What category is the universal property of the Free Group a diagram in?

As you say, $S$ is a set, so this is a diagram in $\text{Set}$. The fact that we force $\varphi$ to be a homomorphism of groups is extra structure that isn't captured by the diagram alone.

You might consider this unsatisfying, so alternatively we can explicitly name the forgetful functor $U : \text{Grp} \to \text{Set}$ from groups to sets, which is being implicitly applied to $G$ here, and regard $f$ as a morphism $f : S \to U(G)$ in $\text{Set}$, then talk about the universal property in terms of the adjunction

$$\text{Hom}_{\text{Grp}}(F(S), G) \cong \text{Hom}_{\text{Set}}(S, U(G)).$$


As the definition mentions, $f$ and the unnamed inclusions are just functions while $\varphi$ is a group homomorphism. Hence the diagram is not in $\mathbf{Grp}$, nor actually in $\mathbf{Set}$ (in the sense that the diagram in $\mathbf{Set}$ would not force $\varphi$ to be a group homomorphism).

The construction gives in fact a functor from $\mathbf{Set}$ to $\mathbf{Grp}$ assigning to each set $S$ the free group $F_S$, and to each function $g:S\to T$ the morphism $\varphi_g:F_S \to F_T$ associated to the map $f=\iota_T\circ g:S\to F_T$ by the universal property (where $\iota_T:T\to F_T$ is the inclusion).


I often think of the free group over $S$ to be the initial object in the category of groups with $S$ specified points, or more formally the category of groups $(G, *)$ along with a specified function from $S$ to $G$, where morphisms consist of a group homomorphism that makes the functions from $S$ agree.

This definition captures what the diagram is trying to convey: it is showing initiality of $F_S$ where the morphism consists of the whole wedge coming out of $S$.

As other answers have said, this construction results in a functor from $\mathbf{Set}$ to $\mathbf{Grp}$, which is left adjoint to the forgetful functor, but I don't believe this context (while fascinating, and points to many interesting generalizations) is necessary to understand the free group.

Tags:

Category Theory

Group Theory

Universal Property

Related

Sum of Telescopic Series $|z-1|+|z-2|+|z-3|=6$ in the Argand Plane Is it true that ${\rm Tr}((A^{1/2}BA^{1/2})^{1/2}) = {\rm Tr}((BA)^{1/2})$ for positive semidefinite matrices $A,B$? What is the probability that the total score after throwing darts is divisible by $3$. Categories whose coproducts are not disjoint How can we prove that there is a number $a$ such that $\lim_{h\to 0}\frac{a^{h} -1}{h}=1$? $f(x+y)=f(x)+f(y)+99$, $f(100)=101$, find $f(7)$. If $P(x)=\sum_{i=0}^da_i\left(\prod_{j=i}^{d+i-1}(x+j)\right)$ is linear, what is its constant term? How to calculate the area of $\triangle ABC$ when the distance from $BC$ to the circumcircle at $G$ is 10? Why doesn't $f'(x) \to 0$ at $x \to \infty$ imply that $\lim_{x \to \infty}{f(x)}$ exists? Is this a correct natural deduction proof? Why are proofs not written as collections of logic symbols but are instead written in sentences?

Recent Posts