nLab closed monoidal category

Idea

A closed monoidal category is a monoidal category $\big( \mathcal{C}, \, \mathbb{1},\, \otimes \big)$ that is also a closed category, in a compatible way, i.e. such that for each object $Y \,\in\, \mathcal{C}$

1. the functor

   $$(-) \otimes Y \;\colon\; C \longrightarrow C$$

   of forming the tensor product with $Y$,

2. has a right adjoint functor

   $$[Y,-] \;\colon\; C \longrightarrow C$$

   forming the internal-hom out of $Y$

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

$$Hom_{\mathcal{C}}(X \otimes Y,\, Z) \;\simeq\; Hom_{\mathcal{C}}\big(X, [Y,Z] \big) \,.$$

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 $(x,y) \mapsto f(x,y)$ of two variables is equivalently a function of a single variable with values in functions of the other variable: $x \mapsto \big(y \mapsto f(x,y) \big)$. In formal logic this is known as “currying”.

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.

Definitions

Symmetric closed monoidal category

A symmetric monoidal category $C$ is closed (cf. symmetric monoidal closed category) if for all objects $b \in C_0$ the tensor product functor $b \otimes - \colon C \to C$ has a right adjoint functor $[b,-] \colon C \to C$.

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

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

$$Hom_C(a \otimes b, c) \;\simeq\; Hom_C(a, [b,c]) \,,$$

natural in all arguments.

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

Cartesian closed monoidal category

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

Left-, right- and bi-closed monoidal category

If $C$ is monoidal but not necessarily symmetric or even braided, then left and right tensor product $-\otimes b$ and $b\otimes -$ 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.

Properties

Proposition

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

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

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

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

Proof

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

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

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

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

Examples

Example

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.

Example

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

Example

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

Example

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

Example

In generalization of Exp. (which is the case $R =$ $\mathbb{Z}$), for $R$ a commutative ring, the category $R$Mod of $R$-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.

Example

In specialization of Exp. to the case that $R = \mathbb{K}$ is a field, the category $\mathbb{K}$-Vect of $\mathbb{K}$-vector spaces and $\mathbb{K}$-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):

$$V,\,W \,\in\, \mathbb{K}Vect \;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\; [V,W] \;\coloneqq\; \left( \mathbb{K}Vec(V,W) ,\, \begin{array}{l} k \cdot f \,\colon\, v \mapsto k \cdot f(v) \\ f + g \,\colon\, v \mapsto f(v) + g(v) \end{array} \right)$$

Example

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

Example

Given a closed monoidal category $C$ with equalizers and coequalizers and a commutative monad $T$, the category of $T$-algebras inherits a closed monoidal structure. See at tensor product of algebras over a commutative monad.

Example

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

Proof

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

$$\array{ [I,C] & \overset{F\otimes - }{\longrightarrow} & [I,C] \\ \big\downarrow && \big\downarrow \\ C^{ob I} & \underset{F_0 \otimes -}{\longrightarrow} & C^{ob I} }$$

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

Example

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.

Example

The delooping $\mathbf{B}M$ of a commutative monoid $M$ 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 $f \otimes g = [f,g] = f g$. 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 $X$, $Y$ the set of continuous maps $X \to Y$ 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 $2 Cat$ 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 $\omega$-categories is also biclosed monoidal, under the Crans-Gray tensor product.

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

• The category of species, with the monoidal structure given by substitution product of species, is closed monoidal (each functor $- \circ G$ 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.

Related concepts

• monoidal category, monoidal (∞,1)-category

• symmetric monoidal category, symmetric monoidal (∞,1)-category

• closed monoidal category , closed monoidal (∞,1)-category

  • doubly closed monoidal category

  • closed monoidal functor

  • internal hom

  • indexed closed monoidal category

  • cartesian closed category, exponential object

• string diagram, Kelly-Mac Lane diagram, natural language syntax

References

• Samuel Eilenberg, G. Max Kelly, §III of: Closed Categories, in: Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]

Textbook accounts:

• Saunders MacLane, VII.7 in: Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, vol 2 section 6.1 of: Handbook of Categorical Algebra, Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]

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.