# Closed monoidal category A **closed monoidal category** is a monoidal category in which the tensor product functor with any fixed object has a right adjoint, so that the category carries an internal hom object \\[b,c\\] classifying morphisms out of a tensor product. Equivalently, for all objects \\(a, b, c\\) there is a natural bijection of hom-sets \\[ \\mathrm{Hom}_{C}(a \\otimes b, c) \\;\\simeq\\; \\mathrm{Hom}_{C}(a, [b,c]), \\] natural in all arguments; this is the statement, attributed to Mac Lane (1971), §IV.7, theorem 3, that \\( - \\otimes b \\) is left adjoint to \\[b,-\\][1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory). The object \\[b,c\\] is called the **internal hom** of \\(b\\) and \\(c\\), and is also commonly written \\(\\underline{\\mathrm{hom}}(b,c)\\)[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory) • [2](https://web.stanford.edu/~dkim04/trace-methods/230411/). | Key fact | Detail | |---|---| | Defining adjunction | For every object \\(b\\), the functor \\( - \\otimes b \\) has a right adjoint \\[b,-\\], giving \\(\\mathrm{Hom}(a \\otimes b, c) \\cong \\mathrm{Hom}(a, [b,c])\\) naturally[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory) | | Internal hom | The value \\[b,c\\] of the right adjoint is the internal hom, also written \\(\\underline{\\mathrm{hom}}(b,c)\\)[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory) | | Handedness | Left closed: \\( - \\otimes b\\) has a right adjoint; right closed: \\(b \\otimes -\\) has a right adjoint; biclosed: both do[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory) • [3](https://math.jhu.edu/~tclingm1/store/cats.pdf) | | Symmetric case | A symmetric closed monoidal category is automatically biclosed; in a braided monoidal category, left-, right-, and biclosedness are equivalent[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory) • [3](https://math.jhu.edu/~tclingm1/store/cats.pdf) | | Cartesian case | If the monoidal structure is cartesian, the category is cartesian closed and the internal hom is an exponential object written \\(c^b\\)[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory) | | Standard example | Modules over a commutative ring satisfy the Hom-tensor adjunction \\(\\mathrm{Hom}_R(M \\otimes_R N, P) \\cong \\mathrm{Hom}_R(M, \\mathrm{Hom}_R(N,P))\\), hence so do abelian groups[4](https://alphabetamath.cc/category-theory/closed-monoidal-categories-and-the-internal-hom-examples) | | Non-symmetric example | Bimodules over a non-commutative ring form a biclosed monoidal category that is not symmetric monoidal[5](https://planetmath.org/closedmonoidalcategory) | ## The tensor–hom adjunction A monoidal category \\(C\\) comes with a tensor product \\(\\otimes\\), a unit object \\(e\\), and coherent associativity and unit isomorphisms. Closedness adds the requirement that tensoring on a fixed side be a left adjoint. In the symmetric case the definition asks that for all objects \\(b\\) the functor \\( - \\otimes b \\colon C \\to C\\) have a right adjoint \\[b,-\\] \\colon C \\to C\\)[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory). The adjunction supplies an **evaluation morphism**, and the internal hom is determined by its evaluation-transposition bijection[4](https://alphabetamath.cc/category-theory/closed-monoidal-categories-and-the-internal-hom-examples). The familiar set-level prototype is the bijection between functions \\(f \\colon A \\times B \\to C\\) and functions \\(A \\to C^B\\), given by currying \\(f\\) to \\((\\lambda f)(a)(b) = f(a,b)\\) with the evaluation map recovering \\(f\\)[3](https://math.jhu.edu/~tclingm1/store/cats.pdf). The general definition recognizes that \\((C^B, \\mathrm{ev})\\) is a universal arrow from the functor \\( \\times B \\) to \\(C\\), and that having such universal arrows for every object is exactly an adjunction[3](https://math.jhu.edu/~tclingm1/store/cats.pdf). ## Left, right, and biclosed When \\(C\\) is monoidal but not symmetric or braided, the two one-variable tensor functors \\( - \\otimes b\\) and \\(b \\otimes -\\) may be inequivalent, and either or both may have an adjoint. The terminology, though not fully standard, distinguishes **left closed** (the left tensor product functor has a right adjoint), **right closed** (the right tensor product functor has a right adjoint), and **biclosed** (both do); some authors simply say closed for biclosed[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory). In the right-closed definition, for every \\(A, B, C\\) one has \\(C(A \\otimes B, C) \\cong C(A, [B,C])\\) naturally in \\(A\\) and \\(C\\), and the image of \\[C,-\\] is called the internal morphism object; dually, a left-closed category has a right adjoint \\[ -, C\\] to \\(C \\otimes -\\)[3](https://math.jhu.edu/~tclingm1/store/cats.pdf). Braiding collapses the distinctions: if the monoidal category is braided, it is left-closed if and only if it is right-closed if and only if it is biclosed, and one simply speaks of a closed [braided monoidal category](https://www.edgepedia.org/braided-monoidal-category), with the isomorphism of external morphism objects natural in all arguments[3](https://math.jhu.edu/~tclingm1/store/cats.pdf). In particular, every symmetric closed monoidal category is biclosed[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory). ## Cartesian closed categories If the monoidal structure of \\(C\\) is cartesian, meaning the tensor product is a categorical product with the terminal object as unit, then a closed such category is called **cartesian closed**. The internal hom is then usually called an exponential object and written \\(c^b\\)[1](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory). Cartesian closed categories are the categorical semantics of the \\(\\\\&\\),\\(\\Rightarrow\\)-fragment of intuitionistic propositional calculus with truth, and decision procedures for them were obtained using ordinary typed lambda terms[6](https://academic.oup.com/book/53996/chapter/422204016). ## Examples - **Sets.