closed monoidal category


Monoidal categories



A closed monoidal category CC is a monoidal category 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 an isomorphism

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

naturally in all three arguments, 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.


Symmetric closed monoidal category

A symmetric monoidal category CC is closed if for all objects bC 0b \in C_0 the functor b:CC - \otimes b : C \to C has a right adjoint functor [b,]:CC[b,-] : C \to 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 often 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 not necessarily symmetric, then left and right tensor product b-\otimes b and bb\otimes - may be non-equivalent 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.

(So in particular a symmetric closed monoidal category is automatically biclosed.)



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]] \,.

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]] \,.


  • The tautological example is the category Set of sets with its Cartesian product: the collection of functions between any two sets is itself a set – the function set. More generally, any topos is cartesian closed monoidal.

  • The category Ab of abelian groups with its tensor product of abelian groups is closed: for any two abelian groups A,BA, B the set of homomorphisms ABA \to B carries (pointwise defined) abelian group structure.

  • A discrete monoidal category (i.e., a monoid) is left closed iff it is right closed iff every object has an inverse (i.e., it is a group).

  • Certain nice categories of topological spaces are cartesian closed: for any two nice enough topological spaces XX, YY the set of continuous maps XYX \to Y can be equipped with a topology to become a nice topological space itself.

  • Certain nice categories of pointed/based topological spaces are closed symmetric monoidal. The monoidal structure is the smash product and the internal-hom is the set of basepoint-preserving maps with topology induced from the space of unbased ones.

  • The category Cat is cartesian closed: the internal-hom is the functor category of functors and natural transformations.

  • The category 2Cat2 Cat of strict 2-categories and strict 2-functors is closed symmetric monoidal under the Gray tensor product. The internal-hom is the 2-category of strict 2-functors, pseudo natural transformations, and modifications.

  • The category of strict ω\omega-categories is also biclosed monoidal, under the Crans-Gray tensor product.

  • If MM is a monoidal category and Set M opSet^{M^{op}} is endowed with the tensor product given by the induced Day convolution product, then the category of presheaves Set M opSet^{M^{op}} is biclosed monoidal.

  • The category of species, with the monoidal structure given by substitution product of species, is closed monoidal (each functor G- \circ G admits a right adjoint) but not biclosed monoidal.

  • The category of modules over any Hopf monoid in a closed monoidal category, or more generally algebras for any Hopf monad, is again a closed monoidal category. In particular, the category of modules over any group object in a cartesian closed category is (cartesian) closed monoidal. For more on this phenomenon see at Tannaka duality.

Functor categories


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).


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.


Original articles studying monoidal biclosed categories are

  • Joachim Lambek, Deductive systems and categories, Mathematical Systems Theory 2 (1968), 287-318.

  • Joachim Lambek, Deductive systems and categories II, Lecture Notes in Math. 86, Springer-Verlag (1969), 76-122.

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

  • Max Kelly, Basic concepts of enriched category theory, section 1.5, (tac)

has a chapter on just closed monoidal categories.

See also the article

on the concept of closed categories.

Revised on June 15, 2016 07:48:35 by Urs Schreiber (