category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
There are many ways to describe a $*$-autonomous category; here are a few:
In particular, it has two monoidal structures $\otimes$ and $\parr$, as in a linearly distributive category. However, because of the dualization operation $(-)^\ast$, each of them determines the other by a “multiplicative de Morgan duality”: $A\parr B \cong (A^\ast \otimes B^\ast)^\ast$. Thus, in the definition we only have to refer to one monoidal structure.
A $\ast$-autonomous category is a special case of the notion of star-autonomous pseudomonoid (a.k.a. Frobenius pseudomonoid, since it categorifies a Frobenius algebra) in a monoidal bicategory.
There are two useful equivalent formulation of the definition
A $*$-autonomous category is a symmetric closed monoidal category $\langle C,\otimes, I,\multimap\rangle$ with a global dualizing object: 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.
It may not be clear from the above definitions what the appropriate notion of “$\ast$-autonomous functor” is. In fact, from at least one perspective it suffices to consider ordinary lax monoidal functors, at least in the symmetric case; no interaction with the $\ast$-autonomy is required.
Let $\ast Aut$ denote the full sub-2-category of $SymMonCat_{lax}$ on the (symmetric) $\ast$-autonomous categories; hence its morphisms and 2-cells are lax symmetric monoidal functors and monoidal natural transformations. Let $SymLinDist$ denote the 2-category of symmetric linearly distributive categories, symmetric linear functors, and linear transformations (defined at linearly distributive category).
There is a functor $\ast Aut \to SymLinDist$ that is 2-fully-faithful, i.e. an equivalence on hom-categories, and has both a left and a right 2-adjoint.
A linearly distributive functor consists of a functor $F_\otimes$ that is lax monoidal for $\otimes$ and a functor $F_\parr$ that is lax monoidal for $\parr$. Recalling that $\parr$ is defined in a $\ast$-autonomous category as $A\parr B = (A^* \otimes B^*)^*$, if $F$ is a lax monoidal functor between $\ast$-autonomous categories we define $F_\otimes = F$ and $F_\parr(A) = (F(A^*))^*$. See Cockett-Seely 1999 for details.
In the non-symmetric case, we need to additionally require of an “$\ast$-autonomous functor” that $^*(F(A^*)) \cong (F({}^*A))^*$, and define $F_\parr$ to be their common value.
On the other hand, if we consider instead “Frobenius linear functors” between linearly distributive categories, then such functors necessarily preserve duals. Thus, the corresponding notion of $\ast$-autonomous functor would need to preserve the dualization functors as well, $F(A^*) \cong (F(A))^*$.
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. These are closely related to a general construction of $*$-autonomous categories (and related types of categories) called poset-valued sets.
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.
Suppose $\langle C,\otimes, I,\multimap\rangle$ is a closed symmetric monoidal category equipped with a “pre-dualizing object” $\bot$, in the sense that the contravariant self-adjunction $(-\multimap\bot) \dashv (-\multimap\bot)$ is idempotent, i.e. the double-dualization map $A \to (A \multimap \bot) \multimap \bot$ is an isomorphism whenever $A$ is of the form $B\multimap\bot$. (Note that idempotence is automatic if $C$ is a poset.) Then the category of fixed points of this adjunction, i.e. the full subcategory of objects of the form $B\multimap\bot$, is $*$-autonomous. For it is closed under $\multimap$, as $(A\multimap (B\multimap \bot)) \cong (A\otimes B\multimap \bot)$, and reflective with reflector $(-\multimap\bot)\multimap\bot$; hence it is closed symmetric monoidal with tensor product $((A\otimes B)\multimap\bot)\multimap\bot$, and all its double-dualization maps are isomorphisms by assumption. A historically important example is Girard’s phase semantics of linear logic.
Phase semantics is also a special case of a ternary frame, which is the case of the previous example when $\langle C,\otimes, I,\multimap\rangle$ has the Day convolution structure induced from a promonoidal poset. There is also another way to obtain negation in a ternary frame involving a “compatibility relation”…
The unit interval $[0,1]$ is a semicartesian $\ast$-autonomous poset with the monoidal structure $x \otimes y = \max(x+y-1,0)$ and dualizing object $0$. The involution is $x^\ast = 1-x$ and the dual multiplicative disjunction is $x\parr y = \min(x+y,1)$. This is known as Lukasiewicz logic? and is used in fuzzy logic?; it is also a special case of a t-norm.
The notion is originally due to
See also
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:
Relation to linearly distributive categories: