nLab
hom-functor preserves limits

Contents

Context

Limits and colimits

Category theory

Contents

Idea

One of the basic facts of category theory is that the hom-functor on a category 𝒞\mathcal{C} preserve limits in both variables (remembering that a limit in the first variable, due to contravariance, is actually a colimit in 𝒞\mathcal{C}).

Statement

Ordinary hom-functor

Proposition

(hom-functor preserves limits)

Let 𝒞\mathcal{C} be a category and write

Hom 𝒞:𝒞 op×𝒞Set Hom_{\mathcal{C}} \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Set

for its hom-functor. This preserves limits in both its arguments (recalling that a limit in the opposite category 𝒞 op\mathcal{C}^{op} is a colimit in 𝒞\mathcal{C}).

More in detail, let X :𝒞X_\bullet \colon \mathcal{I} \longrightarrow \mathcal{C} be a diagram. Then:

  1. If the limit lim iX i\underset{\longleftarrow}{\lim}_i X_i exists in 𝒞\mathcal{C} then for all Y𝒞Y \in \mathcal{C} there is a natural isomorphism

    Hom 𝒞(Y,lim iX i)lim i(Hom 𝒞(Y,X i)), Hom_{\mathcal{C}}\left(Y, \underset{\longleftarrow}{\lim}_i X_i \right) \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( Y, X_i \right) \right) \,,

    where on the right we have the limit over the diagram of hom-sets given by

    Hom 𝒞(Y,)X:X𝒞Hom 𝒞(Y,)Set. Hom_{\mathcal{C}}(Y,-) \circ X \;\colon\; \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{Hom_{\mathcal{C}}(Y,-) }{\longrightarrow} Set\,.
  2. If the colimit lim iX i\underset{\longrightarrow}{\lim}_i X_i exists in 𝒞\mathcal{C} then for all Y𝒞Y \in \mathcal{C} there is a natural isomorphism

    Hom 𝒞(lim iX i,Y)lim i(Hom 𝒞(X i,Y)), Hom_{\mathcal{C}}\left(\underset{\longrightarrow}{\lim}_i X_i ,Y\right) \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( X_i , Y\right) \right) \,,

    where on the right we have the limit over the diagram of hom-sets given by

    Hom 𝒞(,Y)X: opX𝒞 opHom 𝒞(,Y)Set. Hom_{\mathcal{C}}(-,Y) \circ X \;\colon\; \mathcal{I}^{op} \overset{X}{\longrightarrow} \mathcal{C}^{op} \overset{Hom_{\mathcal{C}}(-,Y) }{\longrightarrow} Set\,.
Proof

We give the proof of the first statement. The proof of the second statement is formally dual.

First observe that, by the very definition of limiting cones, maps out of some YY into them are in natural bijection with the set Cones(Y,X )Cones\left(Y, X_\bullet \right) of cones over the diagram X X_\bullet with tip YY:

Hom(Y,lim iX i)Cones(Y,X ). Hom\left( Y, \underset{\longleftarrow}{\lim}_{i} X_i \right) \;\simeq\; Cones\left( Y, X_\bullet \right) \,.

Hence it remains to show that there is also a natural bijection like so:

Cones(Y,X )lim i(Hom(Y,X i)). Cones\left( Y, X_\bullet \right) \;\simeq\; \underset{\longleftarrow}{\lim}_{i} \left( Hom(Y,X_i) \right) \,.

Now, again by the very definition of limiting cones, a single element in the limit on the right is equivalently a cone of the form

{ * const p i const p j Hom(Y,X i) X α() Hom(Y,X j)} i,jObj(),αHom (i,j). \left\{ \array{ && \ast \\ & {}^{\mathllap{const_{p_i}}}\swarrow && \searrow^{\mathrlap{const_{p_j}}} \\ Hom(Y,X_i) && \underset{X_\alpha \circ (-)}{\longrightarrow} && Hom(Y,X_j) } \right\}_{i, j \in Obj(\mathcal{I}), \alpha \in Hom_{\mathcal{I}}(i,j) } \,.

This is equivalently for each object ii \in \mathcal{I} a choice of morphism p i:YX ip_i \colon Y \to X_i , such that for each pair of objects i,ji,j \in \mathcal{I} and each αHom (i,j)\alpha \in Hom_{\mathcal{I}}(i,j) we have X αp i=p jX_\alpha \circ p_i = p_j. And indeed, this is precisely the characterization of an element in the set Cones(Y,X }) Cones\left( Y, X_\bullet\} \right).

Internal hom-functor

Proposition

(internal hom-functor preserves limits)

Let 𝒞\mathcal{C} be a symmetric closed monoidal category with internal hom-bifunctor [,][-,-]. Then this bifunctor preserves limits in the second variable, and sends colimits in the first variable to limits:

[X,limj𝒥Y(j)]limj𝒥[X,Y(j)] [X, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Y(j)] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [X, Y(j)]

and

[limj𝒥Y(j),X]limj𝒥[Y(j),X] [\underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j),X] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j),X]
Proof

For X𝒞X \in \mathcal{C} any object, [X,][X,-] is a right adjoint by definition, and hence preserves limits by adjoints preserve (co-)limits.

For the other case, let Y:𝒞Y \;\colon\; \mathcal{L} \to \mathcal{C} be a diagram in 𝒞\mathcal{C}, and let C𝒞C \in \mathcal{C} be any object. Then there are isomorphisms

Hom 𝒞(C,[limj𝒥Y(j),X]) Hom 𝒞(Climj𝒥Y(j),X) Hom 𝒞(limj𝒥(CY(j)),X) limj𝒥Hom 𝒞((CY(j)),X) limj𝒥Hom 𝒞(C,[Y(j),X]) Hom 𝒞(C,limj𝒥[Y(j),X]) \begin{aligned} Hom_{\mathcal{C}}(C, [ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ] ) & \simeq Hom_{\mathcal{C}}( C \otimes \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ) \\ & \simeq Hom_{\mathcal{C}}( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( C, [Y(j), X] ) \\ & \simeq Hom_{\mathcal{C}}( C, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] ) \end{aligned}

which are natural in C𝒞C \in \mathcal{C}, where we used that the ordinary hom-functor respects (co)limits as shown (see at hom-functor preserves limits), and that the left adjoint C()C \otimes (-) preserves colimits (see at adjoints preserve (co-)limits).

Hence by the fully faithfulness of the Yoneda embedding, there is an isomorphism

[limj𝒥Y(j),X]limj𝒥[Y(j),X]. \left[ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X \right] \overset{\simeq}{\longrightarrow} \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] \,.

Last revised on August 1, 2018 at 08:27:24. See the history of this page for a list of all contributions to it.