nLab
adjoints preserve (co-)limits

Contents

Context

Category theory

Limits and colimits

Contents

Idea

One of the basic facts of category theory is that left/right adjoint functors preserves co/limits, respectively.

Statement

Proposition

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be two categories and let

(LR):𝒞RL𝒟 (L \dashv R) \;\colon\; \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}

be a pair of adjoint functors between them.

Then

  1. If X:𝒞X \colon \mathcal{I} \to \mathcal{C} is a diagram whose limit lim iX i\underset{\longleftarrow}{\lim}_{i} X_i exists in 𝒞\mathcal{C}, then this limit is preserved by the right adjoint RR in that there is a natural isomorphism

    R(lim i(X i))lim i(R(X i)), R \left( \underset{\longleftarrow}{\lim}_i \left(X_i\right) \right) \;\simeq\; \underset{\longleftarrow}{\lim}_i \left( R(X_i) \right) \,,

    where on the right we have the limit in 𝒟\mathcal{D} over the diagram RX:X𝒞R𝒟R \circ X \colon \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{R}{\longrightarrow} \mathcal{D}.

  2. If X:𝒟X \colon \mathcal{I} \to \mathcal{D} is a diagram whose colimit lim iX i\underset{\longrightarrow}{\lim}_{i} X_i exists in 𝒟\mathcal{D}, then this colimit is preserved by the left adjoint ll in that there is a natural isomorphism

    L(lim i(X i))lim i(L(X i)), L \left( \underset{\longrightarrow}{\lim}_i \left(X_i\right) \right) \;\simeq\; \underset{\longrightarrow}{\lim}_i \left( L(X_i) \right) \,,

    where on the right we have the colimit in 𝒞\mathcal{C} over the diagram LX:X𝒟L𝒞L \circ X \colon \mathcal{I} \overset{X}{\longrightarrow} \mathcal{D} \overset{L}{\longrightarrow} \mathcal{C}.

Proof

We show the first statement, the proof of the second is formally dual.

We use the following facts

  1. There is a natural isomorphism, Hom 𝒞(L(d),c)Hom 𝒟(d,R(c))Hom_{\mathcal{C}}(L(d),c) \simeq Hom_{\mathcal{D}}(d,R(c)); this equivalently characterizes the fact that (LR)(L \dashv R) is a pair of adjoint functors;

  2. (hom-functor preserves limits) The hom-functor sends colimits in the first argument and limits in the second argument to limits of hom-sets

    Hom(X,lim iX i)lim iHom(X,X i) Hom\left( X, \underset{\longleftarrow}{\lim}_i X_i \right) \simeq \underset{\longleftarrow}{\lim}_i Hom\left(X,X_i\right)

    and

    Hom(lim iX i,X)lim(Hom(X i,X)). Hom\left(\underset{\longrightarrow}{\lim}_i X_i, X\right) \simeq \underset{\longleftarrow}{\lim} \left(Hom\left(X_i,X\right) \right) \,.

    Again, this is essentially by definition of limits/colimits.

  3. (Yoneda lemma) If for two objects XX and YY in some category the hom-sets out of or into these objects (their representable functors) are naturally isomorphic, then the two objects are isomorphic.

Now using the first two items, we obtain the following chain of natural isomorphisms, for every object Y𝒟Y \in \mathcal{D}:

Hom 𝒟(Y,R(lim iX i)) Hom 𝒞(L(Y),lim iX i) lim i(Hom 𝒞(L(Y),X i)) lim i(Hom 𝒟(Y,R(X i))) Hom 𝒟(Y,lim i(R(X i))). \begin{aligned} Hom_{\mathcal{D}}\left( Y, R \left( \underset{\longleftarrow}{\lim}_i X_i \right) \right) & \simeq Hom_{\mathcal{C}}\left( L(Y), \underset{\longleftarrow}{\lim}_i X_i\right) \\ & \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left(L\left(Y\right), X_i\right)\right) \\ & \simeq \underset{\longleftarrow}{\lim}_i \left(Hom_{\mathcal{D}}\left(Y, R\left(X_i\right)\right)\right) \\ & \simeq Hom_{\mathcal{D}}\left(Y, \underset{\longleftarrow}{\lim}_i \left(R\left(X_i\right) \right) \right) \end{aligned} \,.

Hence the third item above, the Yoneda lemma, implies the claim.

Last revised on August 1, 2018 at 07:27:39. See the history of this page for a list of all contributions to it.