Created: Tue, 5 Feb 2019
Last updated: Fri, 22 Feb 2019

For a fixed field $K$, consider the functors

$\begin{xy} \xymatrix{ \textbf{Set}\ar@<1ex>[r]^V & \textbf{Vct}_K\ar[l]^U } \end{xy}$

$U$ is the forgetful functor which takes the vector field to the corresponding set of vectors, while $V$ takes $\textbf{Set}$ to a basis. For set $X$, $V(X)$ is the finite formal set of linear combinations $\sum r_i x_i$, with scalar coefficents $r_i$, and $x_i \in X$. For vector space $W$, $g : X \rightarrow U(W)$ extends to $f : V(X) \rightarrow W$, and hence there is a bijection $\varphi : \textbf{Set}(X, U(W)) \cong \textbf{Vec}_K(V(X), W)$

Setting $h : X' \rightarrow X, h^* g = g \circ h$, naturality in $X, W$ are obtained from the following diagram:

$\begin{xy} \xymatrix{ \textbf{Set}(X, U(W))\ar[r]^\varphi\ar[d]^{Vh^*} & \textbf{Vec}_K(V(X), W)\ar[d]^{h^*} \\ \textbf{Set}(X', U(W))\ar[r]^\varphi & \textbf{Vec}_K(V(X'), W) } \end{xy}$

As an example, consider the [free category](/ct/1#graphs-and-free-categories) over a small graph $G$, $C = FG$, and $D : \textbf{Grph} \rightarrow \textbf{Cat}$ given by; it is related to the forgetful functor $U : \textbf{Cat} \rightarrow \textbf{Grph}$ since $D : G \rightarrow UB$ extends to $D' : FG \rightarrow B$. Moreover $D \mapsto D'$ is a natural isomorphism: $\textbf{Cat}(FG, B) \cong \textbf{Set}(G, UB)$

In the category of small sets, the function $g : T \times R \rightarrow S$, a function of two variables, can be considered as $\varphi g : T \rightarrow \text{Hom}(R, S)$, a function of one variable, so that the following bijection holds: $\varphi : \text{Hom}(T \times R, S) \cong \text{Hom}(T, \text{Hom}(R, S))$

For modules $A, B, C$ in a commutative ring $K$, a similar bijection holds: $\text{Hom}(A \otimes_K B, C) \cong \text{Hom}(A, \text{Hom}_K(B, C))$

The adjuction of categories $X, A$, where $F, G$ are functors, is given by the triple $\langle F, G, \varphi \rangle : X \rightharpoonup A$

$\begin{xy} \xymatrix{ X\ar@<1ex>[r]^F & A\ar[l]^G } \end{xy}$

while $\varphi$ is a function such that: $\varphi = \varphi_{x, y} : A(Fx, a) \cong X(x, Ga)$

Here, $A(Fx, a)$ is the bifunctor:

$X^{op} \times A \xrightarrow{F^{op} \times \text{Id}} A^{op} \times A \xrightarrow{hom} \textbf{Set}$

It satisfies the diagrams:

$\begin{xy} \xymatrix{ X(Fx, a)\ar[r]^\varphi\ar[d]^{k_*} & X(x, Ga)\ar[d]^{(Gk)_*} \\ X(Fx, a')\ar[r]^\varphi & X(x, Ga') } \end{xy} \begin{xy} \xymatrix{ X(Fx, a)\ar[r]^\varphi\ar[d]^{h^*} & X(x, Ga)\ar[d]^{(Fh)^*} \\ X(Fx', a)\ar[r]^\varphi & X(x', Ga) } \end{xy}$

Here, $h^* = X(h, Ga), k_* = A(Fx, k)$.

An adjuncion can also be described without talking about hom-sets, directly in terms of arrows. There is a bijection which assigns to each of the arrows $f : Fx \rightarrow a$ arrows $g = \text{rad } f : x \rightarrow Ga$, the right adjunct of $f$, in such a way that the natural compositions hold: $\varphi(k \circ f) = Gk \circ \varphi f \quad \varphi(f \circ Fh) = \varphi f \circ h$

$F$ is said to be right adjoint for $G$, and $G$ is the left adjoint for $F$.

Every adjunction yields a universal arrow. Call the hom-set that contains the identity $1 : Fx \rightarrow Fx$ the $\varphi$-image of $\eta_x$. $\eta_x : x \rightarrow GFx \quad \eta_x = \varphi(1_{Fx})$

The adjunction gives such a universal arrow $\eta_x$ for every object $x$. The function $x \mapsto \eta_x$ is an NT $I_x \rightarrow GF$. We obtain the following diagrams:

$\begin{xy} \xymatrix{ x'\ar[r]^{\eta_{x'}}\ar[d]_h & GFx'\ar[d]^{GFh} \\ x\ar[r]^{\eta_x} & GFx } \end{xy}$

$\begin{xy} \xymatrix{ A(x', Fx')\ar[r]^{(Fh)_*}\ar[d]^\varphi & A(x', Fx)\ar[d]^\varphi & A(x, Fx)\ar[l]_{(Fh)^*}\ar[d]^\varphi \\ X(x', GFx')\ar[r]^{(GFh)_*} & X(x', GFx) & X(x, GFx)\ar[l]_{h^*} } \end{xy}$

where $h_* = X(1, h)$ and $h^* = X(h, 1)$.

