Adjoint functors

R. Virk

Department of Mathematics
University of Wisconsin
Madison, WI 53706

virk@math.wisc.edu

For simplicity, we will assume all categories to be concrete categories, i.e., objects have an underlying set structure and morphisms are completely determined by their effect on the underlying sets.

Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F:\mathcal{C}\to\mathcal{D}$ and $F^{\vee}:\mathcal{D}\to\mathcal{C}$ be functors. We say that $F$ is left adjoint to $F^{\vee}$ (or $F^{\vee}$ is right adjoint to $F$) and that $(F,F^{\vee})$ form an adjoint pair if there is a map of sets

$$\alpha: \mathrm{Hom}_{\mathcal{D}}(F(X), Y) \buildrel \sim\over\to \mathrm{Hom}_{\mathcal{C}}(X,F^{\vee}(Y))$$

that is functorial in $X$ and $Y$, $X\in\mathcal{C}$, $Y\in\mathcal{D}$.

Substituting $Y=F(X)$ we obtain a map

$$\eta_X:=\alpha(\mathrm{id}_{F(X)}):X\to F^{\vee}F(X).$$

Substituting $X=F^{\vee}(Y)$ we obtain a map

$$\varepsilon_Y:=\alpha^{-1}(\mathrm{id}_{F^{\vee}(Y)}):FF^{\vee}(Y)\to Y.$$

The families $\{\eta_X\}$ and $\{\varepsilon_Y\}$ define natural transformations

$$\eta:\mathrm{id}_C\to F^{\vee}F, \qquad \varepsilon:FF^{\vee}\to\mathrm{id}_D$$

called the counit and unit, respectively. Both maps are also called the adjunction maps.

Let $f:F(X)\to Y$ be a map in $\mathcal{D}$, then by functoriality the following diagram commutes:

$$\xymatrix{\mathrm{Hom}_{\mathcal{D}}(F(X),F(X)) \ar[d]_{f_*}\ar[r... ..._{\mathcal{D}}(F(X),Y)\ar[r]^{\sim}&\mathrm{Hom}_{\mathcal{C}}(X,F^{\vee}(Y)) }$$

where $f_*$ and $(F^{\vee}f)_*$ are the induced maps on $\mathrm{Hom}$. In particular

$$\alpha(f)=\alpha(f\circ\mathrm{id}_{F(X)})=F^{\vee}(f)\circ \alpha(\mathrm{id}_{F(X)})=F^{\vee}(f)\circ \eta_X.$$

Similarly, given a map $g:X\to F^{\vee}(Y)$ in $\mathcal{D}$, then

$$\alpha^{-1}(g)=\alpha^{-1}(\mathrm{id}_{F^{\vee}(Y)}\circ g)=\alpha^{-1}(\mathrm{id}_{F^{\vee}(Y)}\circ F(g)=\varepsilon_Y \circ F(g).$$

It follows that the compositions

$$\xymatrixcolsep{3pc} \xymatrix{ F(X) \ar[r]^{F(\eta_X)}&FF^{\vee}F(X)\ar[r]^{\varepsilon_{F(X)}}& F(X)}$$ (0.1)

and

$$\xymatrixcolsep{3pc} \xymatrix{ F^{\vee}(Y)\ar[r]^{\eta_{F^{\vee}(Y)}}& F^{\vee}FF^{\vee}(Y)\ar[r]^{F^{\vee}(\varepsilon_Y)}&F^{\vee}(Y)}$$ (0.2)

are the maps $\mathrm{id}_{F(X)}$ and $\mathrm{id}_{F^{\vee}(Y)}$ respectively.

The existence of of adjunction maps is equivalent to $(F,F^{\vee})$ being an adjoint pair. Namely, let $F:\mathcal{C}\to\mathcal{D}$ and $F^{\vee}:\mathcal{D}\to\mathcal{C}$ be functors with the additional data of natural transformations $\eta:\mathrm{id}_{\mathcal{C}}\to F^{\vee}F$ and $\varepsilon: FF^{\vee}\to\mathrm{id}_{\mathcal{D}}$ that satisfy (0.1) and (0.2). Then $(F,F^{\vee})$ is an adjoint pair. The isomorphism $\alpha$ is given by $\alpha(f)=F^{\vee}(f)\circ \eta_X$ and the inverse $\alpha^{-1}$ is given by $\alpha^{-1}(g)=\varepsilon_Y\circ F(g)$.

