What is significance of this proof of existence of free groups (Lang's Algebra)

Question

There are different proofs of existence of free groups. While reading Lang's Algebra, it caught my attention towards proof of this theorem by first bracket statement in proof:

Later I went on reading proof, and lost after half-part. Then an overlook at the proof suggested me that its significance may be due to the fact that the construction comes within the category of groups. However, in some part of construction of free group, he has moved into (category of) set theory also (I feel) -see Lemma 12.2. This burned the question in mind what is real significance of this proof? Even before or after construction of free group there, it was not mentioned about why this proof was given which he owe to Tits? (I mean, there should be some thought-process and communication between Lang and Tits to give this new proof than earlier proofs)

Answer 1

The significance of this proof is that it is secretly a proof of Freyd's General Adjoint Functor Theorem, which gives sufficient conditions for the existence of a reflection of an object along a limit-preserving functor $\mathcal D\xleftarrow{G}\mathcal E$ from a complete locally small category $\mathcal E$.

Definition. A reflection, or left Kan lift, of an object $Y$ in a category $\mathcal D$ along a functor $\mathcal D\xleftarrow{G}\mathcal E$ is an object $FY$ of $\mathcal E$ together with a unit morphism $Y\xrightarrow{\eta_Y}GFY$ of $\mathcal D$ through which any other morphism $Y\xrightarrow{f}GZ$ in $\mathcal D$ factors as $Y\xrightarrow{\eta_Y}GFY\xrightarrow{G\phi}Z$ for a unique $FY\xrightarrow{\phi}Z$ in $\mathcal E$. Equivalently, a reflection of $Y$ in $\mathcal D$ along $\mathcal D\xleftarrow{G}\mathcal E$ is an initial object in the comma category $(Y\downarrow G)$ whose objects are morphism $Y\to GZ$ and whose morphisms are commutative triangles $Y\to GZ_1\xrightarrow{G\phi}GZ_2$ in $\mathcal D$ with $Z_1\xrightarrow{\phi}Z_2$ in $\mathcal E$.

Example. A free group on a set $S$ is the reflection of a set $S$ along the forgetful functor $\mathcal Set\xleftarrow{U}\mathcal Group$. More generally, any free algebraic structure on a set $S$ is the reflection of the set along the forgetful functor from the category of those algebra structures and homomorphisms between them.

Lang's construction of the reflection $FS$ in $\mathcal Group$ of set $S$ in $\mathcal Set$ along the forgetful functor $\mathcal Set\xleftarrow{U}\mathcal Group$, with all categorical details spelled out, is this.

1. As phrased, Lang's Lemma 12.2, only produces a set of groups $\{G_i\}$ such that if $S\xrightarrow{g}UG$ generates $G$, then it $G_i\cong G$ for some $i$. But if $S\xrightarrow{g}UG$ is arbitrary, then $G_i$ must be isomorphic to the image of $S$ in $UG$, hence we have a morphism $S\to UG_i\xrightarrow{U\gamma}UG$. Thus, the lemma really produces a subset of the set $\{S\xrightarrow{f} UG_{i,f}\}$ that Lang considers after the Lemma. That subset, and the full set $\{S\xrightarrow{f} UG_{i,f}\}$, is a set of objects in the comma category $(S\downarrow U)$ so that each object $S\xrightarrow{g} UG$ in $(S\downarrow U)$ admits a morphism $S\xrightarrow{f} UG_{i,f}\xrightarrow{U\gamma} UG$ from an object $S\xrightarrow{f} UG_{i,f}$ in the set to $S\xrightarrow{g} UG$. This is a so-called solution set of the object $S$ relative to $\mathcal Set\xleftarrow{U}\mathcal Group$; the functor $\mathcal Set\xleftarrow{U}\mathcal Group$ is said to satisfy the solution set condition since every object $S$ in $\mathcal Set$ has a solution set.

2. You can understand the solution set of the object $S$ in $\mathcal Set$ along $\mathcal Set\xleftarrow{U}\mathcal Group$ as an inclusion $\{S\xrightarrow{f} UG_{i,f}\}\xrightarrow{i}(S\downarrow U)$ of a small discrete category such that for every object $S\to UG$ of $(S\downarrow U)$, the comma category $(i\downarrow S\to UG)$ is inhabited, that is, have at least one object.

3. A step which Lang uses implicitly is the replacement of the inclusion of the set $\{S\xrightarrow{f} UG_{i,f}\}$ of objects in $(S\downarrow U)$ with the inclusion $\mathcal K\xrightarrow{j}(S\downarrow U)$ full small subcategory $\mathcal K$ of $(S\downarrow U)$ generated by the set $\{S\to UG_i\}$. Note that the full subcategory is small because $(S\downarrow U)$ is locally small and we were including a small category, and $(S\downarrow U)$ is locally small because $\mathcal Group$ is locally small.

