The category of presheaves are cartesien closed

We will prove the title by using coend calculus, as hinted in Coend Calculus by Fosco Loregian.

For consistence let’s first formulate what to prove:

Theorem. For any (small) category 𝐂\mathbf{C}, its presheaf category 𝐂̂:=[𝐂op,Set]\hat{\mathbf{C}} := [\mathbf{C}^{\mathrm{op}}, {\mathrm{Set}}] is cartesien closed, i.e., for any presheaves P,Q,Rβˆˆπ‚Μ‚P, Q, R \in \hat{\mathbf{C}}, there exist a exponential object RQR^Q whi is right adjoint to product:

𝐂̂(PΓ—Q,R)≅𝐂̂(P,RQ);\hat{\mathbf{C}}(P \times Q, R) \cong \hat{\mathbf{C}}(P, R^Q);

proof: The exponential object PQP^Q is defined as PQ(c):=𝐂̂(𝐲cΓ—Q,R)P^Q(c) := \hat{\mathbf{C}}(\mathbf{y}_c \times Q, R).

The theorems/propositions in the comments after each equation are all in the book Coend Calculus, and 𝐲()\mathbf{y}_{()} is the standard yoneda embedding.

𝐂̂(P,𝐂̂(𝐲()Γ—Q,R))β‰…βˆ«cSet(Pc,𝐂̂(𝐲cΓ—Q,R))(By Thm 1.4.1)β‰…βˆ«cSet(Pc,∫xSet(𝐂(x,c)Γ—Qx,Rx))(By Thm 1.4.1)β‰…βˆ«c∫xSet(Pc,Set(𝐂(x,c)Γ—Qx,Rx))(Representables preserve limits)β‰…βˆ«x∫cSet(Pc,Set(𝐂(x,c)Γ—Qx,Rx))(By Thm 1.3.1)β‰…βˆ«x∫cSet(Pc×𝐂(x,c),Set(Qx,Rx))(Set is cartesien closed)β‰…βˆ«xSet(∫cPc×𝐂(x,c),Set(Qx,Rx))(Representables turn colimts into limits)β‰…βˆ«xSet(Px,Set(Qx,Rx))(By Prop 2.2.1)β‰…βˆ«xSet(PxΓ—Qx,Rx)(Set is cartesien closed)≅𝐂̂(PΓ—Q,R)(By Thm 1.4.1)\begin{align} \hat{\mathbf{C}}(P, \hat{\mathbf{C}}(\mathbf{y}_ {()} \times Q, R)) & \cong \int_c {\mathrm{Set}}(Pc, \hat{\mathbf{C}}(\mathbf{y}_c \times Q, R)) &(\text{By Thm 1.4.1}) \\ & \cong \int_c {\mathrm{Set}} (Pc, \int_x {\mathrm{Set}}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{By Thm 1.4.1}) \\ & \cong \int_c\int_x {\mathrm{Set}} (Pc, {\mathrm{Set}}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{Representables preserve limits}) \\ & \cong \int_x\int_c {\mathrm{Set}} (Pc, {\mathrm{Set}}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{By Thm 1.3.1}) \\ & \cong \int_x\int_c {\mathrm{Set}} (Pc \times \mathbf{C}(x, c), {\mathrm{Set}} (Qx, Rx)) &(\text{Set is cartesien closed}) \\ & \cong \int_x {\mathrm{Set}} (\int^c Pc \times \mathbf{C}(x, c), {\mathrm{Set}} (Qx, Rx)) &(\text{Representables turn colimts into limits}) \\ & \cong \int_x {\mathrm{Set}} (Px, {\mathrm{Set}} (Qx, Rx)) &(\text{By Prop 2.2.1}) \\ & \cong \int_x {\mathrm{Set}} (Px \times Qx, Rx) &(\text{Set is cartesien closed}) \\ & \cong \hat{\mathbf{C}}(P \times Q, R) &(\text{By Thm 1.4.1}) \\ \end{align}

QED.


Source of TeX:

\begin{align}
\hat{\mathbf{C}}(P, \hat{\mathbf{C}}(\mathbf{y}_ {()} \times Q, R)) & \cong \int_c {\rm Set}(Pc, \hat{\mathbf{C}}(\mathbf{y}_c \times Q, R))  &(\text{By Thm 1.4.1}) \\
     & \cong \int_c {\rm Set} (Pc, \int_x {\rm Set}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{By Thm 1.4.1}) \\
     & \cong \int_c\int_x {\rm Set} (Pc,  {\rm Set}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{Representables preserve limits}) \\
     & \cong \int_x\int_c {\rm Set} (Pc,  {\rm Set}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{By Thm 1.3.1}) \\
     & \cong \int_x\int_c {\rm Set} (Pc \times \mathbf{C}(x, c), {\rm Set} (Qx, Rx)) &(\text{Set is cartesien closed}) \\
     & \cong \int_x {\rm Set} (\int^c Pc \times \mathbf{C}(x, c), {\rm Set} (Qx, Rx)) &(\text{Representables turn colimts into limits}) \\
     & \cong \int_x {\rm Set} (Px, {\rm Set} (Qx, Rx)) &(\text{By Prop 2.2.1}) \\
     & \cong \int_x {\rm Set} (Px \times Qx, Rx) &(\text{Set is cartesien closed}) \\
     & \cong \hat{\mathbf{C}}(P \times Q, R) &(\text{By Thm 1.4.1}) \\
        
\end{align}