Adjunctions via Initial Objects

Summary

An adjunction $F⊣G$ is equivalent to: for each $A\in \mathcal{A}$, a universal map (= initial object of the comma category $(A\downarrow G)$). This formulation makes the universal property of the unit transparent, and shows that adjunctions are equivalent to universal constructions. Terminal objects and limits are special cases of this framework.

Overview

The comma category formulation of adjunctions makes the universal-property aspect primary. An adjunction $F⊣G$ says that for each $A$, the map ${\eta }_A:A\to GFA$ is initial among all maps $A\to GB$. This is the categorical meaning of ”$FA$ is the free object on $A$.”

Main Content

Initial and Terminal Objects

Definition: Initial and Terminal Objects

In a category $\mathcal{A}$:

• An initial object $\varnothing$ satisfies: for every $A$, there is exactly one map $\varnothing \to A$.
• A terminal object $1$ satisfies: for every $A$, there is exactly one map $A\to 1$.

Both are unique up to isomorphism when they exist. In Set: $\varnothing$ is initial, $\{\ast \}$ is terminal. In Grp: trivial group is both.

Comma Categories

Definition: Comma Category

For functors $F:\mathcal{A}\to \mathcal{C}$ and $G:\mathcal{B}\to \mathcal{C}$, the comma category $(F\downarrow G)$ has:

• Objects: triples $(A,B,f)$ where $A\in \mathcal{A}$, $B\in \mathcal{B}$, $f:FA\to GB$ in $\mathcal{C}$
• Morphisms $(A,B,f)\to (A^{\prime },B^{\prime },f^{\prime })$: pairs $(h:A\to A^{\prime },k:B\to B^{\prime })$ such that $Gk\circ f=f^{\prime }\circ Fh$

Special cases:

• $(A\downarrow G)$ = comma category of $A$ (viewed as functor from $1$) over $G$. Objects are maps $A\to GB$.
• $(F\downarrow B)$ = objects are maps $FA\to B$.

Adjunctions as Initial Objects in Comma Categories

Theorem: Adjunction ↔ Initial Objects in Comma Categories (BCT, Ch. 2.3)

The following are equivalent:

1. $F⊣G$ (adjunction with unit $\eta$)
2. For each $A\in \mathcal{A}$, the pair $(FA,{\eta }_A)$ is an initial object of the comma category $(A\downarrow G)$.

Explicitly: condition 2 says that for every $g:A\to GB$, there is a unique $\tilde{g}:FA\to B$ such that $G\tilde{g}\circ {\eta }_A=g$:

$$\forall g:A\to GB,\exists !\tilde{g}:FA\to B\text{ s.t. }G\tilde{g}\circ {\eta }_A=g$$

This is the universal property of the unit: ${\eta }_A:A\to GFA$ is the initial map from $A$ into a $G$-image.

Example: Free Group as Initial Object (BCT, Ch. 2.3)

For the free/forgetful adjunction $F⊣U:\mathbf{Grp}\to \mathbf{Set}$:

The universal property says: for any set $X$ and any group homomorphism $g:X\to UG$ (i.e., any function from $X$ to the underlying set of a group $G$), there is a unique group homomorphism $\tilde{g}:FX\to G$ such that $U\tilde{g}\circ {\eta }_X=g$.

This is exactly the universal property of the free group: a function on generators extends uniquely to a homomorphism.

Limits and Terminal Objects

Theorem: Limits as Terminal Objects (BCT, Ch. 2.3)

The limit of a diagram $D:\mathcal{I}\to \mathcal{A}$ is the terminal object of the category of cones over $D$, written $\mathrm{Cone}(D)$.

A cone over $D$ with vertex $A$ is a natural transformation $\lambda :\Delta A\Rightarrow D$, i.e., a family of maps ${\lambda }_I:A\to DI$ commuting with all $Du$.

$\mathrm{Cone}(D)$ has cones as objects and cone morphisms as morphisms. The limit is the terminal cone.

This unifies limits with the adjunction/initial-object perspective: limits are right adjoints to the diagonal functor $\Delta :\mathcal{A}\to [\mathcal{I},\mathcal{A}]$.

Connections

• This formulation underlies the Adjoint Functor Theorems (Adjoint Functor Theorems): to construct a left adjoint to $G$, one constructs initial objects in each $(A\downarrow G)$.
• Universal properties throughout mathematics are initial or terminal objects in appropriate categories (Universal Properties - Introduction).
• The limit $\lim D$ is the terminal object of $\mathrm{Cone}(D)$; the colimit $\mathrm{colim}D$ is the initial object of $\mathrm{CoCone}(D)$.
• The Yoneda lemma (Yoneda Lemma) identifies elements of $X(A)$ with natural transformations $H_A\Rightarrow X$, which is another instance of “elements as maps from representables.”

See Also

• Adjoint Functors — Hom-set definition of adjunction
• Units and Counits — Unit/counit definition
• General Limits — Limits as terminal cones
• Adjoint Functor Theorems — Constructing adjoints via initial objects
• Universal Properties - Introduction — Unification of universal constructions