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A survey of graphical languages for monoidal categories

Peter Selinger

arXiv:0908.3347v1math.CT

TL;DR

Different monoidal-category notions and overlapping terminology make their associated string diagrams difficult to navigate. This survey systematically organizes these notions and graphical languages, presenting coherence results while marking cases with limited literature support.

  • Problem

    The proliferation of monoidal-category notions and multiple names for the same concept makes their associated graphical languages confusing to non-experts and sometimes experts.

  • Method

    The survey systematically overviews main notions, associated graphical languages, and coherence results, adding further notions where needed for systematic coverage.

  • Results

    The survey presents coherence results showing that graphical equations characterize the corresponding categorical axioms in several settings, including planar monoidal categories and free monoidal categories.

  • Takeaways & Limitations

    The survey serves as a reference guide to monoidal categories and string diagrams while indicating where results are established only in special cases.

  • Takeaways & Limitations

    Some general coherence cases remain unsupported in the literature, with a cited general case not appearing in prior work.

Abstract

from arXiv · show

This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs. Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study. Where possible, we provide pointers to more rigorous treatments in the literature. Where we include results that have only been proved in special cases, we indicate this in the form of caveats.

1 Introduction

The survey systematically organizes monoidal categories and their graphical languages while addressing confusing terminology and presenting diagrams as a generalization of index notation. It is detailed enough for applications but intentionally informal about some technical definitions and proofs.

  • Terminological variation complicates the subject because equivalent notions may have different names, such as rigid and autonomous.
  • The survey gives a systematic overview of major monoidal-category notions and their associated graphical languages.
  • The survey retains established terminology while proposing the more flexible term “traced category” for qualified traced notions.
  • Some coherence results are reported with caveats, and unproven results are included as conjectures.
  • Graphical languages are presented informally rather than with full technical definitions and diagram-equivalence details, with references supplied where appropriate.
  • Graphical notation generalizes index-pairing diagrams from linear algebra to settings where indices are formal and maps need not be between vector spaces.

2 Categories

Categories consist of objects, morphisms, identities, and composition, and their graphical language represents these as wires, boxes, and connections. Coherence identifies diagrammatic equality with equality derived from the categorical axioms, while signatures and freeness formalize the correspondence.

  • A category comprises objects, morphisms between objects, identity morphisms, and composition satisfying identity and associativity equations.
  • In categorical diagrams, objects are wires, morphisms are boxes, identities are continuing wires, and composition connects outgoing to incoming wires.
  • Coherence means that an equation follows from the categorical axioms exactly when it holds in the graphical language up to diagram isomorphism.
  • Signatures, variables, terms, and equations: A signature labels wires with object variables and boxes with morphism variables whose domain and codomain determine valid connections.
  • Coherence and free categories: The graphical language over a signature forms the free category, so equations valid in all categories correspond to equations in the graphical language.

3 Monoidal categories

Monoidal categories provide a tensor product and unit, represented graphically by parallel wires, boxes, and stacking. Their coherence theorems characterize axiomatic equality through diagram isotopy, with extensions to spacial, braided, balanced, and symmetric settings.

  • Monoidal categories add an associative unital tensor product to a category, with object unit I and corresponding morphism operations.
  • Graphically, tensor products are parallel wires, the unit is zero wires, morphisms are boxes, and tensoring morphisms stacks diagrams.
  • Planar coherence holds exactly when well-formed morphism equations hold up to planar isotopy, subject to the stated recumbent-isotopy caveat.The proof technically applies only when intermediate diagrams remain progressive; arbitrary planar deformations are conjectured to work as well.
  • The general coherence theorem is stronger than traditional commutativity results because it characterizes all formal equations implied by the axioms.
  • The graphical language over a monoidal signature forms a free monoidal category up to planar isotopy.
  • Braided, balanced, symmetric, and spacial variants use respectively three-dimensional isotopy, framed isotopy, diagram isomorphism, and a stated conjectural coherence result.

