# nLab star-autonomous category

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Definition

A $*$-autonomous category [1] is a symmetric closed monoidal category $⟨C,\otimes ,I⟩$ with a dualizing object: an object $\perp$ such that the canonical morphism

${d}_{A}:A\to \left(A⊸\perp \right)⊸\perp ,$d_A: A \to (A \multimap \bot) \multimap \bot ,

which is the transpose of the evaluation map

${\mathrm{ev}}_{A,\perp }:\left(A⊸\perp \right)\otimes A\to \perp ,$ev_{A,\bot}: (A \multimap \bot) \otimes A \to \bot ,

is an isomorphism for all $A$. (Here, $⊸$ denotes the internal hom.)

Define a functor $\left(-{\right)}^{*}=\left(-\right)⊸\perp$. The map ${d}_{A}$ is natural in $A$, so that there is a natural isomorphism $d:1⇒\left(-{\right)}^{**}$. We also have

$\begin{array}{rl}\mathrm{hom}\left(A\otimes B,{C}^{*}\right)& =\mathrm{hom}\left(A\otimes B,C⊸\perp \right)\\ & \cong \mathrm{hom}\left(\left(A\otimes B\right)\otimes C,\perp \right)\\ & \cong \mathrm{hom}\left(A\otimes \left(B\otimes C\right),\perp \right)\\ & \cong \mathrm{hom}\left(A,\left(B\otimes C\right)⊸\perp \right)\\ & =\mathrm{hom}\left(A,\left(B\otimes C{\right)}^{*}\right)\end{array}$\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}

The functor $\left(-{\right)}^{*}$, together with these two isomorphisms, can be taken as an alternative definition of a $*$-autonomous category. The dualizing object $\perp$ is then defined as ${I}^{*}$.

## Examples

• 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 $\mathrm{hom}\left(A,B\right)$ is itself a sup-lattice, so this category is closed. The free sup-lattice on a poset $X$ is the internal hom of posets $\left[{X}^{\mathrm{op}},\Omega \right]$; in particular the poset of truth values $\Omega$ is a unit for the closed structure. Define a duality $\left(-{\right)}^{*}$ on sup-lattices, where ${X}^{*}={X}^{\mathrm{op}}$ is the opposite poset (inf-lattices are sup-lattices), and where ${f}^{*}:{Y}^{*}\to {X}^{*}$ is the left adjoint of ${f}^{\mathrm{op}}:{X}^{\mathrm{op}}\to {Y}^{\mathrm{op}}$. In particular, take as dualizing object $D={\Omega }^{\mathrm{op}}$. Some simple calculations show that under the tensor product defined by the formula $\left(X⊸{Y}^{*}{\right)}^{*}$, 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 $A=T\left(V\right)/I$, where $V$ is a finite-dimensional vector space in degree 1, $T\left(V\right)$ 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 $\left(V,R\right)$. A morphism of quadratic algebras is a morphism of graded algebras; alternatively, a morphism $\left(V,R\right)\to \left(W,S\right)$ is a linear map $f:V\to W$ such that $\left(f\otimes f\right)\left(R\right)\subseteq S$. Define the dual ${A}^{*}$ of a quadratic algebra given by a pair $\left(V,R\right)$ to be that given by $\left({V}^{*},{R}^{\perp }\right)$ 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

$0\to {R}^{\perp }\to {V}^{*}\otimes {V}^{*}\stackrel{{i}^{*}}{\to }{R}^{*}\to 0$0 \to R^\perp \to V^* \otimes V^* \overset{i^*}{\to} R^* \to 0

Define a tensor product by the formula

$\left(V,R\right)\otimes \left(W,S\right)=\left(V\otimes W,\left({1}_{V}\otimes \sigma \otimes {1}_{W}\right)\left(R\otimes S\right)\right)$(V, R) \otimes (W, S) = (V \otimes W, (1_V \otimes \sigma \otimes 1_W)(R \otimes S))

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 $A⊸B=\left(A\otimes {B}^{*}{\right)}^{*}$, 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 space?s, 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.

## References

• Barr, Michael, $*$-Autonomous Categories. LNM 752, Springer-Verlag 1979.

More references to come…

Revised on August 22, 2012 21:58:31 by Toby Bartels (98.19.40.130)