aa r X i v : . [ m a t h . C T ] J a n The smash product of monoidal theories
Amar Hadzihasanovic
Tallinn University of Technology
Abstract.
The tensor product of props was defined byHackney and Robertson as an extension of the Boardman–Vogt product of operads to more general monoidal theor-ies. Theories that factor as tensor products include thetheory of commutative monoids and the theory of bialgeb-ras. We give a topological interpretation (and vast gen-eralisation) of this construction as a low-dimensional pro-jection of a “smash product of pointed directed spaces”.Here directed spaces are embodied by combinatorial struc-tures called diagrammatic sets, while Gray products replacecartesian products. The correspondence is mediated by aweb of adjunctions relating diagrammatic sets, pros, probs,props, and Gray-categories. The smash product applies topresentations of higher-dimensional theories and systemat-ically produces higher-dimensional coherence cells.
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Contents
1. Some higher structures . . . . . . . . . 9
2. Monoidal theories . . . . . . . . . . . . . . .
3. Combinatorial results . . . . . . . . . . .
4. Pros and diagrammatic sets . . .
5. The smash product . . . . . . . . . . . . .
6. Higher-dimensional cells . . . . . . . Introduction
In a categorical tradition of universal algebra dating back to F. WilliamLawvere’s thesis [Law63], algebraic theories are embodied by cartesian mon-oidal categories whose objects are freely generated from a set of sorts. Models amar hadzihasanovic of the theory, embodied by strong monoidal functors, may live in an arbit-rary cartesian monoidal category: we specialise to the category of sets andfunctions to recover the classical notion of model.Following this fundamental shift in perspective, and considering that cartesian-ness of a monoidal structure can be defined equationally [Fox76], it is a relat-ively small step to consider more general monoidal theories whose models livein arbitrary monoidal categories.Monoidal theories are embodied by structures called pros [ML65]. Interme-diate between monoidal theories and algebraic theories, there are braided and symmetric monoidal theories, embodied respectively by probs and props . Thefamiliar term-algebraic calculus is inadequate for these generalised theories,and is commonly replaced by a calculus of string diagrams [Sel10].It is common for a mathematical object to have both a structure of T -modeland of S -model for some theories T, S , satisfying some compatibility condi-tion. For example, a bimodule is both a left and a right module, in such away that the left and right actions commute. A natural question is: can wesystematically compose theories, so that a model of the composite of T and S is an object with compatible T and S -model structures?In a line of work that has been attracting attention in theoretical computerscience [BSZ14], composition of monoidal theories is mediated by distributivelaws which specify a factorisation system between operations of T and S , asdescribed by Steve Lack [Lac04].A less flexible, yet more uniform composition is the tensor product of props ,which applies to all props and does not require additional data. The tensorproduct was defined and studied by Philip Hackney and Marcy Robertson[HR15], who also proved that it extends, in a precise sense, the product ofsymmetric operads introduced by J. M. Boardman and R. M. Vogt [BV06].Some intuition about the tensor product may be gained as follows. If M is a symmetric monoidal category, the category of T -models in M inherits asymmetric monoidal structure: to compose two models, we “run their opera-tions in parallel”, using the symmetric structure of M to rearrange inputs andoutputs as needed. The data of a model in M of the tensor product T ⊗ S S isequivalent to the data of an S -model in the category of T -models in M . As remarked in [Lac04, §4.2], there is something mysterious about the rôle ofsymmetric braidings in the composition of monoidal theories. From a certain In some sources, the term pro or PRO is reserved for a one-sorted theory, and themulti-sorted variant is called a coloured pro. Although its possibility was noticed earlier by John C. Baez [Bae06]. Baez’s lectures arealso a nice survey of the relations between pros, probs, props, and algebraic theories. Or, symmetrically, a T -model in the category of S -models in M . he smash product of monoidal theories perspective, a symmetric braiding is just another operation in a pro, yet itplays an inescapable structural rôle in the tensor product.Consider the theories of monoids and comonoids . These are planar monoidaltheories, naturally embodied by a pro: in the corresponding prop, symmetricbraidings are added freely, so models in the sense of props are equivalent tomodels in the sense of pros. Nevertheless, their tensor product — the theoryof bialgebras — features the non-planar equation= (1)where the symmetric braiding in the left-hand side cannot clearly be attributedto either factor.In particular, the tensor product of props does not restrict to a monoidalstructure on pros. At most, as shown in Section 5.1, we can define an “ex-ternal” tensor product which takes two pros and returns a prob, from whichwe can then universally reconstruct the tensor product of props.A few years ago, we noticed that equation (1) admits the following topo-logical interpretation. Take the string diagrams corresponding to monoidmultiplication and comonoid comultiplication, and extend them along per-pendicular directions in the plane so that they form branching surfaces :, .Intersect the two branching surfaces and “slide” one past another along thevertical axis. As one branching slides past the other branching, the intersection— a “string diagram in 3-dimensional space” — evolves as in the followingfigure: . (2)The two sides of (1) arise as planar projections of the two sides of (2). A similar observation was made, around the same time, by J. Scott Carter [Car18]. amar hadzihasanovic
