nLab star-autonomous category

Idea

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

• it is a monoidal category in which all objects have “duals”, but in a weaker sense than in a compact closed category.
• it is a closed monoidal category in which the internal-hom can be expressed in terms of the tensor product in a particular way.
• it is a linearly distributive category in which all objects have “duals” in an appropriate linearly distributive sense.
• it is a semantics for classical multiplicative linear logic.
• it is a representable star-polycategory.
• it is a sort of categorified Frobenius algebra.

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.

(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.)

Definition

There are two useful equivalent formulation of the definition

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

$$d_A \;\colon\; A \to (A \multimap \bot) \multimap \bot ,$$

which is the transpose of the evaluation map

$$ev_{A,\bot} \;\colon\; (A \multimap \bot) \otimes A \to \bot\,,$$

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

Definition

A $*$-autonomous category is a symmetric monoidal category $\mathcal{C}$ equipped with a full and faithful functor

$$(-)^\ast \colon \mathcal{C}^{op} \longrightarrow \mathcal{C}$$

such that there is a natural isomorphism

$$\mathcal{C}(A \otimes B, C^\ast) \simeq \mathcal{C}(A, (B\otimes C)^\ast) \,.$$

Proposition

These two definitions, def. and def. , are indeed equivalent.

Proof

Given def. , define the dualization functor as the internal hom into the dualizing object

$$(-)^ \ast \coloneqq (-)\multimap\bot \,.$$

Then the morphism $d_A$ is natural in $A$, so that there is a natural isomorphism $d:1\Rightarrow(-)^{**}$. We also have

$$\begin{aligned} \hom(A\otimes B,C^*)& = \hom(A\otimes B, C\multimap\bot) \\ & \cong \hom((A\otimes B)\otimes C,\bot) \\ & \cong \hom(A\otimes(B\otimes C), \bot) \\ & \cong \hom(A,(B\otimes C)\multimap\bot) \\ & = \hom(A,(B\otimes C)^*) \end{aligned}$$

This yields the structure of def. .

Conversely, given the latter then the dualizing object $\bot$ is defined as the dual of the tensor unit $\bot \coloneqq I^*$ and the internal hom by $A \multimap B \coloneqq A^* \parr B$ (where $\parr$ is defined as below).

(See this discussion on the categories mailing list for alternative definitions and remarks about coherence.)

Remark

A $\ast$-autonomous category in which the tensor product is compatible with duality in that there is a natural isomorphism

$$(A \otimes B)^\ast \simeq A^\ast \otimes B^\ast$$

is a compact closed category. This follows from a stronger result of Dold and Puppe [DP83].

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

$$A \parr B \coloneqq (A^\ast \otimes B^\ast)^\ast \,.$$

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.

=–

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).

Theorem

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.

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 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, $F(A^*) \cong (F(A))^*$.

Properties

Internal logic

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.

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 $hom(A, B)$ 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

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

  Define a tensor product by the formula

  $$(V, R) \otimes (W, S) = (V \otimes W, (1_V \otimes \sigma \otimes 1_W)(R \otimes S))$$

  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 operads.

• 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.

• 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.

• 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.

• If $V$ is a closed monoidal category with finite products, then $V \times V^{op}$ is $*$-autonomous (Barr 1996). This is a special case of the Chu construction where the dualizing object is terminal.

• 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 properties of the original space $V$. Write $(V,V')$ for an object of $Chu(Vect,k)$ and $\langle v,w \rangle$ the evaluation of $w$ on $v$. We say that $(V,V')$ is separated if for each $v \neq 0 \in V$, there exists $v' \in V'$ such that $\langle v,v' \rangle \neq 0$ and we say that it is extensional if for each $v' \neq 0 \in V'$, there exists $v \in V$ such that $\langle v,v' \rangle \neq 0$. Then, the full subcategory $chu(Vect,k)$ of separated, extensional pairs is $*$-autonomous.

• 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 $(-\multimap\bot)\multimap\bot$, 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 not the restriction of the tensor product of $Chu(C,\bot)$.

• 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.

• A summary of many different ways to construct examples is in Hyland-Schalk.

Relation to other structures

• A $\ast$-autonomous category is a linearly distributive category with the cotensor product $x\parr y = (x^* \otimes y^*)^*$. Conversely, any linearly distributive category with “duals” in the appropriate sense is $\ast$-autonomous.

• Since a linearly distributive category is a representable polycategory, a $\ast$-autonomous category is a representable polycategory with duals. If the duals are strictly involutive, then it is a *-polycategory.

• As noted above, a compact closed category is a degenerate sort of $\ast$-autonomous category.

• It is shown in HH13 that a $\ast$-autonomous category that is traced must be compact closed.

