# nLab cartesian closed functor

category theory

## Applications

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Definition

###### Definition

A cartesian closed functor is a functor $F:𝒞\to 𝒟$ between cartesian closed categories which preserves both products and exponential objects/internal homs (all the structure of cartesian closed categories).

More precisely, if $F:C\to D$ preserves products, then the canonical morphisms $F\left(A×B\right)\to FA×FB$ (for all objects $A,B𝒞$) are isomorphisms, and we therefore have canonical induced morphism $F\left[A,B\right]\to \left[FA,FB\right]$ — the adjuncts of the composite $F\left[A,B\right]×FA\stackrel{\cong }{\to }F\left(\left[A,B\right]×A\right)\to FB$. $F$ is cartesian closed if these maps $F\left[A,B\right]\to \left[FA,FB\right]$ are also isomorphisms.

###### Remark

When cartesian closed categories are identified with cartesian monoidal categories that are also closed monoidal, a cartesian closed functor can be identified with a strong monoidal functor which is also strong closed.

## Properties

###### Proposition

(Frobenius reciprocity)

Let $R:𝒞\to 𝒟$ be a functor between cartesian closed categories with a left adjoint $L$. Then $R$ is cartesian closed precisely if the natural transformation

$\left(L{\pi }_{1},{ϵ}_{A}L{\pi }_{2}\right):L\left(B×R\left(A\right)\right)\to L\left(B\right)×A$(L \pi_1, \epsilon_A L \pi_2) : L(B \times R(A)) \to L(B) \times A

is an isomorphism.

###### Proof

The above natural transformation is the mate of the exponential comparison natural transformation $R\left[A,B\right]\to \left[RA,RB\right]$ under the composite adjunctions

$𝒞\underset{\left[RA,-\right]}{\overset{-×RA}{⇄}}𝒞\underset{R}{\overset{L}{⇄}}𝒟$\mathcal{C} \underoverset{[R A, -]}{- \times R A}{\rightleftarrows} \mathcal{C} \underoverset{R}{L}{\rightleftarrows} \mathcal{D}

and

$𝒞\underset{R}{\overset{L}{⇄}}𝒟\underset{\left[A,-\right]}{\overset{A×-}{⇄}}𝒟$\mathcal{C} \underoverset{R}{L}{\rightleftarrows} \mathcal{D} \underoverset{[A,-]}{A\times -}{\rightleftarrows} \mathcal{D}

This is called the Frobenius reciprocity law. It is discussed, for instance, as (Johnstone, lemma 1.5.8).

Let still $R$ and $L$ be as above.

###### Corollary

If $R$ is full and faithful and $L$ preserves binary products, then $R$ is cartesian closed.

For instance (Johnstone, corollary A1.5.9).

## Examples

###### Proposition

For $𝒞$ a locally cartesian closed category and $f:{X}_{1}\to {X}_{2}$ a morphism, the base change/pullback functor between the slice categories

${f}^{*}:{𝒞}_{/{X}_{2}}\to {𝒞}_{/{X}_{1}}$f^* : \mathcal{C}_{/X_2} \to \mathcal{C}_{/X_1}

is cartesian closed.

In particular the inverse image functor of an étale geometric morphism between toposes is cartesian closed.

###### Proof

The functor ${f}^{*}$ has a left adjoint

$\sum _{f}:{𝒞}_{/{X}_{1}}\to {𝒞}_{/{X}_{2}}$\sum_f : \mathcal{C}_{/X_1} \to \mathcal{C}_{/X_2}

given by postcomposition with $f$ (the dependent sum along $f$). Therefore by prop. 1 it is sufficient to show that for all $\left(A\to {X}_{2}\right)$ in ${𝒞}_{/{X}_{2}}$ and $\left(B\stackrel{b}{\to }{X}_{1}\right)\in {𝒞}_{/{X}_{1}}$ that

$B{×}_{{X}_{1}}{f}^{*}A\simeq B{×}_{{X}_{2}}A$B \times_{X_1} f^* A \simeq B \times_{X_2} A

in $𝒞$. But this is the pasting law for pullbacks in $𝒞$, which says that the two consecutive pullbacks on the left of

$\begin{array}{ccccc}B{×}_{{X}_{1}}{f}^{*}A& \to & {f}^{*}A& \to & A\\ ↓& & ↓& & ↓\\ B& \stackrel{b}{\to }& {X}_{1}& \stackrel{f}{\to }& {X}_{2}\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\simeq \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccccc}\left(b\circ f{\right)}^{*}A& \to & & \to & A\\ ↓& & & & ↓\\ B& \stackrel{b}{\to }& {X}_{1}& \stackrel{f}{\to }& {X}_{2}\end{array}$\array{ B \times_{X_1} f^* A &\to& f^* A &\to& A \\ \downarrow && \downarrow && \downarrow \\ B &\stackrel{b}{\to}& X_1 &\stackrel{f}{\to}& X_2 } \;\;\; \simeq \;\;\; \array{ (b \circ f)^* A &\to& &\to& A \\ \downarrow && && \downarrow \\ B &\stackrel{b}{\to}& X_1 &\stackrel{f}{\to}& X_2 }

are isomorphic to the direct pullback along the composite on the right.

## References

For instance section A1.5 of

Revised on November 14, 2012 02:15:55 by Urs Schreiber (82.169.65.155)