nLab
cartesian closed functor

Context

Category theory

Monoidal categories

Contents

Definition

Definition

A cartesian closed functor is a functor F:π’žβ†’π’ŸF\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:Cβ†’DF\colon C\to D preserves products, then the canonical morphisms F(AΓ—B)β†’FAΓ—FBF(A\times B) \to F A \times F B (for all objects A,Bπ’žA,B \mathcal{C}) are isomorphisms, and we therefore have canonical induced morphism F[A,B]β†’[FA,FB]F[A,B] \to [F A, F B] β€” the adjuncts of the composite F[A,B]Γ—FAβ†’β‰…F([A,B]Γ—A)β†’FBF[A,B] \times F A \xrightarrow{\cong} F([A,B] \times A) \to F B. FF is cartesian closed if these maps F[A,B]β†’[FA,FB]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:π’žβ†’π’ŸR : \mathcal{C} \to \mathcal{D} be a functor between cartesian closed categories with a left adjoint LL. Then RR is cartesian closed precisely if the natural transformation

(LΟ€ 1,Ο΅ ALΟ€ 2):L(BΓ—R(A))β†’L(B)Γ—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[A,B]β†’[RA,RB]R[A,B] \to [R A, R B] under the composite adjunctions

π’žβ‡„[RA,βˆ’]βˆ’Γ—RAπ’žβ‡„RLπ’Ÿ \mathcal{C} \underoverset{[R A, -]}{- \times R A}{\rightleftarrows} \mathcal{C} \underoverset{R}{L}{\rightleftarrows} \mathcal{D}

and

π’žβ‡„RLπ’Ÿβ‡„[A,βˆ’]AΓ—βˆ’π’Ÿ \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€œWirthmΓΌller contexts” in

Let still RR and LL be as above.

Corollary

If RR is full and faithful and LL preserves binary products, then RR is cartesian closed.

For instance (Johnstone, corollary A1.5.9).

Examples

Proposition

For π’ž\mathcal{C} a locally cartesian closed category and f:X 1β†’X 2f : X_1 \to X_2 a morphism, the base change/pullback functor between the slice categories

f *:π’ž /X 2β†’π’ž /X 1 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 *f^* has a left adjoint

βˆ‘ f:π’ž /X 1β†’π’ž /X 2 \sum_f : \mathcal{C}_{/X_1} \to \mathcal{C}_{/X_2}

given by postcomposition with ff (the dependent sum along ff). Therefore by prop. 1 it is sufficient to show that for all (Aβ†’X 2)(A \to X_2) in π’ž /X 2\mathcal{C}_{/X_2} and (Bβ†’bX 1)βˆˆπ’ž /X 1(B \stackrel{b}{\to} X_1) \in \mathcal{C}_{/X_1} that

BΓ— X 1f *A≃BΓ— X 2A B \times_{X_1} f^* A \simeq B \times_{X_2} A

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

BΓ— X 1f *A β†’ f *A β†’ A ↓ ↓ ↓ B β†’b X 1 β†’f X 2≃(b∘f) *A β†’ β†’ A ↓ ↓ B β†’b X 1 β†’f X 2 \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)