4 Autonomous categories

Autonomous categories equip monoidal categories with left and right duals, represented by wires that may reverse direction and by half-turn units and counits. Their coherence is characterized by planar isotopy, including one-sided autonomous categories via embedding.

  • Autonomous graphical languages represent duals by wires running right-to-left and units and counits by half turns or multiple turning wires.
  • An exact pairing consists of unit and counit morphisms satisfying two adjunction triangles, making the paired objects duals.
  • Right autonomous categories give every object a right dual, left autonomous categories give every object a left dual, and autonomous categories have both.
  • For any morphism, adjoint mates reverse the direction of the corresponding dual morphisms.
  • Planar autonomous coherence equates axiomatic morphism equations with graphical equality up to planar isotopy, without allowing box rotations.
  • Right and left autonomous categories embed fully and faithfully into autonomous categories, extending the same coherence theorem to one-sided cases.

Technicalities

The survey defines autonomous signatures and develops graphical coherence results for autonomous and pivotal categories, while marking several results as proved only in restricted cases.

  • Autonomous categories: An autonomous signature comprises object variables, morphism variables, and domain and codomain functions into the algebra of object terms.Object terms are generated using tensor, the unit, and left and right dual operations.
  • Functoriality: Monoidal functors preserve exact pairings and induce canonical isomorphisms between the image of a dual and the dual of the image.Monoidal natural transformations between strong monoidal functors are constrained by duality, and in autonomous categories they are invertible.
  • Autonomous categories: Winding numbers label autonomous diagrams so that, up to planar isotopy, they form the free autonomous category over an autonomous signature.The labels encode object variables and edge winding, with parity determined by segment orientation.
  • Pivotal categories: A pivotal category is autonomous with an isomorphism A → A∗∗, represented graphically as an identity and reducing winding numbers modulo 2.Its coherence theorem uses planar isotopy including box rotations, but only special cases have been proved in the literature.
  • Pivotal categories: Pivotal diagrams permit rotated boxes, which represent adjoint mates and can rotate composite diagrams to obtain dual morphisms.The graphical language retains a marked box corner to track natural orientation under rotations.

4.3 Spherical pivotal categories

The survey distinguishes spherical, spacial, and braided autonomous structures by the isotopy notions their graphical languages support, including a conjectural spacial coherence result.

  • Spherical pivotal categories: Spherical axioms express the intuition that diagrams live on a 2-sphere, where loops can move across the sphere’s back.This motivates identifying the two sides of the spherical axiom geometrically.
  • Spherical pivotal categories: The spherical axiom is unsound for diagrams embedded in the 2-sphere because that embedding notion is incompatible with composition or tensor.A consequence of the axiom does not hold up to isotopy in the 2-sphere.
  • Spacial pivotal categories: Adding the spacial axiom causes equivalence of diagrams to collapse to isomorphism rather than an isotopy notion.Spacial pivotal categories satisfy both the spacial and spherical axioms.
  • Spacial pivotal categories: For spacial pivotal categories, the graphical language uses planar pivotal diagrams with equivalence taken up to isomorphism, and completeness is conjectured.Soundness is clear from the axioms, whereas the converse remains Conjecture 4.16.
  • Braided autonomous categories: Braided autonomous categories are autonomous exactly when they are right autonomous, but their induced A∗∗ ≅ A is generally noncanonical and does not define a pivotal structure.Thus a general braided autonomous category need not be pivotal.
  • Braided autonomous categories: Braided autonomous coherence identifies well-formed equations with graphical equality up to regular isotopy, not planar or full three-dimensional isotopy.Regular isotopy omits Reidemeister move R1; the theorem is stated with a caveat that only simple signatures have been proved.

4.6 Braided pivotal categories

Braided pivotal categories identify twists with pivotal structures and obtain regular-isotopy coherence, while regular isotopy excludes some isotopic equations and leaves the notion non-spherical in general.

