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hom-functor preserves limits in nLab

Proposition

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

Hom 𝒞:𝒞 op×𝒞⟶Set 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 •:ℐ⟶𝒞X_\bullet \colon \mathcal{I} \longrightarrow \mathcal{C} be a diagram. Then:

If the limit lim⟵ iX i\underset{\longleftarrow}{\lim}_i X_i exists in 𝒞\mathcal{C} then for all Y∈𝒞Y \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:ℐ⟶X𝒞⟶Hom 𝒞(Y,−)Set. Hom_{\mathcal{C}}(Y,-) \circ X \;\colon\; \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{Hom_{\mathcal{C}}(Y,-) }{\longrightarrow} Set\,.
If the colimit lim⟶ iX i\underset{\longrightarrow}{\lim}_i X_i exists in 𝒞\mathcal{C} then for all Y∈𝒞Y \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:ℐ op⟶X𝒞 op⟶Hom 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,j∈Obj(ℐ),α∈Hom ℐ(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 i∈ℐi \in \mathcal{I} a choice of morphism p i:Y→X ip_i \colon Y \to X_i , such that for each pair of objects i,j∈ℐi,j \in \mathcal{I} and each α∈Hom ℐ(i,j)\alpha \in Hom_{\mathcal{I}}(i,j) we have X α∘p 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).

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,lim⟵j∈𝒥Y(j)]≃lim⟵j∈𝒥[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

[lim⟶j∈𝒥Y(j),X]≃lim⟵j∈𝒥[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 X∈𝒞X \in \mathcal{C} any object, [X,−][X,-] is a right adjoint by definition, and hence preserves limits as adjoints preserve (co-)limits.

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

Hom 𝒞(C,[lim⟶j∈𝒥Y(j),X]) ≃Hom 𝒞(C⊗lim⟶j∈𝒥Y(j),X) ≃Hom 𝒞(lim⟶j∈𝒥(C⊗Y(j)),X) ≃lim⟵j∈𝒥Hom 𝒞((C⊗Y(j)),X) ≃lim⟵j∈𝒥Hom 𝒞(C,[Y(j),X]) ≃Hom 𝒞(C,lim⟵j∈𝒥[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 C∈𝒞C \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

[lim⟶j∈𝒥Y(j),X]⟶≃lim⟵j∈𝒥[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] \,.