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\section*{star-autonomous category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{functors_and_transformations}{Functors and transformations}\dotfill \pageref*{functors_and_transformations} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{InternalLogic}{Internal logic}\dotfill \pageref*{InternalLogic} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

There are many ways to describe a $*$-autonomous category; here are a few:

\begin{itemize}%
\item it is a [[monoidal category]] in which all objects have ``duals'', but in a weaker sense than in a [[compact closed category]].
\item it is a [[closed monoidal category]] in which the [[internal-hom]] can be expressed in terms of the tensor product in a particular way.
\item it is a [[linearly distributive category]] in which all objects have ``duals'' in an appropriate linearly distributive sense.
\item it is a semantics for classical multiplicative [[linear logic]].
\item it is a representable [[star-polycategory]].
\item it is a sort of categorified [[Frobenius algebra]].

\end{itemize}
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's laws|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]].

(Regarding the use of ``autonomous'', this was once used as a bare adjective to describe a [[closed monoidal category]], or sometimes a [[compact closed monoidal category]], but is rarely used in this way today. It has been suggested that in these cases with internal-hom objects the category is autonomous or ``sufficient unto itself'' without needing hom-sets, or suchlike.)

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

There are two useful equivalent formulation of the definition

\begin{defn}
\label{DefByDualizingObject}\hypertarget{DefByDualizingObject}{}
A \emph{$*$-autonomous category} is a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] $\langle
C,\otimes, I,\multimap\rangle$ with a \emph{[[dualizing object in a closed category|global dualizing object]]}: an object $\bot$ such that the canonical morphism

\begin{displaymath}
d_A: A \to (A \multimap \bot) \multimap \bot ,
\end{displaymath}
which is the transpose of the [[evaluation map]]

\begin{displaymath}
ev_{A,\bot}: (A \multimap \bot) \otimes A \to \bot ,
\end{displaymath}
is an [[isomorphism]] for all $A$. (Here, $\multimap$ denotes the [[internal hom]].)

\end{defn}
\begin{defn}
\label{DefByDualization}\hypertarget{DefByDualization}{}
A \emph{$*$-autonomous category} is a [[symmetric monoidal category]] $\mathcal{C}$ equipped with a [[full and faithful functor|full and faithful]] [[functor]]

\begin{displaymath}
(-)^\ast \colon \mathcal{C}^{op} \longrightarrow \mathcal{C}
\end{displaymath}
such that there is a [[natural isomorphism]]

\begin{displaymath}
\mathcal{C}(A \otimes B, C^\ast) \simeq \mathcal{C}(A, (B\otimes C)^\ast)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
These two definitions, def. \ref{DefByDualizingObject} and def. \ref{DefByDualization}, are indeed equivalent.

\end{prop}
\begin{proof}
Given def. \ref{DefByDualizingObject}, define the dualization functor as the [[internal hom]] into the [[dualizing object in a closed category|dualizing object]]

\begin{displaymath}
(-)^ \ast  \coloneqq (-)\multimap\bot
  \,.
\end{displaymath}
Then the morphism $d_A$ is natural in $A$, so that there is a [[natural isomorphism]] $d:1\Rightarrow(-)^{**}$. We also have

\begin{displaymath}
\begin{aligned}
\hom(A\otimes B,C^*)& = \hom(A\otimes B, C\multimap\bot) \\
& \cong \hom((A\otimes B)\otimes C,\bot) \\
& \cong \hom(A\otimes(B\otimes C), \bot) \\
& \cong \hom(A,(B\otimes C)\multimap\bot) \\
& = \hom(A,(B\otimes C)^*)
\end{aligned}
\end{displaymath}
This yields the structure of def. \ref{DefByDualization}.

Conversely, given the latter then the [[dualizing object]] $\bot$ is defined as the dual of the [[tensor unit]] $\bot \coloneqq I^*$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
A $\ast$-autonomous category in which the [[tensor product]] is compatible with duality in that there is a [[natural isomorphism]]

\begin{displaymath}
(A \otimes B)^\ast \simeq A^\ast \otimes B^\ast
\end{displaymath}
is a [[compact closed category]].

More generally, even if the $\ast$-autonomous category is not compact closed, then by this linear ``[[de Morgan duality]]'' the tensor product induces a second binary operation

\begin{displaymath}
A \parr B \coloneqq  (A^\ast \otimes B^\ast)^\ast
  \,.
\end{displaymath}
making it into a [[linearly distributive category]]. Here the notation on the left is that used in [[linear logic]], see below at \emph{\hyperlink{InternalLogic}{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.

\end{remark}
\hypertarget{functors_and_transformations}{}\subsection*{{Functors and transformations}}\label{functors_and_transformations}

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]]).

\begin{utheorem}
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]].

\end{utheorem}
\begin{proof}
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 \hyperlink{CockettSeely99}{Cockett-Seely 1999} for details.

\end{proof}
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, $F(A^*) \cong (F(A))^*$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{InternalLogic}{}\subsubsection*{{Internal logic}}\label{InternalLogic}

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]] (\hyperlink{Seely89}{Seely 89, prop. 1.5}). See also at \emph{[[relation between type theory and category theory]]}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item A simple example of a $*$-autonomous category is the category of finite-dimensional [[vector space]]s 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 vector space|dual space]] of linear maps into $k$.


