nLab cartesian closed functor

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 $F\colon \mathcal{C}\to \mathcal{D}$ preserves products, then the canonical morphisms $F(A\times B) \to F A \times F B$ (for all objects $A,B \in \mathcal{C}$) are isomorphisms, and we therefore have canonical induced morphisms $F[A,B] \to [F A, F B]$ — the adjuncts of the composites $F[A,B] \times F A \xrightarrow{\cong} F([A,B] \times A) \to F B$. $F$ 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

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

and

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

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 “Wirthmüller contexts” 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 étale geometric morphism between toposes is cartesian closed and hence constitutes a cartesian Wirthmüller 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. 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.

Related concepts

• strong functor

• cartesian closed category, locally cartesian closed category

• cartesian closed functor, locally cartesian closed functor

• cartesian closed model category, locally cartesian closed model category

• cartesian closed (∞,1)-category, locally cartesian closed (∞,1)-category

References

For instance section A1.5 of

• Peter Johnstone, Sketches of an Elephant

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)