nLab
closed monoidal category

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

A closed monoidal category is a monoidal category CC that is also a closed category, in a compatible way:

it has for each object XX a functor ()X:CC(-) \otimes X : C \to C of forming the tensor product with XX, as well as a functor [X,]:CC[X,-] : C \to C of forming the internal-hom with XX, and these form a pair of adjoint functors.

The strategy for formalizing the idea of a closed category, that “the collection of morphisms from aa to bb can be regarded as an object of CC itself”, is to mimic the situation in Set where for any three objects (sets) aa, bb, cc we have hom-isomorphism

Hom(ab,c)Hom(a,[b,c]), Hom(a \otimes b, c) \simeq Hom(a, [b,c]) \,,

natural in all three variables,

where =×\otimes = \times is the standard cartesian product of sets. This natural isomorphism is called currying.

Currying can be read as a characterization of the internal hom Hom(b,c)Hom(b,c) and is the basis for the following definition.

A closed monoidal category is a special case of the notion of closed pseudomonoid in a monoidal bicategory.

Definitions

Symmetric closed monoidal category

A symmetric monoidal category CC is closed if for all objects bC 0b \in C_0 the tensor product functor b:CC b \otimes - : C \to C has a right adjoint functor [b,]:CC[b,-] \colon C \to C.

bCC[b,]b()C. \underset{b \in C}{\forall} \;\; C \underoverset {\underset{[b,-]}{\longrightarrow}} {\overset{b \otimes (-)}{\longleftarrow}} {\;\;\;\;\;\bot\;\;\;\;\;} C \,.

This means that for all a,b,cC 0a,b,c \in C_0 we have a natural bijection

Hom C(ab,c)Hom C(a,[b,c]), Hom_C(a \otimes b, c) \;\simeq\; Hom_C(a, [b,c]) \,,

natural in all arguments.

The object [b,c][b,c] is called the internal hom of bb and cc. This is commonly also denoted by lower case hom(b,c)hom(b,c) (and then sometimes underlined).

Cartesian closed monoidal category

If the monoidal structure of CC is cartesian (and so in particular symmetric monoidal), then CC is called cartesian closed. In this case the internal hom is often called an exponential object and written c bc^b.

Left-, right- and bi-closed monoidal category

If CC is monoidal but not necessarily symmetric or even braided, then left and right tensor product b-\otimes b and bb\otimes - may be inequivalent functors, and either one or both may have an adjoint. The terminology here is less standard, but many people use left closed, right closed, and biclosed monoidal category to indicate, respectively, that the left tensor product, the right tensor product functors or both have right adjoints. (Other authors simply say closed instead of biclosed.) So in particular a symmetric closed monoidal category is automatically biclosed.

The analogue of exponential objects for monoidal categories are left and right residuals.

Properties

Proposition

For (𝒞,,1)(\mathcal{C}, \otimes, 1) a closed monoidal category with internal hom denoted [,][-,-], then not only are there natural bijections

Hom 𝒞(XY,Z)Hom 𝒞(X,[Y,Z]) Hom_{\mathcal{C}}(X \otimes Y, Z) \simeq Hom_{\mathcal{C}}(X, [Y,Z])

but these isomorphisms themselves “internalize” to isomorphisms in 𝒞\mathcal{C} of the form

[XY,Z][X,[Y,Z]]. [X \otimes Y, Z] \simeq [X,[Y,Z]] \,.
Proof

By the external natural bijections there is for every A𝒞A \in \mathcal{C} a composite natural bijection

Hom 𝒞(A,[XY,Z])Hom 𝒞(A(XY),Z)Hom 𝒞((AX)Y,Z)Hom 𝒞(AX,[Y,Z])Hom 𝒞(A,[X,[Y,Z]]). Hom_{\mathcal{C}}(A, [X \otimes Y, Z]) \simeq Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z) \simeq Hom_{\mathcal{C}}((A \otimes X) \otimes Y, Z) \simeq Hom_{\mathcal{C}}(A \otimes X, [Y,Z]) \simeq Hom_{\mathcal{C}}(A,[X,[Y,Z]]) \,.

Since this holds for every A𝒞A \in \mathcal{C}, the Yoneda lemma (namely the fully faithfulness of the Yoneda embedding) implies that there is already an isomorphism

[XY,Z][X,[Y,Z]]. [X \otimes Y, Z] \simeq [X,[Y,Z]] \,.

Examples

Functor categories

Theorem

Let CC be a complete closed monoidal category and II any small category. Then the functor category [I,C][I,C] is closed monoidal with the pointwise tensor product, (FG)(x)=F(x)G(x)(F\otimes G)(x) = F(x) \otimes G(x).

Proof

Since CC is complete, the category [I,C][I,C] is comonadic over C obIC^{ob I}; the comonad is defined by right Kan extension along the inclusion obIIob I \hookrightarrow I. Now for any F[I,C]F\in [I,C], consider the following square:

[I,C] F [I,C] C obI F 0 C obI\array{[I,C] & \overset{F\otimes - }{\to} & [I,C] \\ \downarrow && \downarrow\\ C^{ob I}& \underset{F_0 \otimes -}{\to} & C^{ob I}}

This commutes because the tensor product in [I,C][I,C] is pointwise (here F 0F_0 means the family of objects F(x)F(x) in C obIC^{ob I}). Since CC is closed, F 0F_0 \otimes - has a right adjoint. Since the vertical functors are comonadic, the (dual of the) adjoint lifting theorem implies that FF\otimes - has a right adjoint as well.

References

Textbook account for symmetric closed categories:

Original articles studying monoidal biclosed categories are

For more historical development see at linear type theory – History of linear categorical semantics.

In enriched category theory the enriching category is taken to be closed monoidal. Accordingly the standard textbook on enriched category theory

has a chapter on just closed monoidal categories.

See also the article

on the concept of closed categories.