nLab
star-autonomous category

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

$$d_A:A\to (A⊸\perp )⊸\perp ,$$d_A: A \to (A \multimap \bot) \multimap \bot ,

which is the transpose of the evaluation map

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

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

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

$$\begin{array}{rl}\mathrm{hom}(A\otimes B,C^{*})& =\mathrm{hom}(A\otimes B,C⊸\perp )\\ & \cong \mathrm{hom}((A\otimes B)\otimes C,\perp )\\ & \cong \mathrm{hom}(A\otimes (B\otimes C),\perp )\\ & \cong \mathrm{hom}(A,(B\otimes C)⊸\perp )\\ & =\mathrm{hom}(A,(B\otimes C{)}^{*})\end{array}$$\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 $(-{)}^{*}$, 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 $\mathrm{hom}(A,B)$ is itself a sup-lattice, so this category is closed. The free sup-lattice on a poset $X$ is the internal hom of posets $[X^{\mathrm{op}},\Omega ]$; in particular the poset of truth values $\Omega$ is a unit for the closed structure. Define a duality $(-{)}^{*}$ on sup-lattices, where $X^{*}=X^{\mathrm{op}}$ 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 $(X⊸Y^{*}{)}^{*}$, 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

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

  Define a tensor product by the formula

  $$(V,R)\otimes (W,S)=(V\otimes W,(1_V\otimes \sigma \otimes 1_W)(R\otimes S))$$(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 $A⊸B=(A\otimes B^{*}{)}^{*}$, 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 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…