nLab closed monoidal category



Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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.


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.



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.

  • The category of modules over a given commutative ring with its usual tensor product is closed monoidal. This is one of the first hom-tensor adjunctions that appeared in algebra.

  • Generalizing the examples above, given a closed monoidal category CC with equalizers and coequalizers and a commutative monad TT, the category of T T -algebras inherits a closed monoidal structure. See tensor product of algebras over a commutative monad.

  • 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).

  • The delooping BM\mathbf{B}M of a commutative monoid MM is a closed monoidal (in fact, compact closed) category with one object, with both the tensor and internal hom defined (on morphisms) using the multiplication operation fg=[f,g]=fgf \otimes g = [f,g] = f g. Conversely, any closed monoidal category with one object must be isomorphic to one constructed from a commutative monoid. (See Eilenberg and Kelly (1965), IV.3, p.553.)

  • 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.

  • The category of locally convex topological vector spaces with the inductive tensor product and internal hom the space of continuous linear maps with the topology of pointwise convergence is symmetric closed monoidal.

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.


Textbook account for symmetric closed categories:

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, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp. (ISBN:9780521287029);

    republished as:

    Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 (tac:tr10, pdf)

has a chapter on just closed monoidal categories.

See also the article

on the concept of closed categories.

Last revised on August 1, 2022 at 17:03:00. See the history of this page for a list of all contributions to it.