** The category of sets with the cartesian product is cartesian closed: the exponential \\(C^B\\) is the set of functions from \\(B\\) to \\(C\\), with evaluation as above[3](https://math.jhu.edu/~tclingm1/store/cats.pdf). - **Modules.** For modules over a commutative ring \\(R\\), the Hom-tensor adjunction \\(\\mathrm{Hom}_{R}(M \\otimes_{R} N, P) \\cong \\mathrm{Hom}_{R}(M, \\mathrm{Hom}_{R}(N, P))\\) holds, so the category is closed symmetric monoidal; abelian groups are the case \\(R = \\mathbb{Z}\\)[4](https://alphabetamath.cc/category-theory/closed-monoidal-categories-and-the-internal-hom-examples). - **Bimodules.** The category of bimodules over a non-commutative ring is an example of a biclosed monoidal category that is not symmetric monoidal[5](https://planetmath.org/closedmonoidalcategory). - **Relations.** In the category **Rel** of sets and relations, the closed structure behaves differently from the hom-sets: \\(A \\multimap B\\) is the cartesian product of \\(A\\) and \\(B\\) in **Set**, whereas \\(\\mathrm{Hom}(A,B)\\) is the powerset of that product, so the internal hom is not a structure built on the external hom[7](https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf). - **Preorders.** The unit interval \\[0,1\\] with its usual order is a closed symmetric monoidal preorder, the one-object-thin analog of a closed symmetric monoidal category[8](https://people.cs.umass.edu/~mahadeva/papers/Lecture4.pdf). A symmetric monoidal category may equivalently be described as closed when the internal hom \\(\\underline{\\mathrm{Hom}}(X,Y)\\) exists for all objects \\(X, Y\\)[2](https://web.stanford.edu/~dkim04/trace-methods/230411/). ## Related structures There is also a notion of **closed category**, due to Eilenberg and Kelly, that does not require the category to already be monoidal: a closed category is a symmetric-monoidal-like structure in which each functor \\( - \\otimes b\\) has a specified right adjoint \\( (-)^b\\)[9](https://encyclopediaofmath.org/wiki/Closed_monoidal_category). Any closed monoidal category gives a closed category by forgetting the tensor product and remembering only the internal hom[10](https://ncatlab.org/nlab/show/closed%2Bcategory), and the [2-category](https://www.edgepedia.org/2-category) of closed categories in the sense of Eilenberg and Kelly is equivalent to a suitable full 2-subcategory of the 2-category of closed multicategories[11](https://arxiv.org/abs/0904.3137). Closedness also interacts with duality. Since it is defined by the tensor product having a right adjoint, the internal-hom functor, one can ask whether a closed monoidal category is rigid, that is, whether its objects have duals, by verifying that the internal hom is tensor-representable[12](https://fis.tu-dresden.de/portal/en/publications/duality-in-monoidal-categories%28a3c3709c-eb0a-416e-9054-ceec306c95ff%29.html). ## Coherence In the 1960s [Saunders Mac Lane](https://www.edgepedia.org/saunders-mac-lane), the Harvard-born mathematician who formulated the standard axioms for monoidal categories, conjectured a coherence theorem for closed symmetric monoidal categories; Sergei Soloviev, a computer science researcher based in Toulouse, proved it in the 1990s. The theorem asserts that a diagram drawn in a free closed symmetric monoidal category commutes precisely when every one of its instances in vector spaces commutes, with the vector spaces allowed to be over an arbitrary fixed field; Soloviev's theorem is in fact broader, supplying an axiomatic characterization of the test categories in which it remains valid.[13](https://ncatlab.org/nlab/show/coherence%2Btheorem%2Bfor%2Bclosed%2Bsymmetric%2Bmonoidal%2Bcategories) ## References 1. [Closed monoidal category, nLab](https://ncatlab.org/nlab/show/closed%2Bmonoidal%2Bcategory) 2. [Symmetric monoidal categories, Dongryul Kim, Stanford](https://web.stanford.edu/~dkim04/trace-methods/230411/) 3. [1.2. Closed monoidal categories, Clingman, Johns Hopkins University lecture notes](https://math.jhu.edu/~tclingm1/store/cats.pdf) 4. [Closed Monoidal Categories and the Internal Hom, Examples, Alphabeta Math](https://alphabetamath.cc/category-theory/closed-monoidal-categories-and-the-internal-hom-examples) 5. [Closed monoidal category, PlanetMath](https://planetmath.org/closedmonoidalcategory) 6. [Lambda Terms with Functional Symbols and Decidability in Certain Closed Categories, in Substructural Logics, Oxford Academic](https://academic.oup.com/book/53996/chapter/422204016) 7. [Categories with monoidal structure, Barr and Wells, McGill University](https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf) 8. [Lecture 4, University of Massachusetts Amherst](https://people.cs.umass.edu/~mahadeva/papers/Lecture4.pdf) 9. [Closed monoidal category, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Closed_monoidal_category) 10. [Closed category, nLab](https://ncatlab.org/nlab/show/closed%2Bcategory) 11. [Closed categories vs. closed multicategories, arXiv:0904.3137](https://arxiv.org/abs/0904.3137) 12. [Duality in Monoidal Categories (Halbig), TU Dresden publication portal](https://fis.tu-dresden.de/portal/en/publications/duality-in-monoidal-categories%28a3c3709c-eb0a-416e-9054-ceec306c95ff%29.html) 13. [Coherence theorem for closed symmetric monoidal categories, nLab](https://ncatlab.org/nlab/show/coherence%2Btheorem%2Bfor%2Bclosed%2Bsymmetric%2Bmonoidal%2Bcategories) --- *Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Logical calculi and logical syntax › Lambda calculus and type theory › Categorical and realizability semantics* *Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026* *Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.* License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license