With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
There are many ways to describe a -autonomous category; here are a few:
In particular, it has two monoidal structures and , as in a linearly distributive category. However, because of the dualization operation , each of them determines the other by a “multiplicative de Morgan duality”: . Thus, in the definition we only have to refer to one monoidal structure.
A -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.)
There are two useful equivalent formulation of the definition
A -autonomous category is a symmetric closed monoidal category with a global dualizing object: an object such that the canonical morphism
which is the transpose of the evaluation map
is an isomorphism for all . (Here, denotes the internal hom.)
A -autonomous category is a symmetric monoidal category equipped with a full and faithful functor
such that there is a natural isomorphism
Given def. , define the dualization functor as the internal hom into the dualizing object
Then the morphism is natural in , so that there is a natural isomorphism . We also have
This yields the structure of def. .
Conversely, given the latter then the dualizing object is defined as the dual of the tensor unit and the internal hom by (where is defined as below).
(See this discussion on the categories mailing list for alternative definitions and remarks about coherence.)
A -autonomous category in which the tensor product is compatible with duality in that there is a natural isomorphism
is a compact closed category. This follows from a stronger result of Dold and Puppe [DP83].
More generally, even if the -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 -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 but , where and 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 -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 “-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 -autonomy is required.
Let denote the full sub-2-category of on the (symmetric) -autonomous categories; hence its morphisms and 2-cells are lax symmetric monoidal functors and monoidal natural transformations. Let denote the 2-category of symmetric linearly distributive categories, symmetric linear functors, and linear transformations (defined at linearly distributive category).
There is a functor 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 that is lax monoidal for and a functor that is lax monoidal for . Recalling that is defined in a -autonomous category as , if is a lax monoidal functor between -autonomous categories we define and . See Cockett-Seely 1999 for details.
In the non-symmetric case, we need to additionally require of an “-autonomous functor” that , and define 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 -autonomous functor would need to preserve the dualization functors as well, .
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 . In this case itself plays the role of the dualizing object, so that for an f.d. vector space , is the usual dual space of linear maps into .
More generally, any compact closed category is -autonomous with the unit 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 is itself a sup-lattice, so this category is closed. The free sup-lattice on a poset is the internal hom of posets ; in particular the poset of truth values is a unit for the closed structure. Define a duality on sup-lattices, where is the opposite poset (inf-lattices are sup-lattices), and where is the left adjoint of . In particular, take as dualizing object . Some simple calculations show that under the tensor product defined by the formula , 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 is a graded algebra , where is a finite-dimensional vector space in degree 1, is the tensor algebra (the free -algebra generated by ), and is a graded ideal generated by a subspace in degree 2; so , and determines the pair . A morphism of quadratic algebras is a morphism of graded algebras; alternatively, a morphism is a linear map such that . Define the dual of a quadratic algebra given by a pair to be that given by where is the kernel of the transpose of the inclusion , i.e., there is an exact sequence
Define a tensor product by the formula
where 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 , 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 is a closed monoidal category with finite products, then 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 , then is the category of vector spaces equipped with a specified “dual” having no further structure than an evaluation map . One often wants to impose nondegeneracy conditions on this “dual”, which in turn can be reflected as topological properties of the original space . Write for an object of and the evaluation of on . We say that is separated if for each , there exists such that and we say that it is extensional if for each , there exists such that . Then, the full subcategory of separated, extensional pairs is -autonomous.
A quantale (see there) is a -autonomous category if it has a dualizing object.
Suppose is a closed symmetric monoidal category equipped with a “pre-dualizing object” , in the sense that the contravariant self-adjunction is idempotent, i.e. the double-dualization map is an isomorphism whenever is of the form . (Note that idempotence is automatic if is a poset.) Then the category of fixed points of this adjunction, i.e. the full subcategory of objects of the form , is -autonomous. For it is closed under , as , and reflective with reflector , and it contains since . Hence it is closed symmetric monoidal with tensor product , 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 closed under duality — indeed, it is the intersection of the two embeddings of and therein — but its tensor product is not the restriction of the tensor product of .
Phase semantics is also a special case of a ternary frame, which is the case of the previous example when 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 is a semicartesian -autonomous poset with the monoidal structure and dualizing object . The involution is and the dual multiplicative disjunction is . 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.
A -autonomous category is a linearly distributive category with the cotensor product . Conversely, any linearly distributive category with “duals” in the appropriate sense is -autonomous.
Since a linearly distributive category is a representable polycategory, a -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 -autonomous category.
It is shown in HH13 that a -autonomous category that is traced must be compact closed.
We can weaken the requirement that the canonical morphism is an isomorphism to the one that it is a monomorphism: weak (star)-autonomous category.
The notion is originally due to:
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, -Autonomous categories: once more around the track, Theory and Applications of Categories 6 1 (1999) 5-24 [tac:6-01]
Michael Barr, Topological -autonomous categories, revisited, rewrite of TAC, 16 (2006), 700-708 (arXiv:1609.04241)
Michael Barr, John Kennison, Robert Raphael, On -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
further discussed in
Michael Barr, -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
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:
Examples using toplogical vector spaces are given here:
Relation to linearly distributive categories:
Combination with traces yields compact closure:
A wide-ranging summary of different model constructions:
The fact that a -autonomous category for which there is a natural isomorphism is a compact closed category has a purely string diagrammatic proof, but it also follows from the implication (a) (b) of Theorem 1.3 in this paper:
According to Tobias Fritz (on the Category Theory Community Server):
They apparently didn’t know about -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 only, but they also require the induced morphisms to be isomorphisms.
See also:
Michael Barr, The Chu construction, Theory Appl. Categories 2 2 (1996) 17–35 [tac:2-02]
Michael Barr, Topological -autonomous categories, Theory Appl. Categories, 16 (2006) 700–708 [pdf, pdf]
Michael Barr, Topological -autonomous categories, revisited, Tbilisi Math. J. 10 3 (2017) 51–64 [arXiv:1609.04241]
Michael Barr, John Kennison, Robert Raphael, The -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):
Last revised on April 24, 2024 at 01:49:13. See the history of this page for a list of all contributions to it.