  • Structures: In a braided autonomous category, giving a twist is equivalent to giving a pivotal structure, so braided pivotal categories are the same as balanced autonomous categories.The equivalence relates structures defined from the braided and autonomous components, respectively.
  • Structures: The twist–pivotal correspondence is not canonical, and multiple related correspondences coincide exactly in tortile categories.The survey notes a countable family of similar one-to-one correspondences.
  • Graphical language and coherence: Braided pivotal coherence characterizes well-formed equations by graphical equality up to regular isotopy.The theorem carries a literature caveat: only simple-signature cases have been proved.
  • Graphical language and coherence: Some equations hold under regular isotopy using Reidemeister moves R2 and R3, whereas others require isotopy and therefore fail in braided pivotal categories.Regular isotopy preserves total curvature, motivating the introduction of tortile categories.
  • Graphical language and coherence: A braided pivotal category is not generally spherical or spacial, supporting the survey’s assessment that these structures are not natural in general.Only a weaker relation holds up to regular isotopy instead of the spherical axiom.

4.7 Tortile categories

Tortile categories refine braided pivotal structure so that ribbon diagrams support framed three-dimensional isotopy, with coherence established in the survey subject to a restricted-proof caveat.

  • Definition: A tortile category is a braided pivotal category satisfying the condition that makes its alternative twist definitions coincide.Equivalently, it is a balanced autonomous category satisfying the corresponding condition; it is also called a ribbon category.
  • Graphical language: Tortile graphical languages represent morphisms by ribbons rather than wires, extending the braided pivotal graphical language.Units and counits receive the corresponding ribbon representations.
  • Graphical language: The twist map has several equivalent graphical representations, but in a merely braided pivotal category the latter representations need not be equal.They become equal in a tortile category under framed three-dimensional isotopy.
  • Coherence: Tortile coherence identifies well-formed equations with graphical equality up to framed 3-dimensional isotopy.The cited theorem has only been proved for diagrams over simple signatures in the literature.

4.8 Compact closed categories

Compact closed categories are characterized as right autonomous symmetric monoidal categories, with graphical languages governed by symmetries rather than braidings. The survey states a coherence theorem for these diagrams while flagging that the general case is not established in the literature.

  • A compact closed category is defined as a right autonomous symmetric monoidal category.
  • Symmetry does not generally force the twist to be trivial, although it does imply θ_A^2 = id_A.
  • Compact closed categories include Rel, finite-dimensional vector spaces, finite-dimensional Hilbert spaces, and oriented cobordisms as examples.
  • Their graphical language removes framing and twist maps from tortile diagrams and uses symmetries instead of braidings.
  • A coherence theorem states that equations follow from the axioms exactly when they hold up to isomorphism of diagrams.
  • The general coherence result is caveated because the literature proves only the special case of diagrams over a simple signature.

5 Traced categories

Traced categories extend monoidal graphical languages with loops under progressively different directional and topological constraints. The survey develops right, planar, braided, balanced, and symmetric traced notions, with coherence results ranging from conjectures to theorems.

  • Traced categories: Traced diagrams permit loops, but their wires must point left-to-right at the endpoints, unlike autonomous diagrams.
  • Right traced categories: Right traced categories add a right trace to a monoidal category, represented graphically by looping an output X back to an input X.
  • Right traced categories: The right-traced coherence statement remains a weak conjecture because there is little empirical evidence and no obvious proof strategy.
  • Planar traced categories: Planar traced categories combine left and right traces with additional interchange and pivoting axioms, and their coherence remains conjectural up to planar isotopy.
  • Braided traced categories: Braided traced categories have coherence up to regular isotopy, and the survey proves both a fully faithful embedding into braided pivotal categories and a coherence theorem.
  • Balanced and symmetric traced categories: Balanced traced categories are both spacial traced and braided traced, and symmetric traced categories embed into compact closed categories with coherence up to diagram isomorphism.
  • Balanced traced categories: Balanced traced categories embed fully and faithfully into tortile categories, while their coherence is characterized by framed isotopy in 3 dimensions.

