With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Just as a functor is a homomorphism between categories, so a closed functor is a homomorphism between closed categories. Like monoidal functors, closed functors come in varying levels of strictness and strength.
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” (since their definition violates the principle of equivalence), 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 $F\colon C\to D$ together with:
A transformation $\hat{F}\colon F([X,Y]_C) \to [F(X),F(Y)]_D$, natural in $X$ and $Y$.
A morphism $F^0\colon I_D \to F(I_C)$.
which satisfy the following axioms.
The following diagram commutes for any $X$.
The following diagram commutes for any $X$.
The following diagram commutes for any $X,Y,Z$.
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.
Suppose that $C$ and $D$ are closed monoidal categories. Then any lax monoidal functor $F\colon C\to D$ gives rise to a lax closed functor by defining $F^0$ to be the unit constraint of $F$, and $\hat{F}\colon F([X,Y]) \to [F(X),F(Y)]$ to be the adjunct under the internal-hom adjunction of the composite
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
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 (or at least for which $\hat{F}$ is an isomorphism).
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 $F\colon C\to D$ 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.)
The inverse image base change in an etale geometric morphism between toposes is a cartesian closed functor (see there).
An adjoint triple of functors between symmetric closed monoidal categories is a “Wirthmüller context of six operations” when the middle functor is closed monoidal.
A class of examples of strong closed functors that are not strong monoidal comes from the inclusions of reflective exponential ideals. Day's reflection theorem implies that if $C$ is a closed monoidal category and $E\subseteq C$ is a reflective subcategory that is an exponential ideal in the monoidal sense (i.e. $x\in C$ and $y\in E$ imply $[x,y]\in E$), then $E$ is a closed monoidal category with the induced internal-hom and a reflected tensor product $x\otimes_E y = R(x\otimes_C y)$ where $R$ is the reflector. Then the inclusion functor $E\hookrightarrow C$ preserves internal-homs but is only lax monoidal (in contrast to its left adjoint, the reflection, which is strong monoidal). It doesn’t in general preserve the unit strongly, but there are cases when it does.
For instance, let $C = P A$ be a presheaf category with a Day convolution monoidal structure induced by a monoidal structure on a small category $A$, and let $E$ be the subcategory of $\Phi$-continuous presheaves for some set $\Phi$ of colimits in $A$ that are preserved on both sides by the tensor product (e.g. if $A$ is itself closed). Then $E$ is a reflective exponential ideal, and contains the unit object since the latter is a representable presheaf (at the unit object of $A$) hence preserves all colimits.
Samuel Eilenberg, G. Max Kelly, §I.3 of: Closed Categories, in: Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]
Geoffrey Lewis, “Coherence for a Closed Functor” UNSW Ph.D. thesis 1974
Last revised on August 23, 2023 at 08:40:10. See the history of this page for a list of all contributions to it.