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A Kan extension computing the free-group functor

Example

Let $1$ be the one-object category, so $[1^{o p}, S e t] ≅ S e t$ . Let $y : 1 \to S e t$ be the Yoneda embedding. Choose a free group $F ({*})$ on the one-element set ${*}$ , and let $F : 1 \to G r p$ send the unique object to that group.

Then the left Kan extension ${Lan}_{y} F : S e t \to G r p$ is the free-group functor.

Facts & Assumptions

Given: The one-object category $1$ , the Yoneda embedding $y$ , and the functor whose value is a chosen free group on ${*}$ .

[L1]

The free-cocompletion theorem makes ${Lan}_{y} F$ left adjoint to $G r p (F -, -)$ (The presheaf category on a small category is the free cocompletion).

[L2]

Choosing a free group on every set gives a free-group functor left adjoint to the underlying-set functor; at ${*}$ its adjunction bijection is $G r p (F ({*}), G) ≅ U (G)$ naturally in $G$ (The free-group functor is left adjoint to the underlying-set functor).

[L3]

Two left adjoints to the same functor are uniquely naturally isomorphic compatibly with their adjunction data (Adjoints are unique up to a unique natural isomorphism compatible with the adjunction data).

[F1]

The presheaf category on $1$ is $S e t$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding, Sets and functions form the large locally small category $S e t$ ).

[F2]

Groups and group homomorphisms form the locally small category $G r p$ , and $G r p$ has all small colimits (Groups and group homomorphisms form the large locally small category $G r p$ , Grp is complete and cocomplete).

Verification

technique · direct

1.1F1F2L1

By [F1], the functor ${Lan}_{y} F$ has domain $S e t$ , and [F2] verifies the locally small and cocomplete target hypotheses of [L1]. Hence [L1] makes it left adjoint to the functor $G \mapsto G r p (F ({*}), G)$ .

2.1L2step 1.1

By [L2], the functor $G \mapsto G r p (F ({*}), G)$ is naturally isomorphic to the underlying-set functor on groups. Therefore ${Lan}_{y} F$ is a left adjoint to the underlying-set functor.

3.1L2L3step 2.1∎

The published free-group functor is also a left adjoint to the underlying-set functor by [L2], so [L3] identifies ${Lan}_{y} F$ with that free functor.

Depends on

Used by

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Dependency tree · two levels

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Sources

E. Riehl, Category Theory in Context, 2nd ed., Proposition 6.5.11 (standard reference, not scraped)