6 Products, coproducts, and biproducts

The section extends graphical languages from monoidal categories to products, coproducts, and biproducts, where diagrammatic equations become essential. It also characterizes traced variants through iteration, fixed-point, and repetition operators, while noting limits on generality.

  • Products: A finite product category consists of a chosen terminal object and chosen products, equivalently a symmetric monoidal category with natural copy and erase maps.Products provide projections and a unique pairing morphism satisfying the universal property.
  • Products: Graphical languages for finite product categories require diagrammatic equations in addition to isomorphism or isotopy of diagrams.Copy and erase maps extend the symmetric monoidal graphical language, and Table 8 represents selected product axioms.
  • Products: Coherence for finite product categories holds exactly up to diagram isomorphism and the diagrammatic manipulations listed in Table 8.The result follows from symmetric monoidal coherence together with the additional product axioms represented diagrammatically.
  • Coproducts: Finite coproduct categories and their graphical languages are obtained by reversing the arrows, maps, and axioms used for products.Coproducts use an initial object, injections, merge maps, and initial maps in the dual presentation.
  • Biproducts: Biproduct categories combine product and coproduct structures, with π_i ◦ ι_j = δ_ij ensuring that their symmetric monoidal structures coincide.Their graphical language combines the product and coproduct languages, with the relevant equalities and dual axioms.
  • Biproducts and traced variants: Coherence for biproducts and traced product, coproduct, and biproduct categories is characterized by diagram isomorphisms plus the product axioms and their duals.For biproducts, the relevant manipulations are those in Table 8 and their duals; traced variants add the applicable diagrammatic equations.
  • Traced variants: In finite coproducts, traces are equivalent to iteration operators, while in finite products they are equivalent to fixed-point operators and in biproducts to repetition operations.These correspond to control-flow iteration, data-flow fixed points, and the analogous biproduct operation.
  • Uniformity and regular trees: Uniform tracing is characterized by compatibility conditions for strict morphisms in traced coproduct and biproduct categories.For coproducts, the condition relates iter_X(f) and iter_Y(g); for biproducts, it relates f* and g* through h.

7 Dagger categories

The section develops dagger structures for categories and monoidal categories, representing adjoints as mirror images in graphical languages. It establishes coherence results for several dagger variants and identifies relationships among dagger autonomous, pivotal, braided, traced, and biproduct settings.

  • 7.1 Dagger categories: A dagger category equips a category with an involutive, identity-on-objects, contravariant functor, sending each morphism f : A → B to an adjoint f † : B → A.Hilbert spaces provide the motivating example, while unitary and self-adjoint morphisms are defined using the dagger.
  • 7.1 Dagger categories: The adjoint of a diagram is represented by its mirror image, with wires reversed in location but not direction.This graphical extension distinguishes mirrored morphism-variable boxes by marking their upper-left corners.
  • 7.2 Dagger monoidal categories: Coherence holds for dagger categories up to isomorphism of diagrams and for planar dagger monoidal categories up to planar isotopy.The proof removes dagger operations except when they apply directly to morphism variables, reducing to ordinary category or monoidal coherence.
  • 7.2 Dagger monoidal categories: Dagger monoidal categories require the dagger to be compatible with the monoidal structure, including unitary associators and unitors.Their graphical language extends monoidal diagrams, with adjoints again represented by mirror images.
  • 7.2 Dagger progressive notions: Dagger variants of braided, balanced, and symmetric monoidal categories require unitary braidings, twists, or symmetries, and inherit corresponding coherence theorems.The unitarity requirements allow dagger operations to be removed from terms except at morphism variables; the graphical dagger remains mirror reflection.
  • 7.3 Dagger autonomous and pivotal categories: A dagger structure on an autonomous category induces a canonical A ∼= A∗∗ isomorphism, so dagger autonomous and dagger pivotal categories coincide under mild assumptions.Equivalent characterizations involve unitarity of canonical duality isomorphisms and the relation f ∗† = f †∗; dagger pivotal coherence includes box rotations by multiples of 180 degrees.
  • 7.4–7.6 Further dagger notions: Dagger coherence extends across the surveyed notions, while dagger braided, balanced, and symmetric traced categories embed fully and faithfully into corresponding pivotal, tortile, and compact closed categories.Dagger biproduct categories have coherence up to diagram isomorphism together with the listed diagrammatic manipulations and their duals.