Let $(F,F^{\vee})$ and $(G,G^{\vee})$ be adjoint pairs with units $\eta$, $\overline{\eta}$ respectively and counits $\varepsilon$, $\overline{\varepsilon}$ respectively. Let $\varphi\in\mathrm{Hom}(F,G)$. Then, we define $\varphi^{\vee}:G^{\vee}\to F^{\vee}$ as the composition

$$\varphi^{\vee}: \xymatrixcolsep{4pc} \xymatrix{ G^{\vee}(Y)\ar[r]... ...{\vee}GG^{\vee}(Y) \ar[r]^{F^{\vee}(\overline{\varepsilon}_Y)} & F^{\vee}(Y), }$$

$Y\in\mathcal{D}$. This is the unique map making the following diagram commutative, for any $X\in\mathcal{C}$ and $Y\in\mathcal{D}$:

$$\xymatrix{\mathrm{Hom}_{\mathcal{D}}(F(X),Y)\ar[r]^{\sim}&\mathrm... ...varphi^*}& \mathrm{Hom}_{\mathcal{D}}(X,G^{\vee}(Y))\ar[u]_{\varphi^{\vee}_*}}$$

Using the construction of $\varphi^{\vee}$ it follows that if a functor has a left/right adjoint then this adjoint is unique upto isomorphism.

The following gives useful criteria for showing exactness of functors.

Lemma 0.0.1   Let $\mathcal{C}$ be an abelian category, then a sequence $\xymatrix{ A \ar[r]^{\alpha}& B \ar[r]^{\beta}& C}$ is exact in $\mathcal{C}$, provided that for every $X\in\mathcal{C}$ the following sequence is exact:

$$\xymatrix{ \mathrm{Hom}_{\mathcal{C}}(X,A) \ar[r]^{\alpha_*} & \mathrm{Hom}_{\mathcal{C}}(X,B)\ar[r]^{\beta_*} & \mathrm{Hom}_{\mathcal{C}}(X,C)}.$$

Proof. Put $X=A$ to get that $\beta\circ\alpha=\beta_*\circ\alpha_*(\mathrm{id}_A)=0$, so $\mathrm{im}(\alpha)\subseteq \ker(\beta)$. Now put $X=\ker(\beta)$ and let $\iota:\ker(\beta)\to B$ be the inclusion map. Then $\beta_*(\iota)=\beta\circ\iota=0$, so there exists $\varphi \in \mathrm{Hom}_{\mathcal{C}}(\ker(\beta), A)$ such that $\alpha\circ\varphi=\alpha^*(\varphi)=\iota$. So $\ker(\beta)\subseteq \mathrm{im}(\alpha)$. $\qedsymbol$

Proposition 0.0.2   Let $\mathcal{C}, \mathcal{D}$ be abelian categories and let $F:\mathcal{C}\to\mathcal{D}$ be a functor left adjoint to $F^{\vee}:\mathcal{D}\to\mathcal{C}$. Then $F$ is a right exact functor and $F^{\vee}$ is a left exact functor.

Proof. Let $0\to A \to B \to C$ be exact in $\mathcal{C}$ and let $X\in\mathcal{C}$, then we have the following commutative diagram

$$\xymatrix{ 0 \ar[r] & \mathrm{Hom}(F(X), A)\ar[r]\ar[d]_{\sim} & ... ...)) \ar[r] & \mathrm{Hom}(X, F^{\vee}(B)) \ar[r] & \mathrm{Hom}(X,F^{\vee}(C))}$$

The top row is exact as the $\mathrm{Hom}$ functor is left exact, thus the bottom row is also exact. By 0.0.1, $0\to F^{\vee}(A) \to F^{\vee}(B) \to F^{\vee}(C)$ must be exact. This proves that every right adjoint is left exact. In particular $F^{op}: \mathcal{C}^{op}\to \mathcal{D}^{op}$ (which is a right adjoint) is left exact, i.e, $F$ is right exact. $\qedsymbol$