This interpretation extends to all “compatibility” equations in the tensorproduct of props, and recasts the tensor product as a dimension-raising con-struction: given planar diagrams, it produces equations of 3-dimensional dia-grams. This solves our conundrum about braidings: they are absent in the3-dimensional picture, and only appear in the 2-dimensional picture as anartefact of planar projection.What is going on? As first suggested in [Had17, Section 2.3], the correctinterpretation of (2) is that it arises from a smash product of pointed directedspaces , in a sense that we will soon explain.Our model of directed space is a diagrammatic set [Had20b]. We developedthe theory of diagrammatic sets partly as a foundation for this work, whichrequires the ability to do rewriting and diagrammatic reasoning in weak highercategories of arbitrary dimension, to an extent that pre-existing frameworksdid not seem to support.The aim of this article is the statement and proof of Theorem From these, thesmash product generates non-invertible higher-dimensional cells rather thanequations.Already when applied to 3-dimensional presentations of monoidal theories[Mim14], not only this construction produces a presentation of their tensorproduct, that is, it produces oriented equations, or rewrites; it also producesinteresting higher-dimensional coherence cells, or syzygies, up to dimension 6.In higher-dimensional rewriting and universal algebra [GM16, Gui19], co-herence is usually pursued with analytic methods of rewriting theory such asthe computation of critical branchings. We believe that our results may be agateway to new synthetic and compositional methods. As opposed to structures used in homotopical algebra, such as ∞ -operads, that embodytheories with invertible higher data. he smash product of monoidal theories Monoidal theories, directed spaces, and diagrammatic sets
The connection between monoidal theories and directed spaces is based onfour conceptual leaps. The first leap, as mentioned, is the realisation thatmonoidal categories can embody algebraic theories.The second leap is John C. Baez and James Dolan’s formulation of the periodic table of n -categories , by which a monoidal category is equivalent to abicategory with a single 0-cell, but a braided monoidal category is equivalentto a tricategory with a single 0-cell and 1-cell [BD95]. A variant of this resultimplies that a pro is a special kind of 2-category, while a prob is a special kindof Gray-category, a semistrict notion of tricategory [GPS95]. This matchesthe intuition that tensoring pros to obtain a prob is dimension-raising.The third leap is Grothendieck’s homotopy hypothesis , that “spaces”, moreprecisely homotopy types, are modelled in a precise combinatorial sense byhigher groupoids. In models where higher groupoids are higher categorieswhose cells are all invertible in a weak sense, this leaves open the possibilityof interpreting higher categories as “spaces of directed cells”.The fourth and final leap is Albert Burroni’s observation that various notionsof presentations by generators and relations, or rewrite systems, can be unifiedas presentations of “cell complexes in a category of higher categories”, a notionof directed space with combinatorial structure [Bur93].Following the sequence, we can reinterpret a monoidal theory with its setof sorts as a kind of directed 2-dimensional space containing a 1-dimensionalcell complex. A braided monoidal theory is the same thing one dimension up.These spaces are canonically pointed with the unique 0-cell in the cell com-plex structure. It is natural, at this point, to wonder about a directed coun-terpart of the classical smash product of pointed spaces. The correct gener-alisation replaces the cartesian product of spaces with a version of the Grayproduct [Gra74].In [Had17, Section 2.3], we considered smash products in the context ofBurroni and Ross Street’s theory of polygraphs, based on strict ω -categor-ies. This had the advantage that a theory of Gray products had already beendeveloped [Ste04, AM20], and that we could identify a pro directly with apointed 2-category. However, in this context the smash product of pros pro-duces a strict 3-category equivalent not to a braided monoidal category, butto a highly degenerate commutative monoidal category. In [Had20b], based on an abandoned idea of Mikhail Kapranov and VladimirVoevodsky, we developed the theory of diagrammatic sets as an alternative to This is connected to the known failure of the homotopy hypothesis for strict ω -categories,see [Sim09, Chapter 4]. amar hadzihasanovic polygraphs that would avoid this pitfall and support rewriting and diagram-matic reasoning in weak higher categories.While the model of a directed cell in a polygraph is algebraic, diagrammaticsets adopt a combinatorial model. Roughly, a model of a directed n -cell isthe face poset of a regular CW-decomposition of the topological closed n -ball,together with an orientation subdividing the boundary of each cell into an input and output half, in such a way that the input and output half are alsoface posets of regular CW-balls, and their orientations determine a composablepasting diagram in a strict ω -category. Common higher-categorical shapessuch as oriented simplices and cubes appear as special cases.A pleasant outcome of the transition to the combinatorial setup is that Grayproducts and smash products are much easier to define and compute. On theother hand, the identification of pros or probs with certain pointed diagram-matic sets is non-trivial. The technical core of this article is the definitionof a full and faithful diagrammatic nerve of pros (Section 4.1), and then of anon-trivial realisation functor of diagrammatic sets in Gray-categories (Sec-tion 4.2), which allows us to recover the tensor product of two pros as therealisation of the smash product of their nerves. Diagrammatic sets are related to (nice) topological spaces by a nerve andrealisation pair [Had20b, Section 8.3], where the nerve realises a version of thehomotopy hypothesis. As detailed in Section 5.2, the geometric realisationsends Gray products to cartesian products, so it sends smash products tosmash products.Altogether, our results amount to the surprising fact that the tensor productof pros and the smash product of pointed spaces are two facets of the sameconstruction.Related work
