nLab cartesian closed functor



Category theory

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monoidal categories

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In higher category theory




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:π’žβ†’π’ŸF\colon \mathcal{C}\to \mathcal{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,Bβˆˆπ’žA,B \in \mathcal{C}) are isomorphisms, and we therefore have canonical induced morphisms F[A,B]β†’[FA,FB]F[A,B] \to [F A, F B] β€” the adjuncts of the composites 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.


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.



(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.


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‡†RLπ’Ÿβ‡†[RA,βˆ’]βˆ’Γ—RAπ’Ÿ \mathcal{C} \underoverset{R}{L}{\leftrightarrows} \mathcal{D} \underoverset{[R A, -]}{- \times R A}{\leftrightarrows} \mathcal{D}


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

Let still RR and LL be as above.


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

For instance (Johnstone, corollary A1.5.9).



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.


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. it is sufficient to show that for all (Aβ†’X 2)βˆˆπ’ž /X 2(A \to X_2) \in \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.


For instance section A1.5 of


  • 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)

Last revised on February 1, 2024 at 17:18:47. See the history of this page for a list of all contributions to it.