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^{[citation needed]} 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.^{[citation needed]}

• 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 symmetric 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.

Counterexamples edit

• The category of rings is a symmetric, monoidal category under the tensor product of rings, with ${\displaystyle \mathbb {Z} }$  serving as the unit object. This category is not closed. If it were, there would be exactly one homomorphism between any pair of rings: ${\displaystyle \operatorname {Hom} (R,S)\cong \operatorname {Hom} (\mathbb {Z} \otimes R,S)\cong \operatorname {Hom} (\mathbb {Z} ,R\Rightarrow S)\cong \{\bullet \}}$ . The same holds for the category of R-algebras over a commutative ring R.

• Isbell conjugacy

• Kelly, G.M. (1982). Basic Concepts of Enriched Category Theory (PDF). London Mathematical Society Lecture Note Series. Vol. 64. Cambridge University Press. ISBN 978-0-521-28702-9. OCLC 1015056596.
• Melliès, Paul-André (2009). "Categorical Semantics of Linear Logic" (PDF). Panoramas et Synthèses. 27: 1–197. CiteSeerX 10.1.1.62.5117.
• Closed monoidal category at the nLab