The bijection $\varphi$ can be expressed as: $\varphi(f) = G(f)\eta_x \quad f = Fx \rightarrow a$

and we obtain:

$\begin{xy} \xymatrix{ A(Fx, Fx)\ar[r]^\varphi\ar[d]^{f_*} & X(x, GFx)\ar[d]^{(Gf)_*} \\ A(Fx, a)\ar[r]^\varphi & X(x, Ga) } \end{xy} \begin{xy} \xymatrix{ 1\ar@{|->}[r]\ar@{|->}[d] & \eta_x\ar@{|->}[d] \\ f \circ 1 \mapsto \varphi f\ar@{}[r]|{=} & Gf \circ \eta_x } \end{xy}$

To summarize, every adjunction $\langle F, G, \varphi \rangle : X \rightharpoonup A$ determines:

1. A natural transform $\eta : I_x \overset{\bullet}{\rightarrow} GF$ such that for each object $x$, the arrow $\eta_x$ is universal from $x$ to $G$, and right adjunct $f : Fx \rightarrow a$ is detemined by $\varphi f = Gf \circ \eta_x : x \rightarrow Ga$
2. Dually, another natural transform $\epsilon : FG \overset{\bullet}{\rightarrow} I_x$ such that every arrow $\epsilon_a$ is universal from $G$ to $a$, and left adjunct $g : x \rightarrow Ga$ is determined by $\varphi^{-1} g = \epsilon_a \circ Fg : Fx \rightarrow a$

Moreover, the following composites are identities: $G \xrightarrow{\eta G} GFG \xrightarrow{G \epsilon} G \quad F \xrightarrow{F \eta} FGF \xrightarrow{\epsilon F} F$

$\eta$ is termed unit and $\epsilon$ is termed counit of the adjunction.

The adjunct is completely determined by one of the following items:

1. Functors $F, G$ and NT $\eta : 1_x \overset{\bullet}{\rightarrow} GF$ such that every $\eta_x : x \rightarrow GFx$ is universal. Then, $\varphi f = Gf \circ \eta_x : x \rightarrow Ga$.
2. Functor $G : A \rightarrow X$, object $F_0 x \in A$, and universal arrow $\eta_x : x \rightarrow GF_0 x$, from $x$ to $G$. Then, functor $F$ has object function $F_0$ and, on arrow $h : x \rightarrow x'$, by $GFh \circ \eta_x \rightarrow \eta_{x'}$.
3. Functors $F, G$, and NT $\epsilon : FG \overset{\bullet}{\rightarrow} I_A$ such that every $\epsilon_a : FGa \rightarrow a$ is universal. Then, $\varphi^{-1} g = \epsilon_a \circ Fg : Fx \rightarrow a$.
4. Functor $F : X \rightarrow A$, object $G_0 \in X$, and universal arrow $\epsilon_a : FG_0 a \rightarrow a$, from $F$ to $a$.
5. Functors $F, G$ and NTs $\eta : I_X \overset{\bullet}{\rightarrow} GF$ and $\epsilon : FG \overset{\bullet}{\rightarrow} I_A$.

Due to (v), we can also write the adjunct $\langle F, G, \varphi \rangle$ as $\langle F, G, \eta, \epsilon \rangle : A \rightharpoonup X$.

To prove (i), or its dual (iii), notice:

$\begin{xy} \xymatrix{ Fx\ar@{.>}[d]_g & x\ar[r]^{\eta_x}\ar[dr]_f & GFx\ar@{.>}[d]^{Gg} \\ a & & Ga } \end{xy}$

To prove (ii), or its dual (iv), notice:

$\begin{xy} \xymatrix{ F_0 x\ar@{.>}[d] & x\ar[r]^{\eta_x}\ar[d]^h & GF_0 x\ar@{.>}[d] \\ F_0 x' & x'\ar[r]^{\eta_{x'}} & GF_0 x' } \end{xy}$

To prove (v), given $\eta, \epsilon$, define:

$\begin{xy} \xymatrix{ A(Fx, a)\ar@<1ex>[r]^\varphi & X(x, Ga)\ar[l]^\theta } \end{xy}$

by $\varphi f = Gf \circ \eta_x$ for all $f : Fx \rightarrow a$, and $\theta g = \epsilon_a \circ Fg$ for all $g : x \rightarrow Ga$. Since $G$ is a functor and $\varphi$ an NT, we have: $\varphi \theta g = G \eta_a \circ GFg \circ \eta_x = G \eta_a \circ \eta_{Ga} \circ g$

Now, since $G \eta_a \circ \eta_{Ga} = 1$, $\varphi \theta$ is identity.