We have paid tribute to our main influences on the conceptual side. On theother hand, this article is technically most indebted to three sources.The first is Hackney and Robertson’s article on the category of props [HR15]:beyond the fact that they defined the tensor product of props, our proofsin Section 2 that certain categories of pros and probs have small limits andcolimits are essentially lifted from their work on props, with minor tweaks.The second is John Power’s work on pasting diagrams [Pow91]. While ourformalisation of diagrams is based on Richard Steiner’s combinatorial frame- There is also a “trivial” functor passing through strict 3-categories. More precisely, of their nerves with an orientation reversal in the second factor, asexplained in Section 5.3. he smash product of monoidal theories work [Ste93], an analogue of Power’s domain replacement condition turns outto be key to the constructions of Section 4, and the technical Section 3 isdevoted to showing that it holds for all our 3-dimensional diagrams.In particular, our Theorem 3.24 is roughly equivalent in meaning to [Pow91,Theorem 4.14]. Interestingly, though, Power’s topological setup seems to havecompletely different strengths and weaknesses compared to our combinator-ial setup: when translating Power’s proofs, we discovered that every singlenon-trivial step in his proofs followed easily from our definitions, whereas thetrivial steps would require non-trivial proofs, as in Proposition 3.21. No formalcomparison has been made, to our knowledge, between Steiner’s and Power’stheory, so we think it is justified to consider our results original.The third is Simon Forest and Samuel Mimram’s article on the rewritingtheory of Gray-categories [FM18]. Not only we learnt from them a convenientaxiomatisation of Gray-categories, but the construction of Section 4.2 drawsdirectly on their ideas and results and can be seen as a continuation of theirwork, showing that every diagrammatic set presents a Gray-category. Structure of the article
Most of the article is aimed at the proof of Theorem 5.29.The statement of this result involves many different structures, related viaa number of ancillary structures, each in need of definition. Some of these areobscure enough that basic technical aspects could not be found in the literatureand had to be developed expressly. That said, we tried to keep redundancyto a minimum by treating a structure as a special case of another wheneverpossible, even if it results in unconventional choices, such as the definition ofreflexive ω -graphs after diagrammatic sets.Section 1 recaps the elementary theory of directed complexes, diagrammaticsets, and strict ω -categories. Section 2 introduces categories of pros, probs,and props, proves some of their properties, and clarifies the relation betweenprobs and Gray-categories. Section 3 proves some technical results aboutdirected complexes in low dimension. Section 4 is the technical core of thearticle, constructing the adjunctions that relate diagrammatic sets, pros, andGray-categories. Section 5 defines the tensor product of props and the smashproduct of pointed diagrammatic sets, then proves the main theorem. Section6 takes the first steps into diagrammatic sets as a framework for higher-di-mensional rewriting and universal algebra.Every reader should get at least acquainted with the definitions in the firsttwo sections. On a first read, they can then skip to Section 5, using thediagram that concludes Section 4 as a reference: most of the time, knowing amar hadzihasanovic that certain functors exist and are left or right adjoints should be enough tofollow the outline of the proof.Some readers may be content with understanding the picture (2) and wantto stop there. Those interested in higher-dimensional rewriting and universalalgebra should move on to Section 6.Section 3 may appeal to the reader who appreciates the combinatorics ofhigher-categorical diagrams. The reader who enjoyed Forest and Mimram’s[FM18] can read Section 4 as a follow-up of sorts.We use the diagrammatic order f ; g for the composition of morphisms f and g in a category, but the “classical” order GF for the composition of functors F and G . Other notational choices are explained when they are introduced. Outlook and open problems
Section 6 is an extended outlook towards our main prospect, namely, theintroduction of new compositional methods in higher-dimensional rewritingand universal algebra.We briefly mention other potential developments. Christoph Dorn, DavidReutter, and Jamie Vicary have defined a semistrict algebraic model of n -cat-egories, called associative n -categories , which is equivalent to Gray-categoriesfor n = n -categories for n >
3. We note, however, that our construction uses a prop-erty, frame acyclicity, which holds in general up to dimension 3 but fails indimension 4 or higher, so it is likely that new ideas will be needed.The theory of diagrammatic sets is based on simple data structures: a cellmodel U can be encoded as the directed graph H o U of §3.12 together with agrading of its vertices; the Gray product is then encoded as a cartesian productof directed graphs, while the degrees of vertices are summed. We expect thatthis setup should lend itself to computational formalisation.This is of particular interest considering that the theory of associative n -cat-egories is formalised in the graphical proof assistant homotopy.io : implement-ing the constructions of Section 4.2 would give us access to visualisations ofGray and smash products through this graphical frontend. With some edges reversed, depending on the degree of first factor. he smash product of monoidal theories
