category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
There are two useful equivalent formulation of the definition
A $*$-autonomous category is a symmetric closed monoidal category $\langle C,\otimes, I\rangle$ with a dualizing object in a closed category: an object $\bot$ such that the canonical morphism
which is the transpose of the evaluation map
is an isomorphism for all $A$. (Here, $\multimap$ denotes the internal hom.)
A $*$-autonomous category is a symmetric monoidal category $\mathcal{C}$ equipped with a full and faithful functor
such that there is a natural isomorphism
Given def. 1, define the dualization functor as the internal hom into the dualizing object
Then the morphism $d_A$ is natural in $A$, so that there is a natural isomorphism $d:1\Rightarrow(-)^{**}$. We also have
This yields the structure of def. 2.
Conversely, given the latter then the dualizing object $\bot$ is defined as the dual of the tensor unit $\bot \coloneqq I^*$.
A $\ast$-autonomous category in which the tensor product is compatible with duality in that there is a natural isomorphism
is a compact closed category.
Conversely, if the $\ast$-autonomous category is not compact closed, then by this linear “ de Morgan duality” the tensor product induces a second binary operation
making it into a linearly distributive category. Here the notation on the left is that used in linear logic, see below at Properties – Internal logic.
A general $\ast$-autonomous category can be thought of as like a compact closed category in which the unit and counit of the dual objects refer to two different tensor products: we have $\top \to A \parr A^*$ but $A^* \otimes A \to \bot$, where $(\otimes,\top)$ and $(\parr,\bot)$ are two different monoidal structures. The necessary relationship between two such monoidal structures such that this makes sense, i.e. such that the triangle identities can be stated, is encoded by a linearly distributive category; then an $\ast$-autonomous category is precisely a linearly distributive category in which all such “mixed duals” (or “negations”) exist.
The internal logic of star-autonomous categories is the multiplicative fragment of classical linear logic, conversely star-autonomous categories are the categorical semantics of classical linear logic (Seely 89, prop. 1.5). See also at relation between type theory and category theory.
A simple example of a $*$-autonomous category is the category of finite-dimensional vector spaces over some field $k$. In this case $k$ itself plays the role of the dualizing object, so that for an f.d. vector space $V$, $V^*$ is the usual dual space of linear maps into $k$.
More generally, any compact closed category is $*$-autonomous with the unit $I$ as the dualizing object.
A more interesting example of a $*$-autonomous category is the category of sup-lattices and sup-preserving maps (= left adjoints). Clearly the poset of sup-preserving maps $hom(A, B)$ is itself a sup-lattice, so this category is closed. The free sup-lattice on a poset $X$ is the internal hom of posets $[X^{op}, \Omega]$; in particular the poset of truth values $\Omega$ is a unit for the closed structure. Define a duality $(-)^*$ on sup-lattices, where $X^* = X^{op}$ is the opposite poset (inf-lattices are sup-lattices), and where $f^*: Y^* \to X^*$ is the left adjoint of $f^{op}: X^{op} \to Y^{op}$. In particular, take as dualizing object $D = \Omega^{op}$. Some simple calculations show that under the tensor product defined by the formula $(X \multimap Y^*)^*$, the category of sup-lattices becomes a $*$-autonomous category.
Another interesting example is due to Yuri Manin: the category of quadratic algebras. A quadratic algebra over a field $k$ is a graded algebra $A = T(V)/I$, where $V$ is a finite-dimensional vector space in degree 1, $T(V)$ is the tensor algebra (the free $k$-algebra generated by $V$), and $I$ is a graded ideal generated by a subspace $R \subseteq V \otimes V$ in degree 2; so $R = I_2$, and $A$ determines the pair $(V, R)$. A morphism of quadratic algebras is a morphism of graded algebras; alternatively, a morphism $(V, R) \to (W, S)$ is a linear map $f: V \to W$ such that $(f \otimes f)(R) \subseteq S$. Define the dual $A^*$ of a quadratic algebra given by a pair $(V, R)$ to be that given by $(V^*, R^\perp)$ where $R^\perp \subseteq V^* \otimes V^*$ is the kernel of the transpose of the inclusion $i: R \subseteq V \otimes V$, i.e., there is an exact sequence
Define a tensor product by the formula
where $\sigma: V \otimes W \to W \otimes V$ is the symmetry. The unit is the tensor algebra on a 1-dimensional space. The hom that is adjoint to the tensor product is given by the formula $A \multimap B = (A \otimes B^*)^*$, and the result is a $*$-autonomous category.
In a similar vein, I am told that there is a $*$-autonomous category of quadratic operad?s.
Girard’s coherence spaces?, developed as models of linear logic, give an historically important example of a $*$-autonomous category.
Hyland and Ong have given a completeness theorem for multiplicative linear logic in terms of a $*$-autonomous category of fair games, part of a series of work on game semantics for closed category theory (compare Joyal’s interpretation of Conway games as forming a compact closed category).
The Chu construction can be used to form many more examples of $*$-autonomous categories.
A quantale (see there) is a $\ast$-autonomous category if it has a dualizing object.
The notion is originally due to
The relation to linear logic was first described in
and a detailed review (also of a fair bit of plain monoidal category theory) is in
Examples from algebraic geometry are given here:
These authors call any closed monoidal category with a dualizing object a Grothendieck-Verdier category, thanks to the examples coming from Verdier duality.
Here it is explained how $*$-autonomous categories give Frobenius pseudomonads in the 2-category where morphisms are profunctors: