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Free algebras by means of a universal property

Consider a vector space VV, when XX is a basis of VV, we say that VV is a free vector space on XX 1. For any vector space VV and WW and a basis XX of VV, if there is a map f:XWf:X\to W then there exists a unique linear map from VV to WW that extends ff. That is, there is a unique linear map making the following diagram commute,

Commutative diagram for the universal property of a free vector space. It shows that for a map i from set X to vector space V and a map f from X to vector space W, there exists a unique linear map g from V to W making the diagram commute.

where ii is the inclusion map i:XVi:X\to V (in fact, any map XVX\to V satisfying this property turns out to be injective) 1. In other words there exists a unique map g:VWg:V\to W such that gi=fgi=f.

We can use this to define free in terms of a universal property. In order to do this we define the following category. Let C\mathbf{C} be a full subcategory of Alg(Ω)\mathbf{Alg}(\Omega) with objects in C\mathbf{C} called C\mathbf{C}-algebras, and let XX be a set. Define C[X]\mathbf{C}[X] to be the category of pairs (A,α)(A,\alpha), in which AA is a C\mathbf{C}-algebra and α\alpha a map from XX to AA. A morphism f:(A,α)(B,β)f:(A,\alpha)\to (B,\beta) in C[X]\mathbf{C}[X] is a homomorphism f:ABf:A\to B such that fα=βf\alpha = \beta 2. That is, a morphism in C[X]\mathbf{C}[X] is a map f:ABf:A\to B making the following diagram commute.

Commutative diagram for a morphism in C[X]. It shows a map alpha from set X to algebra A, a map beta from X to algebra B, and a homomorphism f from A to B such that f composed with alpha equals beta.

A free algebra on XX is then an initial object in C[X]\mathbf{C}[X] 2.

The example of a free KK-vector space above can be constructed in terms of this definition. Let KK be a field and C\mathbf{C} the category of KK-vector spaces. The free C\mathbf{C}-algebra (V,α)(V,\alpha) on a set XX is the free KK-vector space on XX where VV is the set K(X)K^{(X)} of all maps u:XKu: X\to K such that xX:u(x)0\\{x\in X:u(x)\neq 0\\} is finite. The KK-vector space structure on VV is defined by

(u+v)(x)=u(x)+v(x) and (ku)(x)=k(u(x))(u+v)(x)=u(x)+v(x)\text{ and }(ku)(x)=k(u(x))

for u,vVu,v\in V, xXx\in X, and kKk\in K. The map α:XV\alpha: X\to V is defined by

α(x)(y)={1, if x=y0, if xy\alpha(x)(y)=\begin{cases} 1, \text { if } x=y \\\\ 0, \text { if } x \neq y \end{cases}

for all x,yXx,y\in X.

Now, for every map β:XW\beta:X\to W there exists a unique morphism of KK-vector spaces f:VWf:V\to W with fα=βf\alpha = \beta, that is, making the diagram

Commutative diagram characterizing a free algebra as an initial object. It shows that for any map beta from set X to algebra W, there exists a unique morphism f from the free algebra V to W that makes the diagram commute.

commute and this map is explicitly given by

f(u)=xXu(x)β(x)f(u) = \sum_{x\in X}u(x)\beta(x)

2. Hence, VV is initial since this morphism always exists for any object (W,β)(W,\beta) of C[X]\mathbf{C}[X]. In fact, every KK-vector space is the free-semimodule over a field and so every KK-vector space is free.

We can apply this definition to other algebraic structures. Consider C\mathbf{C} as the category of magmas. Then a free magma on XX is an initial object (M,α)(M,\alpha) in C[X]\mathbf{C}[X]. Similarly, for groups, if we consider C\mathbf{C} the category of groups, the free group on XX is the initial object in C[X]\mathbf{C}[X].


  1. George Janelidze. Conversation on free algebras. Personal communica- tion 15/03/22, 2022. ↩︎ ↩︎

  2. George Janelidze. Introduction to abstract algebra. Lecture Notes, 2015. ↩︎ ↩︎ ↩︎