1. Some higher structures
We quickly go through the main definitions, and refer the reader to [Had20b]for an in-depth development. (Graded poset) . Let P be a finite poset with order relation ≤ . For allelements x, y ∈ P , we say that y covers x if x < y and, for all y ′ ∈ X , if x < y ′ ≤ y then y ′ = y .The Hasse diagram of P is the finite directed graph H P with H P := P as set of vertices and H P := { y → x | y covers x } as set of edges.Let P ⊥ be P extended with a least element ⊥ . We say that P is graded if,for all x ∈ P , all directed paths from x to ⊥ in H P ⊥ have the same length.If this length is n + 1, we let dim( x ) := n be the dimension of x . (Closed and pure subsets) . Let P be a poset and U ⊆ P . The closure of U is the subset cl U := { x ∈ P | ∃ y ∈ U x ≤ y } of P . We say that U is closed if U = cl U .Suppose P is graded and U ⊆ P is closed. Then U is graded with the partialorder inherited from P . The dimension dim( U ) of U is max { dim( x ) | x ∈ U } if U is inhabited, − { x } ) = dim( x ).We say that U is pure if its maximal elements all have dimension dim( U ). (Oriented graded poset) . An orientation on a finite poset P is an edge-labelling o : H P → { + , −} of its Hasse diagram.An oriented graded poset is a finite graded poset with an orientation. . We will often let variables α, β range implicitly over { + , −} . (Boundaries) . Let P be an oriented graded poset and U ⊆ P a closedsubset. Then U inherits an orientation from P by restriction.For all α ∈ { + , −} and n ∈ N , we define∆ αn U := { x ∈ U | dim( x ) = n and if y ∈ U covers x , then o ( y → x ) = α } ,∂ αn U := cl(∆ αn U ) ∪ { x ∈ U | for all y ∈ U , if x ≤ y , then dim( y ) ≤ n } , ∆ n U := ∆ + n U ∪ ∆ − n U, ∂ n U := ∂ + n U ∪ ∂ − n U. We call ∂ − n U the input n -boundary and ∂ + n U the output n -boundary of U .If U is ( n + 1)-dimensional, we write ∆ α U := ∆ αn U and ∂ α U := ∂ αn U . Foreach x ∈ P , we write ∆ αn x := ∆ αn cl { x } and ∂ αn x := ∂ αn cl { x } . amar hadzihasanovic (Atoms and molecules) . Let P be an oriented graded poset. We definea family of closed subsets of P , the molecules of P , by induction on propersubsets. If U is a closed subset of P , then U is a molecule if either • U has a greatest element, in which case we call it an atom , or • there exist molecules U and U , both properly contained in U , and n ∈ N such that U ∩ U = ∂ + n U = ∂ − n U and U = U ∪ U .We define ⊑ to be the smallest partial order relation such that, if U and U are molecules and U ∩ U = ∂ + n U = ∂ − n U , then U , U ⊑ U ∪ U .We say n -molecule for an n -dimensional molecule. We say that P itself is amolecule if P ⊆ P is a molecule. (Spherical boundary) . An n -molecule U in an oriented graded poset hasspherical boundary if, for all k < n , ∂ + k U ∩ ∂ − k U = ∂ k − U. (Regular directed complex) . An oriented graded poset P is a regulardirected complex if, for all x ∈ P and α, β ∈ { + , −} ,1. cl { x } has spherical boundary,2. ∂ α x is a molecule, and3. ∂ α ( ∂ β x ) = ∂ αn − x if n := dim( x ) > map f : P → Q of regular directed complexes is a function of their under-lying sets that satisfies ∂ αn f ( x ) = f ( ∂ αn x )for all x ∈ P , n ∈ N , and α ∈ { + , −} . We call an injective map an inclusion .With their maps, regular directed complexes form a category DCpx R . Remark 1.9.
As shown in [Had20b, Section 1.3],
DCpx R has an initial object,a terminal object, and pushouts of inclusions. (Regular molecule) . A regular molecule is a molecule which is a regulardirected complex.By [Had20b, Proposition 1.38], if two regular molecules are isomorphic in DCpx R , they are isomorphic in a unique way. As customary in these situ-ations, we will treat isomorphic regular molecules as “equal” under appropriatecircumstances. (Globe) . For each n ∈ N , let O n be the poset with a pair of elements k + , k − for each k < n and a greatest element n , with the partial order defined he smash product of monoidal theories by j α ≤ k β if and only if j ≤ k . This is a graded poset, with dim( n ) = n anddim( k α ) = k for all k < n .With the orientation o ( y → k α ) := α if y covers k α , O n becomes a regulardirected complex, in particular a regular atom. We call O n the n -globe . (Pasting of molecules) . Let U , U be regular molecules and supposethat ∂ + k U and ∂ − k U are isomorphic in DCpx R . Given an isomorphic copy V of the two, there is a unique span of inclusions V ֒ → U and V ֒ → U whoseimages are, respectively, ∂ + k U and ∂ − k U . We let U k U be the pushout V U k U U U in DCpx R . Then U k U is a regular molecule, decomposing as U ∪ U with U ∩ U = ∂ + k U = ∂ − k U . ( − ⇒ − construction) . Let
U, V be regular n -molecules with sphericalboundary such that ∂ α U is isomorphic to ∂ α V for all α ∈ { + , −} .Form the pushout U ∪ V of the span of inclusions ∂U ֒ → U , ∂U ֒ → V whoseimages are ∂U and ∂V , respectively. We define U ⇒ V to be the orientedgraded poset obtained from U ∪ V by adjoining a greatest element ⊤ with ∂ − ⊤ := U and ∂ + ⊤ := V . Then U ⇒ V is an ( n + 1)-dimensional atom withspherical boundary. ( h−i construction) . Let U be a regular molecule with spherical boundary.Then ∂ − U ⇒ ∂ + U is defined, and we denote it by h U i . . There is a unique 0-atom, namely, the 0-globe 1 := O , which is alsothe terminal object of DCpx R .We define a sequence { I n } n> of 1-molecules by I := O , I n := I n − O for n > . Every regular 1-molecule is of the form I n for some n > n, m >
0, let U n,m := ( I n ⇒ I m ). Every regular 2-atom is ofthe form U n,m for some n, m >
0. Regular 2-molecules are then generated by I and the U n,m under the pasting operations , . (Diagrammatic set) . We write for a skeleton of the full subcategoryof
DCpx R on the atoms of every dimension.A diagrammatic set is a presheaf on . Diagrammatic sets and their morph-isms of presheaves form a category Set . amar hadzihasanovic Comment 1.17.
