n$-multilinear maps	Exterior Powers
Exterior algebra $\Lambda(M)$	the quotient of the tensor algebra by $x \otimes x$	The Exterior Algebra
Clifford algebra $\mathrm{Cl}(V,Q)$	the quotient of $T(V)$ by $x \otimes x - Q(x)$	Clifford Algebras
Non-example: the tensor product over a non-commutative ring without a bimodule structure	the candidate fails: $M \otimes_R N$ requires one side left and one right	The Balanced Product over an Algebra

Localizations and Fraction Objects

A localization inverts a chosen set of elements and is universal among rings in which they become units. Its two extreme cases are the fraction field of an integral domain, the initial field in which the domain embeds, and the local ring at a prime ideal; for modules the same construction is the localisation of a module. The non-commutative analogue is the division ring of fractions of an Ore domain.

Object	The property it has	Introduced in
Localization $S^{-1}R$	the universal ring in which every element of $S$ becomes a unit	Localization and the Fraction Field
Fraction field $\operatorname{Frac}(R)$	the initial field in which an integral domain embeds	Localization and the Fraction Field
Local ring $R_\mathrm{P}$	the localization at a prime ideal, with a single maximal ideal	Localization and the Fraction Field
Localization of a module $S^{-1}M$	the universal module over $S^{-1}R$ receiving $M$	Localization and Completion of Modules
Total ring of fractions	the localization inverting the nonzero non-zero-divisors	Localization and the Fraction Field
Division ring of fractions of an Ore domain	the non-commutative analogue of the fraction field	Ore Domains and Division Rings of Fractions
Non-example: $\operatorname{Frac}(R)$ for a ring with zero divisors	the candidate fails: the fraction field exists only for a domain	Localization and the Fraction Field
Non-example: a division ring of fractions for the free algebra	the candidate fails: the free algebra is not an Ore domain	Ore Domains and Division Rings of Fractions

Completions

A completion is the universal complete object into which the given object embeds densely. The corpus meets the algebraic $I$-adic completion — of a ring, of a module and, for $I = (x_1,\dots,x_n)$, of the polynomial algebra as the formal power series algebra — and, once a distance is available, the metric completion, the profinite completion and the Dedekind completion.

Object	The property it has	Introduced in
$I$-adic completion $\hat A = \varprojlim A/I^k$	the universal complete ring receiving $A$	Localization and Completion of Modules
Formal power series algebra $R[[x_1,\dots,x_n]]$	the $I$-adic completion of the polynomial algebra	Formal Power Series and Completion
Associated graded algebra $\operatorname{gr}_{\mathrm{M}} A$	the graded object attached to an $I$-adic filtration	Formal Power Series and Completion
Metric completion	the universal complete metric space receiving a metric space	Metric, Uniform and Complete Spaces
Profinite completion	the inverse limit of the finite quotients, universal among profinite groups	Profinite Groups and the Krull Topology
Dedekind completion	the universal complete ordered field extension of an ordered field	Real-Closed and Complete Ordered Fields
Non-example: the category of fields	the coproduct fails to exist there: $\mathbb{C} \otimes_\mathbb{R} \mathbb{C}$ is a ring but not a field	Tensor Products of Algebras

Summary

The list gathers the objects of the corpus that are defined by a universal property. The free objects are the free monoid, the free group, the free abelian group, the free module, the free algebra, the free commutative algebra, the polynomial algebra, the free product and the free operad; the products and coproducts are the initial and terminal objects, the product, the coproduct, the equaliser and coequaliser, the limit and colimit, the direct sum and product and the pushout; the quotients are the quotient set, group, ring, module and algebra, with the abelianisation and the normal closure; the tensor constructions are the balanced product, the tensor product of algebras, the bimodule tensor, the tensor, symmetric and exterior powers and algebras and the Clifford algebra; the localizations are the localization of a ring, the fraction field, the local ring, the localization of a module, the total ring of fractions and the division ring of fractions of an Ore domain; and the completions are the $I$-adic completion, the formal power series algebra, the metric completion, the profinite completion and the Dedekind completion. The non-examples — the non-free module, the non-free presented group, the infinite direct product, the quotient by a non-normal subgroup, the quotient by a one-sided ideal, the tensor product over a non-commutative ring without a bimodule structure, the fraction field of a ring with zero divisors, the division ring of fractions of the free algebra and the missing coproduct in the category of fields — name the hypothesis that each fails.

Summary of Notation

The article denotes its objects by name; the symbols appearing in the tables are those of the introducing articles.

Symbol	Meaning
$X^*$	free monoid on a set $X$
$F(X)$	free group on a set $X$
$R^{(I)}$, $R^n$	free module on the index set $I$
$\mathbb{Z}$, $\mathbb{R}$, $\mathbb{C}$	the integers, the reals, the complex numbers
$T(V)$, $\operatorname{Sym}(V)$, $\Lambda(V)$, $\mathrm{Cl}(V,Q)$	tensor, symmetric, exterior and Clifford algebras
$V^{\otimes n}$, $\operatorname{Sym}^n M$, $\Lambda^n M$	tensor, symmetric and exterior powers
$M \otimes_R N$	balanced product over $R$
$A \sqcup B$, $A \otimes_C B$	free product and pushout of algebras
$S^{-1}R$, $S^{-1}M$	localizations of a ring and of a module
$\operatorname{Frac}(R)$	fraction field
$R_\mathrm{P}$	localization at a prime ideal
$\hat A = \varprojlim A/I^k$	$I$-adic completion
$R[[x_1,\dots,x_n]]$	formal power series algebra
$\bigoplus_i M_i$, $\prod_i M_i$	direct sum and direct product
$G_1 * G_2$, $G_1 *_H G_2$	free product and free product with amalgamation
$\langle X \mid R\rangle$, $\langle\langle R\rangle\rangle$	presentation; normal closure of the relations