Closed monoidal category

A closed monoidal category is a monoidal category ${\displaystyle {\mathcal {C}}}$ such that for every object ${\displaystyle B}$ the functor given by right tensoring with ${\displaystyle B}$

${\displaystyle A\mapsto A\otimes B}$

has a right adjoint, written

${\displaystyle A\mapsto (B\Rightarrow A).}$

This means that there exists a bijection, called 'currying', between the Hom-sets

${\displaystyle {\text{Hom}}_{\mathcal {C}}(A\otimes B,C)\cong {\text{Hom}}_{\mathcal {C}}(A,B\Rightarrow C)}$

that is natural in both A and C. In a different, but common notation, one would say that the functor

${\displaystyle -\otimes B:{\mathcal {C}}\to {\mathcal {C}}}$

has a right adjoint

${\displaystyle [B,-]:{\mathcal {C}}\to {\mathcal {C}}}$

Equivalently, a closed monoidal category ${\displaystyle {\mathcal {C}}}$ is a category equipped, for every two objects A and B, with

• an object ${\displaystyle A\Rightarrow B}$,
• a morphism ${\displaystyle \mathrm {eval} _{A,B}:(A\Rightarrow B)\otimes A\to B}$,

satisfying the following universal property: for every morphism

${\displaystyle f:X\otimes A\to B}$

there exists a unique morphism

${\displaystyle h:X\to A\Rightarrow B}$

such that

${\displaystyle f=\mathrm {eval} _{A,B}\circ (h\otimes \mathrm {id} _{A}).}$

It can be shown that this construction defines a functor ${\displaystyle \Rightarrow$ :{\mathcal {C}}^{op}\times {\mathcal {C}}\to {\mathcal {C}}} . This functor is called the internal Hom functor, and the object ${\displaystyle A\Rightarrow B}$ is called the internal Hom of ${\displaystyle A}$ and ${\displaystyle B}$. Many other notations are in common use for the internal Hom. When the tensor product on ${\displaystyle {\mathcal {C}}}$ is the cartesian product, the usual notation is ${\displaystyle B^{A}}$ and this object is called the exponential object.

Strictly speaking, we have defined a right closed monoidal category, since we required that right tensoring with any object ${\displaystyle A}$ has a right adjoint. In a left closed monoidal category, we instead demand that the functor of left tensoring with any object ${\displaystyle A}$

${\displaystyle B\mapsto A\otimes B}$

have a right adjoint

${\displaystyle B\mapsto (B\Leftarrow A)}$

A biclosed monoidal category is a monoidal category that is both left and right closed.

A symmetric monoidal category is left closed if and only if it is right closed. Thus we may safely speak of a 'symmetric monoidal closed category' without specifying whether it is left or right closed. In fact, the same is true more generally for braided monoidal categories: since the braiding makes ${\displaystyle A\otimes B}$ naturally isomorphic to ${\displaystyle B\otimes A}$, the distinction between tensoring on the left and tensoring on the right becomes immaterial, so every right closed braided monoidal category becomes left closed in a canonical way, and vice versa.

We have described closed monoidal categories as monoidal categories with an extra property. One can equivalently define a closed monoidal category to be a closed category with an extra property. Namely, we can demand the existence of a tensor product that is left adjoint to the internal Hom functor. In this approach, closed monoidal categories are also called monoidal closed categories.

• Every cartesian closed category is a symmetric, monoidal closed category, when the monoidal structure is the cartesian product structure. The internal Hom functor is given by the exponential object ${\displaystyle B^{A}}$.
  • In particular, the category of sets, Set, is a symmetric, closed monoidal category. Here the internal Hom ${\displaystyle A\Rightarrow B}$ is just the set of functions from ${\displaystyle A}$ to ${\displaystyle B}$.
• The category of modules, R-Mod over a commutative ring R is a non-cartesian, symmetric, monoidal closed category. The monoidal product is given by the tensor product of modules and the internal Hom ${\displaystyle M\Rightarrow N}$ is given by the space of R-linear maps ${\displaystyle \operatorname {Hom} _{R}(M,N)}$ with its natural R-module structure.
  • In particular, the category of vector spaces over a field ${\displaystyle K}$ is a symmetric, closed monoidal category.
  • Abelian groups can be regarded as Z-modules, so the category of abelian groups is also a symmetric, closed monoidal category.
• A compact closed category is a symmetric, monoidal closed category, in which the internal Hom functor ${\displaystyle A\Rightarrow B}$ is given by ${\displaystyle A^{*}\otimes B}$. The canonical example is the category of finite-dimensional vector spaces, FdVect.

• Isbell conjugacy

• Kelly, G.M. "Basic Concepts of Enriched Category Theory", London Mathematical Society Lecture Note Series No.64 (C.U.P., 1982)
• Paul-André Melliès, "Categorical Semantics of Linear Logic", Panoramas et Synthèses 27, Société Mathématique de France, 2009
• Closed monoidal category in nLab