The definition in [Had20b] is relative to a fixed “convenient”class of molecules; for simplicity, here we pick the class of all molecules withspherical boundary. . We identify with a full subcategory ֒ → Set via the Yoneda em-bedding. With this identification, we use morphisms in
Set as our notationfor both elements and structural operations of a diagrammatic set X : • x ∈ X ( U ) becomes x : U → X , and • for each map f : V → U in , X ( f )( x ) ∈ X ( V ) becomes f ; x : V → X .As described in [Had20b, §4.4], the embedding ֒ → Set extends to anembedding
DCpx R ֒ → Set . (Diagrams and cells) . Let X be a diagrammatic set and U a regularmolecule. A diagram of shape U in X is a morphism x : U → X . It is composable if U has spherical boundary and a cell if U is an atom. For all n ∈ N , we say that x is n -diagram or an n -cell when dim( U ) = n .If U decomposes as U k U , we write x = x k x for x i := ı i ; x , where ı i is the inclusion U i ֒ → U for i ∈ { , } . This extends associatively to n -arydecompositions for n > x : U → X is a diagram in X and f : X → Y a morphism of diagrammaticsets, we may write f ( x ) for the diagram x ; f : U → Y . (Boundaries of diagrams) . Let X be a diagrammatic set, x : U → X a diagram, and let ı αk : ∂ αk U ֒ → U be the inclusions of the k -boundaries of U . The input k -boundary of x is the diagram ∂ − k x := ı − k ; x and the output k -boundary of x is the diagram ∂ + k x := ı + k ; x . We may omit the index k when k = dim( U ) − x : y − ⇒ y + to express that ∂ αk x = y α for each α ∈ { + , −} , andsay that x is of type y − ⇒ y + . We say that two diagrams x , x are parallel ifthey have the same type. . A 1-cell a in a diagrammatic set has shape I . A 2-cell ϕ has shape U n,m for some n, m >
0, so it is of type a . . . a n ⇒ b . . . b m for some 1-cells a , . . . , a n , b , . . . , b m . We may depict such cells as string dia-grams a , a b a a n b b m ϕ , he smash product of monoidal theories where a lighter shade indicates a repeated pattern. Each region bounded bywires corresponds to a potentially different 0-cell; in practice, we will mostlywork with diagrammatic sets that have a single 0-cell. Labels will be omittedwhen irrelevant, or implied by the shape of a cell.A 2-diagram decomposes into 1-cells and 2-cells under the , opera-tions. In string diagrams, is horizontal juxtaposition and is verticaljuxtaposition with the output wires of one diagram connecting to the inputwires of another. For example, ϕ ψ depicts a generic 2-diagram of the form( ϕ a . . . a n ) ( b . . . b m ψ )for some 2-cells ϕ, ψ and 1-cells a , . . . , a n , b , . . . , b m .Observe that, in our setting, there is no need to attribute a topologicalnature to string diagrams, à la Joyal and Street [JS91]: they should insteadbe interpreted as compact encodings of regular molecules – a discrete, com-binatorial structure – and their morphisms to diagrammatic sets.A 2-diagram is composable if and only if it is connected as a string diagram.For example, of the diagrams ϕ ψ , ϕ ψ only the first one is composable.We may depict a 3-diagram as a sequence of rewrites on composable sub-diagrams of a diagram. For example, a diagrammatic set with a single 0-cell,a single 1-cell, and 3-cells ϕ, ψ of the form µ δ ϕ µδ , δ µ ψ µδ admits a 3-diagram of the form ϕ ψ . (3) amar hadzihasanovic where the input boundary of each 3-cell is highlighted in pink.We will also use string diagrams to describe certain regular molecules dir-ectly. This is justified by the interpretation of a molecule U as the “tautolog-ous” diagram id U : U → U in Set . (Dual diagrammatic set) . Let U be a regular atom. The oriented gradedposet U ◦ with the same underlying poset as U and the opposite orientation o ◦ ( y → x ) := − o ( y → x ) is a regular atom. If f : U → V is a map in ,its underlying function also defines a map f ◦ : U ◦ → V ◦ . This determines aninvolution − ◦ on .Let X be a diagrammatic set. Its dual X ◦ is the diagrammatic set definedby X ◦ ( − ) := X ( − ◦ ). This extends to morphisms in the obvious way, andextends the involution on to an involution on Set . Remark 1.23. If x : U → X is a 2-diagram, the depiction of x ◦ : U ◦ → X ◦ instring diagrams is the horizontal and vertical reflection of the depiction of