4. The inclusion $\mathcal K\xleftarrow{j}(S\downarrow U)$ is better because each comma category $(j\downarrow S\to UG)$ is then not only inhabited but also connected: any two objects can be joined by a zig-zag. Indeed, for any two factorizations $S\xrightarrow{f_1} UG_{i,f_1}\xrightarrow{U\phi_1} UG$ and $S\xrightarrow{f_2} UG_{j,f_2}\xrightarrow{U\phi_2} UG$, there is a pullback $H=\{(g_1,g_2)\in G_i\times G_j: \phi_1(g_1)=\phi_2(g_2)\}$. We have a map $S\xrightarrow{f}UH$ given by $s\mapsto(f_1(s),f_2(s))$, which must factor as $S\to UG_k\xrightarrow{U\psi}UH$. It follows that we have a zig-zag from $S\xrightarrow{f_1}UG_i\xrightarrow{U\phi_1} UG$ to $S\xrightarrow{f_2}UG_j\xrightarrow{U\phi_2} UG$ by way of $S\to UG_k\xrightarrow{U\psi}UH\xrightarrow{\pi_1}G_i$ and $S\to UG_k\xrightarrow{U\psi}UH\xrightarrow{\pi_2}G_j$ (since by applying fullness to $UG_k\xrightarrow{U\psi}UH\xrightarrow{\pi_1}G_i$ and $UG_k\xrightarrow{U\psi}UH\xrightarrow{\pi_2}G_j$). Note that this works because $\mathcal Group$ has pullbacks which the forgetful functor $\mathcal Set\xleftarrow{U}\mathcal Group$ preserves.

5. The product $F_0=\prod_{i\in I}\prod_{f\in M_i}G_{i,f}$ together with its natural map $S\xrightarrow{f_0} UF_0$ is actually the limit of $\mathcal K\xrightarrow{j}(S\downarrow U)$ considered as a small diagram in the comma category $(S\downarrow U)$. This small limits in $(S\downarrow U)$ exists because $\mathcal Group$ has small limits and $\mathcal Set\xleftarrow{U}\mathcal Group$ preserves limits.

6. The nice thing about the inclusion $\mathcal K\xrightarrow{j}(S\downarrow U)$, that the comma categories $(j\downarrow S\to UG)$ are inhabited and connected implies that the inclusion is a final functor. This means in that for any other other functor $\mathcal Group\xrightarrow{K}\mathcal C$, the image of its limit $S\to\prod_{i\in I}\prod_{f\in M_i}G_{i,f}$ under a functor $(S\downarrow U)\xrightarrow{J}\mathcal C$ is the limit of the composite $\mathcal K\xrightarrow{j}(S\downarrow U)\xrightarrow{J}\mathcal C$.

7. Lang's argument for constructing a homomorphism $F_0\to G$ from $S\to G$ is I think a compressed version of the proof that the above inclusion is final. The reason is that finality implies that $S\to \prod_{i\in I}\prod_{f\in M_i}G_{i,f}$ is the limit of the identity functor $(S\downarrow U)\xrightarrow{\mathrm{id}}(S\downarrow U)$, and the limit of the identity functor on a category is the same thing as an initial object in the category. Hence, $S\to F_0$ is the initial object in $(S\downarrow U)$, so the reflection of $S$ along $\mathcal Set\xleftarrow{U}\mathcal Group$.

If you read carefully, this is a proof of the hard implication in $1\Rightarrow2$ of

Theorem (General Adjoint Functor Theorem). Whenever a functor $\mathcal D\xleftarrow{G}\mathcal E$ is limit-preserving from a complete locally small category $\mathcal E$, the following are equivalent

1. An object $Y\in\mathcal D$ satisfies the solution set condition.
2. An object $Y\in\mathcal D$ has a reflection.

The easy implication $2\Rightarrow 1$ follows from the fact that a reflection of $Y$ is itself a solution set with a single element.

Remark. The reason the theorem is called an adjoint functor theorem is this. If $\mathcal C\xrightarrow{J}\mathcal D$ is another functor, then a choice of relections $JX\xrightarrow{\eta_X}GFX$ in $\mathcal E$ of each object $JX$ along $\mathcal D\xleftarrow{G}\mathcal E$ extends naturally to functor $\mathcal C\xrightarrow{F}\mathcal E$, with a unit natural transformation $\eta\colon J\Rightarrow GF\colon\mathcal C\to\mathcal D$. This functor and natural transformation is the relative left adjoint, or absolute left Kan lift, of $\mathcal C\xrightarrow{J}\mathcal D$ along $\mathcal D\xleftarrow{G}\mathcal E$ and is denoted $F{}_J\vdash G$. In particular, if all objects of $\mathcal D$ have reflections along $\mathcal D\xleftarrow{G}\mathcal E$, and a choice of such reflections gives an absolute left Kan lift $\mathcal D\xrightarrow{F}\mathcal E$ of $\mathcal D\xrightarrow{\mathrm{id}_{\mathcal D}}\mathcal D$ along $\mathcal D\xleftarrow{G}\mathcal E$, i.e. the left adjoint to $G$, denoted simply $F\vdash G$.