category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A closed monoidal category $C$ is a monoidal category that is also a closed category, in a compatible way:
it has for each object $X$ a functor $(-) \otimes X : C \to C$ of forming the tensor product with $X$, as well as a functor $[X,-] : C \to C$ of forming the internal-hom with $X$, and these form a pair of adjoint functors.
The strategy for formalizing the idea of a closed category, that “the collection of morphisms from $a$ to $b$ can be regarded as an object of $C$ itself”, is to mimic the situation in Set where for any three objects (sets) $a$, $b$, $c$ we have an isomorphism
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 $Hom(b,c)$ 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.
A symmetric monoidal category $C$ is closed if for all objects $b \in C_0$ the functor $- \otimes b : C \to C$ has a right adjoint functor $[b,-] : C \to C$.
This means that for all $a,b,c \in C_0$ we have a natural bijection
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 often underlined).
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 $c^b$.
If $C$ is monoidal not necessarily symmetric, then left and right tensor product $-\otimes b$ and $b\otimes -$ 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. (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 $(\mathcal{C}, \otimes, 1)$ a closed monoidal category with internal hom denoted $[-,-]$, then not only are there natural bijections
but these isomorphisms themselves “internalize” to isomorphisms in $\mathcal{C}$ of the form
By the external natural bijections there is for every $A \in \mathcal{C}$ a composite natural bijection
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
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 $A, B$ the set of homomorphisms $A \to B$ 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).
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.
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.
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, $(F\otimes G)(x) = F(x) \otimes G(x)$.
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:
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.
symmetric monoidal category, symmetric monoidal (∞,1)-category
closed monoidal category , closed monoidal (∞,1)-category
string diagram, Kelly-Mac Lane diagram, natural language syntax
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
has a chapter on just closed monoidal categories.
See also the article
on the concept of closed categories.