x . (Reflexive ω -graph) . Let O be the full subcategory of whose objectsare the globes O n . For all n and k < n , • the k -boundary inclusions ı + k , ı − k are the only inclusions of O k into O n ; • the map τ : O n ։ O k , defined by τ ( n ) , τ ( j α ) := k if j ≥ k and τ ( j α ) := j α if j < k , is the only surjective map from O n onto O k .A reflexive ω -graph is a presheaf X on O . With their morphisms of presheaves,reflexive ω -graphs form a category ω Gph ref . . The embedding O ֒ → induces a restriction functor Set → ω Gph ref with a full and faithful left adjoint ω Gph ref ֒ → Set ; we can thus identifyreflexive ω -graphs with particular diagrammatic sets, and use for them thesame terminology and notation.Because all n -cells in a reflexive ω -graph X have the same shape O n , weleave it implicit and write X n := X ( O n ). (Units) . Let x be a k -cell in a reflexive ω -graph X . For n > k , we let ε n x := τ ; x where τ is the unique surjective map O n ։ O k . We call ε n x a unit on x . We may omit the index when n = k + 1. (Rank of a cell) . Let x be an n -cell in a reflexive ω -graph. The rank rk( x ) of x is defined inductively on n as follows: • if n = 0, then rk( x ) := 0; he smash product of monoidal theories if n >
0, if x = εy for an ( n − y , then rk( x ) := rk( y ), otherwiserk( x ) := n . (Partial ω -category) . A partial ω -category is a reflexive ω -graph X to-gether with partial k -composition operations k : X n × X n ⇀ X n for all n ∈ N and k < n , satisfying the following axioms:1. for all n -cells x, y and all k < n such that x k y is defined, ∂ + k x = ∂ − k y and ε ( x k y ) = εx k εy ;2. for all n -cells x and all k < n , the k -compositions x k ε n ( ∂ + k x ) and ε n ( ∂ − k x ) k x are defined and equal to x ;3. for all ( n + 1)-cells x, y and k < n , whenever the left-hand side is defined,the right-hand side is defined and ∂ − ( x n y ) = ∂ − x,∂ + ( x n y ) = ∂ + y,∂ α ( x k y ) = ∂ α x k ∂ α y ;4. for all cells x, y, z and all k such that both sides are defined,( x k y ) k z = x k ( y k z );5. for all cells x, y, x ′ , y ′ , all n and all k < n such that both sides are defined,( x n x ′ ) k ( y n y ′ ) = ( x k y ) n ( x ′ k y ′ ) . (4)A functor f : X → Y of partial ω -categories is a morphism of the underlyingreflexive ω -graphs such that, for all cells x, y in X , if x n y is defined in X then f ( x ) n f ( y ) is defined and equal to f ( x n y ) in Y . Partial ω -categoriesand their functors form a category p ω Cat . . We will generally confuse the notation for a k -cell and the units onit: for example, if x is an n -cell and y a k -cell, k < n , such that x m ε n y isdefined, we will write x m y := x m ε n y . amar hadzihasanovic ( ω -Precategory) . An ω -precategory is a partial ω -category X such that,for all n -cells x, y in X , the k -composition x k y is defined if and only if ∂ + k x = ∂ − k y and min { rk( x ) , rk( y ) } ≤ k + 1. With their functors, ω -precategor-ies form a category ω PreCat . ( ω -Category) . An ω -category is a partial ω -category such that, for all n -cells x, y in X , the k -composition x k y is defined if and only if ∂ + k x = ∂ − k y .With their functors, ω -categories form a category ω Cat . . The inclusion ω Cat ֒ → p ω Cat has a left adjoint − ∗ : p ω Cat → ω Cat ;if X is a partial ω -category, then X ∗ is the free ω -category on the underlyingreflexive ω -graph of X , quotiented by all the equations involving compositionsthat are defined in X .By [Had20b, Proposition 1.23], if P is a regular directed complex, there isa partial ω -category M oℓP where1. the set M oℓP n of n -cells is the set of molecules U ⊆ P with dim( U ) ≤ n ,2. ∂ αk : M oℓP n → M oℓP k is U ∂ αk U ,3. ε n : M oℓP k → M oℓP n is U U ,4. U k V is defined if and only if U ∩ V = ∂ + k U = ∂ − k V , and in that case itis equal to U ∪ V .We will write W = U k V to indicate that W is a molecule decomposing as U ∪ V , where U and V are molecules with U ∩ V = ∂ + k U = ∂ − k V .As detailed in [Section 7, ibid. ], the assignment P
7→ M oℓP ∗ extends to afunctor M oℓ − ∗ : DCpx R → ω Cat which is faithful and injective on objects. (Principal composition) . Let x, y be n -cells in an ω -precategory or an ω -category, and let k := min { rk( x ) , rk( y ) } −
