# nLab cartesian closed functor

category theory

## Applications

#### Monoidal categories

monoidal categories

# Contents

## Definition

###### Definition

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

More precisely, if $F\colon C\to D$ preserves products, then the canonical morphisms $F(A\times B) \to F A \times F B$ (for all objects $A,B \mathcal{C}$) are isomorphisms, and we therefore have canonical induced morphism $F[A,B] \to [F A, F B]$ β the adjuncts of the composite $F[A,B] \times F A \xrightarrow{\cong} F([A,B] \times A) \to F B$. $F$ is cartesian closed if these maps $F[A,B] \to [F A, F B]$ 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 $R : \mathcal{C} \to \mathcal{D}$ be a functor between cartesian closed categories with a left adjoint $L$. Then $R$ is cartesian closed precisely if the natural transformation

$(L \pi_1, \epsilon_A L \pi_2) : L(B \times R(A)) \to L(B) \times A$

is an isomorphism.

###### Proof

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

$\mathcal{C} \underoverset{[R A, -]}{- \times R A}{\rightleftarrows} \mathcal{C} \underoverset{R}{L}{\rightleftarrows} \mathcal{D}$

and

$\mathcal{C} \underoverset{R}{L}{\rightleftarrows} \mathcal{D} \underoverset{[A,-]}{A\times -}{\rightleftarrows} \mathcal{D}$

This is called the Frobenius reciprocity law. It is discussed, for instance, as (Johnstone, lemma 1.5.8). More generally for closed monoidal categories (not necessarily cartesian monoidal) it is discussed in βWirthmΓΌller contextsβ in

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 $\mathcal{C}$ a locally cartesian closed category and $f : X_1 \to X_2$ a morphism, the base change/pullback functor between the slice categories

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

is cartesian closed.

In particular the inverse image functor of an Γ©tale geometric morphism between toposes is cartesian closed and hence constitutes a cartesian WirthmΓΌller context.

###### Proof

The functor $f^*$ has a left adjoint

$\sum_f : \mathcal{C}_{/X_1} \to \mathcal{C}_{/X_2}$

given by postcomposition with $f$ (the dependent sum along $f$). Therefore by prop. 1 it is sufficient to show that for all $(A \to X_2)$ in $\mathcal{C}_{/X_2}$ and $(B \stackrel{b}{\to} X_1) \in \mathcal{C}_{/X_1}$ that

$B \times_{X_1} f^* A \simeq B \times_{X_2} A$

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

$\array{ B \times_{X_1} f^* A &\to& f^* A &\to& A \\ \downarrow && \downarrow && \downarrow \\ B &\stackrel{b}{\to}& X_1 &\stackrel{f}{\to}& X_2 } \;\;\; \simeq \;\;\; \array{ (b \circ f)^* A &\to& &\to& A \\ \downarrow && && \downarrow \\ B &\stackrel{b}{\to}& X_1 &\stackrel{f}{\to}& X_2 }$

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

## References

For instance section A1.5 of

Also

• H. Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (TAC, pdf)

Revised on December 9, 2013 03:32:37 by Urs Schreiber (82.113.121.146)