Contents

Idea

Just as a functor is a morphism between categories, so a closed functor is a morphism between closed categories. Like monoidal functors, closed functors come in varying levels of strictness and strength.

Definition

A strict closed functor is a functor $F:C\to D$ between closed categories that preserves all the structure on the nose. In particular, it must preserve the internal homs and the unit object:

but it should also preserve all of the additional structure transformations. Strict closed functors are not common “in nature” (due to the problem of evil), but they are sometimes useful to consider in syntactic studies of closed categories, as for example in coherence problems.

More commonly occurring in nature are functors that preserve the closed structure merely up to a transformation, which may perhaps be invertible. Just as lax monoidal functors are often called simply “monoidal functors,” the “lax” sort of closed functor for which the transformation is not invertible are often called simply closed functors, the ones where it is invertible being called strong.

To be precise: a (lax) closed functor between closed categories $C$ and $D$ is a functor $F:C\to D$ together with:

• A transformation $\hat{F}:F([X,Y{]}_C)\to [F(X),F(Y){]}_D$, natural in $X$ and $Y$.

• A morphism $F^0:I_D\to F(I_C)$.

which satisfy the following axioms.

• The following diagram commutes for any $X$.

  $$\begin{array}{ccc}F(I_C)& \stackrel{F(j)}{\to }& F([X,X])\\ {}^{F^0}\uparrow & & {\downarrow }^{\hat{F}}\\ I_D& \underset{j}{\to }& [F(X),F(X)]\end{array}$$\array{F(I_C) & \overset{F(j)}{\to} & F([X,X])\\ ^{\mathllap{F^0}}\uparrow && \downarrow^{\mathrlap{\hat{F}}}\\ I_D & \underset{j}{\to} & [F(X),F(X)]}

• The following diagram commutes for any $X$.

  $$\begin{array}{ccc}F([I,X])& \stackrel{\hat{F}}{\to }& [F(I),F(X)]\\ {}^{F(j)}\downarrow & & {\downarrow }^{[F^0,1]}\\ F(X)& \underset{j}{\to }& [I,F(X)]\end{array}$$\array{F([I,X]) & \overset{\hat{F}}{\to} & [F(I),F(X)]\\ ^{\mathllap{F(j)}}\downarrow && \downarrow^{\mathrlap{[F^0 ,1]}}\\ F(X) & \underset{j}{\to} & [I,F(X)]}

• The following diagram commutes for any $X,Y,Z$.

  $$\begin{array}{ccccc}F([Y,Z])& \stackrel{F(L)}{\to }& F([[X,Y],[X,Z]])& \stackrel{\hat{F}}{\to }& [F([X,Y]),F([X,Z])]\\ {}^{\hat{F}}\downarrow & & & & {\downarrow }^{[1,\hat{F}]}\\ [F(Y),F(Z)]& \underset{L}{\to }& [[F(X),F(Y)],[F(X),F(Z)]]& \underset{[\hat{F},1]}{\to }& [F([X,Y]),[F(X),F(Z)]]\end{array}$$\array{F([Y,Z]) & \overset{F(L)}{\to} & F([[X,Y],[X,Z]]) & \overset{\hat{F}}{\to} & [F([X,Y]),F([X,Z])]\\ ^{\mathllap{\hat{F}}}\downarrow &&&& \downarrow^{\mathrlap{[1,\hat{F}]}}\\ [F(Y),F(Z)] & \underset{L}{\to} & [[F(X),F(Y)],[F(X),F(Z)]] & \underset{[\hat{F},1]}{\to} & [F([X,Y]),[F(X),F(Z)]]}

A strong closed functor is a closed functor such that $\hat{F}$ and $F^0$ are isomorphisms.

Together with closed natural transformations, closed categories and closed functors form a 2-category.

Examples

• Suppose that $C$ and $D$ are closed monoidal categories. Then any lax monoidal functor $F:C\to D$ gives rise to a lax closed functor by defining $F^0$ to be the unit constraint of $F$, and $\hat{F}:F([X,Y])\to [F(X),F(Y)]$ to be the adjunct under the internal-hom adjunction of the composite

  $$F([X,Y])\otimes F(X)\to F([X,Y]\otimes X)\to F(Y).$$F([X,Y]) \otimes F(X) \to F([X,Y] \otimes X) \to F(Y).

  Conversely, from a lax closed functor between closed monoidal categories we can recover a lax monoidal functor, with multiplication constraint $F(X)\otimes F(Y)\to F(X\otimes Y)$ being adjunct to the composite

  $$F(X)\to F([Y,X\otimes Y])\to [F(Y),F(X\otimes Y)]$$F(X) \to F([Y,X\otimes Y]) \to [F(Y), F(X\otimes Y)]

  where the map $X\to [Y,X\otimes Y]$ is adjunct to the identity of $X\otimes Y$. In this way, lax monoidal and lax closed functors between closed monoidal categories are in bijective correspondence.

  Note, however, that for such a functor to be strong monoidal or strong closed are generally independent conditions. Since any lax (or strong) monoidal functor is automatically a lax closed functor, the term closed monoidal functor is usually used to mean one which is strong closed.

• The same idea works more generally for closed unital multicategories, since arbitrary “multifunctors” between multicategories correspond to lax monoidal functors.

• However, there does not seem to be a natural notion of “colax” functor between closed categories. One could, of course, simply ask for transformations in the other direction, but such things do not seem to arise much in practice, and would not correspond to colax monoidal functors in the same way.

• If $C$ and $D$ are cartesian closed categories, then any functor $F:C\to D$ is automatically colax monoidal, and it is strong (hence also lax) monoidal iff it preserves products. By the above argument, any product-preserving functor between cartesian closed categories is automatically a lax closed functor. If it is moreover a strong closed functor, we call it a cartesian closed functor. (See also locally cartesian closed functor.)

References

• Samuel Eilenberg and Max Kelly, Closed categories. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965).