1. If ∂ + k x = ∂ − k y , the principalcomposition of x and y is x y := x k y. Comment 1.34.
Given cells x, y in an ω -precategory, suppose that x k y isdefined. Then either • min { rk( x ) , rk( y ) } = k + 1, in which case x k y = x y , or • rk( x ) ≤ k , in which case x k y = y , or • rk( y ) ≤ k , in which case x k y = x .This implies that ω -precategories admit an axiomatisation involving only prin-cipal compositions, at the cost of explicitly handling some corner cases in theaxioms.Moreover, the two sides of (4) can both be defined only if he smash product of monoidal theories both x and x ′ have rank lower or equal than k + 1, in which case (4) isequivalent to x k ( y n y ′ ) = ( x k y ) n ( x k y ′ ) , or dually • both y and y ′ have rank lower or equal than k + 1, in which case (4) isequivalent to ( x n x ′ ) k y = ( x k y ) n ( x ′ k y ) . These two observations can be used to establish an equivalence between ourdefinition of ω -precategory and the one given in [FM18, Section 4.1]. . There is a forgetful functor U : ω Cat → ω PreCat which makes x k y undefined whenever min { rk( x ) , rk( y ) } > k + 1. Proposition 1.36 —
The functor U : ω Cat → ω PreCat is full and faithful.Its image consists of the ω -precategories satisfying ( x k − ∂ − k y ) k ( ∂ + k x k − y ) = ( ∂ − k x k − y ) k ( x k − ∂ + k y ) (5) for all cells x, y with min { rk( x ) , rk( y ) } = k + 1 and ∂ + k − x = ∂ − k − y .Proof. First of all, observe that both sides of (5) are defined in an ω -precat-egory when x, y satisfy the conditions of the statement. If this precategory isof the form U X for some ω -category X , then both sides are equal to x k − y in X , so (5) is satisfied.Let X ′ be an ω -precategory such that (5) holds for all cells in X ′ in the con-ditions of the statement. We will define an ω -category X such that U X = X ′ ,which necessarily has the same underlying reflexive ω -graph as X ′ .Let x, y be cells such that ∂ + k − x = ∂ − k − y for some k >
0. We must define x k − y in X . If min { rk( x ) , rk( y ) } ≤ k , then x k − y is defined in X ′ , andwe declare it to be the same in X .Otherwise, min { rk( x ) , rk( y ) } = k + 1 + m for some m ≥
0. If m = 0, wedefine x k − y to be equal to either side of (5). For m >
0, observe thatmin { rk( x ) , rk( ∂ αk + m y ) } , min { rk( ∂ αk + m x ) , rk( y ) } ≤ k + m for all α ∈ { + , −} , but ∂ + k − x = ∂ + k − ( ∂ αk + m x ) = ∂ − k − ( ∂ αk + m y ) = ∂ − k − y. We may thus assume, inductively, that x k − ∂ αk + m y and ∂ αk + m x k − y havealready been defined, and let x k − y := ( x k − ∂ − k + m y ) k + m ( ∂ + k + m x k − y ) . amar hadzihasanovic It is an exercise to show, by induction, that this is equal to( ∂ − k + m x k − y ) k + m ( x k − ∂ + k + m y )and derive that X is an ω -category. Because all the definitions are enforcedby the axioms of ω -categories, X is unique with the property that U X = X ′ .Since all compositions in X are defined in terms of compositions in U X ,every functor f ′ : U X → U Y of ω -precategories lifts to a functor f : X → Y of ω -categories. This proves fullness; faithfulness is immediate from the factthat f and U f have the same underlying morphism of reflexive ω -graphs. (cid:4) (Skeleta) . Let X be an ω -(pre)category, n ∈ N . The n -skeleton σ ≤ n X of X is the restriction of X to cells of rank ≤ n . We let σ ≤− X := ∅ , the initial ω -(pre)category. The n -skeleton operation extends functorially to morphismsin the obvious way. ( n -Category) . An ω -(pre)category is an n -(pre)category if it is equal toits n -skeleton. An n -(pre)category is determined by its restriction to k -cellswith k ≤ n .Let n PreCat denote the full subcategory of ω PreCat and n Cat the fullsubcategory of ω Cat on the n -(pre)categories. In both cases, the inclusionof subcategories has a right adjoint and σ ≤ n is the comonad induced by theadjunction.The inclusion also has a left adjoint, inducing a monad τ ≤ n : given X , the n -(pre)category τ ≤ n X is obtained from σ ≤ n X by identifying all pairs of n -cells x, y such that there exists an ( n + 1)-cell e : x ⇒ y in X . Remark 1.39.
Both ω PreCat and ω Cat are categories of algebras for finit-ary monads on ω Gph ref , a presheaf topos. By the Remark at the end of[AR94, §2.78], they are locally finitely presentable, and in particular have allsmall limits and colimits. The same applies to their reflective subcategories n PreCat and n Cat for all n ∈ N . (Polygraph) . Let ∂O n := σ ≤ n − O n .A (pre)polygraph is an ω -(pre)category X together with a set X = P n ∈ N X n of generating cells such that, for all n ∈ N , ` x ∈ X n ∂O n σ ≤ n Xσ ≤ n − X ` x ∈ X n O n ( x ) x ∈ X n is a pushout in ω PreCat or ω Cat . he smash product of monoidal theories An n -(pre)polygraph is a (pre)polygraph whose underlying ω -(pre)categoryis an n -(pre)category. In an n -(pre)polygraph, X m = ∅ for m > n . Remark 1.41.
In every (pre)polygraph ( X, X ), all cells in X n have rank n .The set X is the entire set X of 0-cells. Example 1.42.
Both 1-precategories and 1-categories coincide with small cat-egories; a 1-(pre)polygraph is a category free on a graph.