• We can weaken the requirement that the canonical morphism $d_A: A \to (A \multimap \bot) \multimap \bot$ is an isomorphism to the one that it is a monomorphism: weak (star)-autonomous category.

References

The notion is originally due to:

• Michael Barr, $\ast$-Autonomous Categories, Lecture Notes in Mathematics 752, Springer (1979) [doi:10.1007/BFb0064579]

See also:

• Michael Barr, Heinrich Kleisli, Topological balls, Cahiers Topologie Géométrie Différentielle Catégorique, 40 1 (1999) 3–20 [numdam:CTGDC_1999__40_1_3_0]

• Michael Barr, $\ast$-Autonomous categories: once more around the track, Theory and Applications of Categories 6 1 (1999) 5-24 [tac:6-01]

• Michael Barr, Topological $\ast$-autonomous categories, revisited, rewrite of TAC, 16 (2006), 700-708 (arXiv:1609.04241)

• Michael Barr, John Kennison, Robert Raphael, On $\ast$-autonomous categories of topological modules, Theory Appl. Categories, 24 14 (2010) 278–293 [tac:24-14]

The relation to linear logic was first described in

• R. A. G. Seely, Linear logic, $\ast$-autonomous categories and cofree coalgebras, Contemporary Mathematics 92, 1989. (pdf, ps.gz)

further discussed in

• Michael Barr, $\ast$-Autonomous categories and linear logic, Math. Structures Comp. Sci. 1 2 (1991) 159–178 [doi:10.1017/S0960129500001274, pdf, pdf]

• Michael Barr, Accessible categories and models of linear logic, J. Pure Appl. Algebra 69 (1990) 219–232 [doi:10.1016/0022-4049(91)90020-3pdf, pdf]

and a detailed review (also of a fair bit of plain monoidal category theory) is in

• Paul-André Melliès, Categorial Semantics of Linear Logic, in Interactive models of computation and program behaviour, Panoramas et synthèses 27, 2009 (pdf)

Examples from algebraic geometry are given here:

• Mitya Boyarchenko and Vladimir Drinfeld, A duality formalism in the spirit of Grothendieck and Verdier, arXiv:1108.6020 (pdf)

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:

• Ross Street, Frobenius monads and pseudomonoids, J. Math. Physics 45 (2004) 3930-3948 [pdf]

Examples using toplogical vector spaces are given here:

• Michael Barr, On $*$-autonomous categories of topological vector spaces, Cahiers de topologie et géométrie différentielle catégoriques 41 4 (2000) 243-254 [numdan:CTGDC_2000__41_4_243_0]

Relation to linearly distributive categories:

• Robin Cockett, Robert Seely, Linearly distributive functors (1999) [doi:10.1016/S0022-4049(98)00110-8]

Combination with traces yields compact closure:

• Tamás Hajgató and Masahito Hasegawa, Traced $\ast$-autonomous categories are compact closed, TAC, 2013

A wide-ranging summary of different model constructions:

The fact that a $\ast$-autonomous category for which there is a natural isomorphism $(A \otimes B)^\ast \simeq A^\ast \otimes B^\ast$ is a compact closed category has a purely string diagrammatic proof, but it also follows from the implication (a)$\implies$ (b) of Theorem 1.3 in this paper:

• Albrecht Dold, Dieter Puppe, Duality, Trace and Transfer, Proceedings of the Steklov Institute of Mathematics 154 (1984) 85–103 [mathnet:tm2435, pdf, pdf]

According to Tobias Fritz (on the Category Theory Community Server):

They apparently didn’t know about $\ast$-autonomous categories (which were introduced a few years prior to that), and their statement has slightly weaker assumptions in that they start with a natural isomorphism $\mathscr{C}(A \otimes B, I) \cong \mathscr{C}(A, B^*)$ only, but they also require the induced morphisms $A^* \otimes B^* \to (A \otimes B)^*$ to be isomorphisms.

See also:

• Michael Barr, The Chu construction, Theory Appl. Categories 2 2 (1996) 17–35 [tac:2-02]

• Michael Barr, Topological $\ast$-autonomous categories, Theory Appl. Categories, 16 (2006) 700–708 [pdf, pdf]

• Michael Barr, Topological $\ast$-autonomous categories, revisited, Tbilisi Math. J. 10 3 (2017) 51–64 [arXiv:1609.04241]

• Michael Barr, John Kennison, Robert Raphael, The $\ast$-autonomous category of uniform sup semi-lattices, Theory and Applications of Categories 27 11 (2012) 222–241 [tac:27-11]

With an eye towards apication in 2d CFT (to representations of vertex operator algebras and their bimodules):

• Jürgen Fuchs, Gregor Schaumann, Christoph Schweigert, Simon Wood, Grothendieck-Verdier duality in categories of bimodules and weak module functors [arXiv:2306.17668]