\item More generally, any [[compact closed category]] is $*$-autonomous with the unit $I$ as the dualizing object.


\item A more interesting example of a $*$-autonomous category is the category of [[sup-lattice]]s 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.


\item Another interesting example is due to Yuri Manin: the category of [[quadratic algebra]]s. A \textbf{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

\begin{displaymath}
0 \to R^\perp \to V^* \otimes V^* \overset{i^*}{\to} R^* \to 0
\end{displaymath}
Define a tensor product by the formula

\begin{displaymath}
(V, R) \otimes (W, S) = (V \otimes W, (1_V \otimes \sigma \otimes 1_W)(R \otimes S))
\end{displaymath}
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.


\item In a similar vein, I am told that there is a $*$-autonomous category of [[quadratic operad]]s.


\item 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]].


\item The category of [[finiteness spaces]] and their relations is $*$-autonomous. Probably so is any category of [[arity spaces]], which includes coherence spaces and finiteness spaces.


\item Hyland and Ong have given a completeness theorem for multiplicative linear logic in terms of a $*$-autonomous category of \emph{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).


\item The [[Chu construction]] can be used to form many more examples of $*$-autonomous categories.


\item Various [[subcategories]] of Chu constructions are also $*$-autonomous. For instance, if [[Vect]] is the category of [[vector spaces]] over a [[field]] $k$, then $Chu(Vect,k)$ is the category of vector spaces equipped with a specified ``dual'' having no further structure than an evaluation map $V\otimes W\to k$. One often wants to impose nondegeneracy conditions on this ``dual'', which in turn can be reflected as [[topological vector space|topological]] properties of the original space $V$.


\item A [[quantale]] (see there) is a $\ast$-autonomous category if it has a [[dualizing object]].


\item 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 adjunction|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 point of an adjunction|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 $(-\multimap\bot)\multimap\bot$, and it contains $\bot$ since $\bot\cong (I\multimap \bot)$. 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]]. Note that this category is a full subcategory of $Chu(C,\bot)$ closed under duality --- indeed, it is the intersection of the two embeddings of $C$ and $C^{op}$ therein --- but its tensor product is \emph{not} the restriction of the tensor product of $Chu(C,\bot)$.


\item 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 category|promonoidal]] poset. There is also another way to obtain negation in a ternary frame involving a ``compatibility relation''\ldots{}


\item The [[unit interval]] $[0,1]$ is a [[semicartesian monoidal category|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]].


\item A summary of many different ways to construct examples is in \hyperlink{HylandSchalk01}{Hyland-Schalk}.



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion is originally due to

\begin{itemize}%
\item [[Michael Barr]], \emph{$*$-Autonomous Categories}. LNM 752, Springer-Verlag 1979.

\end{itemize}
See also

\begin{itemize}%
\item [[Michael Barr]], \emph{Topological $\ast$-autonomous categories, revisited}, rewrite of TAC, 16 (2006), 700-708 (\href{http://arxiv.org/abs/1609.04241}{arXiv:1609.04241})

\end{itemize}
The relation to [[linear logic]] was first described in

\begin{itemize}%
\item [[R. A. G. Seely]], \emph{Linear logic, $\ast$-autonomous categories and cofree coalgebras}, \emph{Contemporary Mathematics} 92, 1989. ([[SeelyLinearLogic.pdf:file]], \href{http://www.math.mcgill.ca/rags/nets/llsac.ps.gz}{ps.gz})

\end{itemize}
and a detailed review (also of a fair bit of plain monoidal category theory) is in

\begin{itemize}%
\item [[Paul-André Melliès]], \emph{Categorial Semantics of Linear Logic}, in \emph{Interactive models of computation and program behaviour}, Panoramas et synth\`e{}ses 27, 2009 (\href{http://www.pps.univ-paris-diderot.fr/~mellies/papers/panorama.pdf}{pdf})

\end{itemize}
Examples from algebraic geometry are given here:

\begin{itemize}%
\item Mitya Boyarchenko and Vladimir Drinfeld, A duality formalism in the spirit of Grothendieck and Verdier, arXiv:1108.6020 (\href{http://arxiv.org/pdf/1108.6020v2.pdf}{pdf})

\end{itemize}
These authors call any closed monoidal category with a [[dualizing object in a closed category|dualizing object]] a \textbf{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]]:

\begin{itemize}%
\item Ross Street, Frobenius monads and pseudomonoids, J. Math. Physics 45 (2004) 3930-3948. (\href{http://www.math.mq.edu.au/~street/Frob.pdf}{pdf})

\end{itemize}
Relation to linearly distributive categories:

\begin{itemize}%
\item [[Robin Cockett]] and [[Robert Seely]], \emph{Linearly distributive functors}, 1999 \href{https://doi.org/10.1016/S0022-4049(98}{doi}00110-8)

\end{itemize}
A wide-ranging summary of different model constructions:

\begin{itemize}%
\item [[Martin Hyland]] and [[Andreas Schalk]], \emph{Glueing and orthogonality for models of linear logic}, \href{http://www.cs.man.ac.uk/~schalk/publ/gomll.pdf}{pdf}

\end{itemize}
[[!redirects star-autonomous categories]] [[!redirects \emph{-autonomous category]] [[!redirects}-autonomous categories]] [[!redirects star-autonomous]] [[!redirects \emph{-autonomous]]}



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