nLab closed monoidal category



Monoidal categories

monoidal categories

With braiding

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 (π’ž,πŸ™,βŠ—)\big( \mathcal{C}, \, \mathbb{1},\, \otimes \big) that is also a closed category, in a compatible way, i.e. such that for each object Yβˆˆπ’žY \,\in\, \mathcal{C}

  1. the functor

    (βˆ’)βŠ—Y:C⟢C (-) \otimes Y \;\colon\; C \longrightarrow C

    of forming the tensor product with YY,

  2. has a right adjoint functor

    [Y,βˆ’]:C⟢C [Y,-] \;\colon\; C \longrightarrow C

    forming the internal-hom out of YY

in that for all triples of objects X,Y,ZX,\, Y,\, Z there is a natural hom-isomorphism of the following form:

Hom π’ž(XβŠ—Y,Z)≃Hom π’ž(X,[Y,Z]). Hom_{\mathcal{C}}(X \otimes Y,\, Z) \;\simeq\; Hom_{\mathcal{C}}\big(X, [Y,Z] \big) \,.

The archetypical example is the cartesian closed category of Sets, where the internal hom is given by forming function sets, so that the above hom-isomorphism expresses the fact that a function (x,y)↦f(x,y)(x,y) \mapsto f(x,y) of two variables is equivalently a function of a single variable with values in functions of the other variable: x↦(y↦f(x,y))x \mapsto \big(y \mapsto f(x,y) \big). In formal logic this is known as β€œcurrying”.

More abstractly, from the point of view of 2-category theory, closed monoidal categories may be regarded as a special case of the notion of closed pseudomonoids in a monoidal bicategory.


Symmetric closed monoidal category

A symmetric monoidal category CC is closed (cf. symmetric monoidal closed category) if for all objects b∈C 0b \in C_0 the tensor product functor bβŠ—βˆ’:Cβ†’C b \otimes - \colon C \to C has a right adjoint functor [b,βˆ’]:Cβ†’C[b,-] \colon C \to C.

βˆ€b∈CCβŠ₯⟢[b,βˆ’]⟡bβŠ—(βˆ’)C. \underset{b \in C}{\forall} \;\; C \underoverset {\underset{[b,-]}{\longrightarrow}} {\overset{b \otimes (-)}{\longleftarrow}} {\;\;\;\;\;\bot\;\;\;\;\;} C \,.

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

Hom C(aβŠ—b,c)≃Hom C(a,[b,c]), Hom_C(a \otimes b, c) \;\simeq\; Hom_C(a, [b,c]) \,,

natural in all arguments (see (MacLane (1971), Β§IV.7, theorem 3)).

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 βˆ’βŠ—b-\otimes b and bβŠ—βˆ’b\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 (π’ž,βŠ—,1)(\mathcal{C}, \otimes, 1) a closed monoidal category with internal hom denoted [βˆ’,βˆ’][-,-], then not only are there natural bijections

Hom π’ž(XβŠ—Y,Z)≃Hom π’ž(X,[Y,Z]) Hom_{\mathcal{C}}(X \otimes Y, Z) \simeq Hom_{\mathcal{C}}(X, [Y,Z])

but these isomorphisms themselves β€œinternalize” to isomorphisms in π’ž\mathcal{C} of the form

[XβŠ—Y,Z]≃[X,[Y,Z]]. [X \otimes Y, Z] \simeq [X,[Y,Z]] \,.

By the external natural bijections there is for every Aβˆˆπ’žA \in \mathcal{C} a composite natural bijection

Hom π’ž(A,[XβŠ—Y,Z])≃Hom π’ž(AβŠ—(XβŠ—Y),Z)≃Hom π’ž((AβŠ—X)βŠ—Y,Z)≃Hom π’ž(AβŠ—X,[Y,Z])≃Hom π’ž(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 Aβˆˆπ’žA \in \mathcal{C}, the Yoneda lemma (namely the fully faithfulness of the Yoneda embedding) implies that there is already an isomorphism

[XβŠ—Y,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 (a cartesian closed monoidal category): the collection of functions between any two sets is itself a set – the function set.


In generalization of Exp. and as a special case of Exp. , every category of presheaves, i.e. every functor category of the form Func(π’ž op,Set)Func(\mathcal{C}^{op}, Set), is a cartesian closed monoidal category, details are spelled out at closed monoidal structure on presheaves.


In further generalization of Exp , any topos is a cartesian closed monoidal category.


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


In generalization of Exp. (which is the case R=R = β„€ \mathbb{Z} ), for RR a commutative ring, the category RRMod of RR-modules with its usual tensor product of modules is (non-cartesian!) closed monoidal. This is one of the first hom-tensor adjunctions that appeared in algebra.


In specialization of Exp. to the case that R=𝕂R = \mathbb{K} is a field, the category 𝕂\mathbb{K}-Vect of 𝕂\mathbb{K}-vector spaces and 𝕂\mathbb{K}-linear maps between them is (non-cartesian!) closed monoidal with respect to the usual tensor product of vector spaces. Here the internal hom assigns vector spaces of linear maps (with the vector operations given argument-wise):

V,Wβˆˆπ•‚Vect⊒[V,W]≔(𝕂Vec(V,W),kβ‹…f:v↦kβ‹…f(v) f+g:v↦f(v)+g(v)) V,\,W \,\in\, \mathbb{K}Vect \;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\; [V,W] \;\coloneqq\; \left( \mathbb{K}Vec(V,W) ,\, \begin{array}{l} k \cdot f \,\colon\, v \mapsto k \cdot f(v) \\ f + g \,\colon\, v \mapsto f(v) + g(v) \end{array} \right)


A higher homotopical variant of Exp. : For RR a commutative ring the category of chain complexes over RR is closed monoidal, via the tensor product of chain complexes and the internal hom of chain complexes, see there for more.


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 at tensor product of algebras over a commutative monad.


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, (FβŠ—G)(x)=F(x)βŠ—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 obIβ†ͺIob I \hookrightarrow I. Now for any F∈[I,C]F\in [I,C], consider the following square:

[I,C] ⟢FβŠ—βˆ’ [I,C] ↓ ↓ C obI ⟢F 0βŠ—βˆ’ C obI \array{ [I,C] & \overset{F\otimes - }{\longrightarrow} & [I,C] \\ \big\downarrow && \big\downarrow \\ C^{ob I} & \underset{F_0 \otimes -}{\longrightarrow} & 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 0βŠ—βˆ’F_0 \otimes - has a right adjoint. Since the vertical functors are comonadic, the (dual of the) adjoint lifting theorem implies that FβŠ—βˆ’F\otimes - has a right adjoint as well.


A discrete monoidal category (i.e., a monoid) is left closed iff it is right closed iff every object has an inverse, i.e. iff the monoid (aka semigroup) is in fact 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 fβŠ—g=[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 Xβ†’YX \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. There is exactly one other closed monoidal structure on Cat, given by the funny tensor product.

  • 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 Ο‰\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 βˆ’βˆ˜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.


Textbook accounts:

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.

Last revised on May 17, 2024 at 15:56:15. See the history of this page for a list of all contributions to it.