8 Bicategories

Bicategories generalize monoidal categories by adding colored 0-cells that constrain how objects, tensors, morphisms, and units may be composed. A monoidal category is recovered as a one-object bicategory, while the survey does not develop the many higher-dimensional variants in detail.

  • Bicategorical structure: A bicategory adds 0-cells, visualized as colors, and assigns each object a source and target 0-cell.Diagrams must satisfy coloring constraints so that object and morphism compositions are well-typed.
  • Coloring constraints: Objects may tensor only when the target of the first object equals the source of the second, and morphisms require source and target agreement.Each color also has a unit object Iα : α → α.
  • Relation to monoidal categories: Subject to these coloring constraints, bicategories satisfy the same associativity, unit, identity, and composition axioms as monoidal categories.A monoidal category is exactly a one-object bicategory.
  • Scope: The survey omits a detailed account of bicategories because their variants include 2-, 3-, and 4-dimensional graphical languages, with further extensions for tricategories and beyond.These variations arise from bicategorical versions of braids, twists, adjoints, and traces.

9 Beyond a single tensor product

Categories with multiple tensor products require substantially more elaborate graphical languages than the single-tensor settings surveyed earlier. The section outlines the additional nodes, labeling, equivalences, and correctness criteria needed for such diagrams.

  • Multiple tensor products: Multiple-tensor graphical languages are more complicated, and the survey gives only an outline with references.Linearly distributive and *-autonomous categories are examples, and both model multiplicative linear logic.
  • Multiple tensor products: The two tensors, commonly called tensor and par and written A ⊗ B and A # B, are related by some morphisms but not their converses.The asymmetry of these relationships is part of the structure that the graphical language must represent.
  • Graphical language: Diagrams with multiple tensors label wires by morphism terms and introduce distinct tensor, par, and unit nodes.Diagram equivalence includes axiomatic manipulations such as cut elimination.
  • Correctness: A correctness criterion is required to distinguish well-formed proof-net diagrams from diagrams that are not well formed.The resulting theory is proof nets, first developed for unit-free multiplicative linear logic and later extended to tensor units.

10 Summary

The summary table organizes monoidal notions by their graphical languages, diagram dimensions, isotopy conventions, and coherence status. It also records important omissions and scope boundaries for dagger variants and bicategories.

  • Table 10: Table 10 lists each class of monoidal categories alongside a typical diagram or equation and indicates the forgetful functors between classes.Spherical categories are omitted because they lack a graphical language modulo a natural notion of isotopy.
  • Table 10: The table distinguishes diagram dimension from ambient isotopy dimension and marks special cases including framed, crossing, regular, and rotational isotopy.When ambient dimension exceeds diagram dimension, isotopy coincides with diagram isomorphism.
  • Table 10: Coherence status records whether a theorem is proved, conjectured, folklore or trivial, obtained by Int-construction, or proved only for simple signatures.These labels make the evidential status of the surveyed graphical languages explicit.
  • Scope boundaries: Dagger variants exist for all notions shown in the table except planar autonomous and braided autonomous notions, while bicategories require a separate larger table.The bicategory omission reflects the number of possible higher-dimensional variants.
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