2. Monoidal theories
For us, monoidal theories are embodied by pros . For technical reasons, wetreat these as a special case of a more general structure of bicoloured pro ,whose relation to pros is the same as the relation of bicategories to monoidalcategories. (Bicoloured pro) . A bicoloured pro is a 2-category T together with thestructure of a 1-polygraph ( σ ≤ T, T ) on its 1-skeleton.A morphism f : ( T, T ) → ( S, S ) of bicoloured pros is a functor f : T → S of 2-categories with the property that f ( a ) ∈ ε ( S ) ∪ S for all a ∈ T .Bicoloured pros and their morphisms form a category Pro bi . (Strict monoidal category) . A strict monoidal category is a 2-categorywith a single 0-cell. Comment 2.3.
We are using the characterisation of strict monoidal categoriesin [CG07, Theorem 4.1] as a definition. . When some object has a single 0-cell, we denote that 0-cell by • . (Pro) . A pro is a bicoloured pro with a single 0-cell. We let Pro denotethe full subcategory of
Pro bi on pros. Comment 2.6.
That is, a pro is a bicoloured pro whose underlying 2-categoryis a strict monoidal category.
Comment 2.7.
Equivalently, a bicoloured pro ( T, T ) is a 2-category whose1-cells of type x ⇒ y are finite paths from x to y in the graph T T . ∂ + ∂ − amar hadzihasanovic When x = y , the path is allowed to be of length 0, in which case it is interpretedas the unit εx .If T has a single 0-cell, these are the same as finite ordered lists of elements of T , that is, elements of the free monoid on T . Seeing a pro as the embodimentof a monoidal theory, we interpret the elements of T as sorts , and 2-cells ϕ : ( a , . . . , a n ) ⇒ ( b , . . . , b m )as operations taking n inputs of sorts a , . . . , a n and returning m outputs ofsorts b , . . . , b m .In particular, if the monoidal theory is one-sorted, then T is a singleton,the 1-cells of T are in bijection with natural numbers, and the type of a 2-cellis fixed by the arity of its input and its output. In that case, we may write ϕ : ( n ) ⇒ ( m ) for an operation with n inputs and m outputs.If we forget the structure of 1-polygraph on a pro, we get a strict monoidalcategory T , which we may see as a special kind of monoidal category. Given amonoidal category M , a model in M of the monoidal theory ( T, T ) is a strongmonoidal functor from T to M . Remark 2.8.
Our definition of a morphism of pros allows a sort to be “col-lapsed” by mapping it onto a unit. We make this choice for technical reasons,even though it seems more common to disallow it, as done in [HR15]. Thischoice does not have any impact on models.
Example 2.9.
The monoid N of natural numbers with addition, seen as a strictmonoidal category with no rank-2 cells, is a one-sorted pro. This correspondsto the “trivial” theory of objects in a monoidal category. Example 2.10.
There is a one-sorted pro
Mon whose 1-cell ( n ) is identified withthe finite ordinal { < . . . < n − } for each n ∈ N , and 2-cells ϕ : ( n ) ⇒ ( m ) areorder-preserving maps. The 0-composite of ϕ : ( n ) ⇒ ( m ) and ψ : ( p ) ⇒ ( q ) isgiven by “concatenation”, that is, ϕ ψ : ( n + p ) ⇒ ( m + q ) , k ϕ ( k ) if k < n,m + ψ ( k − n ) if k ≥ n. This corresponds to the theory of monoids . Example 2.11.
If ( T, T ) is a pro, then ( T co , T co ), obtained by reversing theorientation of all 2-cells of T , is also a pro. For example, Mon co is the theoryof comonoids . Example 2.12.
Let
Bimod be the strict monoidal category whose 1-cells are in-jective maps ı : ( k ) ֒ → ( n ) in Mon , 2-cells ϕ : ( ı : ( k ) ֒ → ( n )) ⇒ ( j : ( k ) ֒ → ( m )) he smash product of monoidal theories are commutative triangles ( n ) ( k ) ( m ), ı jϕ and the 0-composite of a pair of 2-cells ϕ : ( ı : ( k ) ֒ → ( n )) ⇒ ( j : ( k ) ֒ → ( m )) ,ψ : ( ı ′ : ( ℓ ) ֒ → ( p )) ⇒ ( j ′ : ( ℓ ) ֒ → ( q ))is the commutative triangle( n + p ) ( k + ℓ ) ( m + q ). ı ı ′ j j ′ ϕ ψ The 1-cells in
Bimod are freely generated under 0-composition by the pair ı : (0) ֒ → (1) , id : (1) → (1) , where ı is the unique inclusion of the empty ordinal, so Bimod admits thestructure of a two-sorted pro.There are morphisms of pros
Mon → Bimod and N → Bimod , sending thegenerating 1-cell to ı and id , respectively. The models of Bimod in a monoidalcategory M are given by an object of M , a monoid in M , and a two-sidedaction of the monoid on the object, making the object a bimodule . . There is an obvious functor U : Pro bi → Cat forgetting the structureof 1-polygraph. This functor has a right adjoint R : 2 Cat → Pro bi , describedas follows.Given a 2-category X , the 1-skeleton of R X is free on the underlying reflexive graph of σ ≤ X : that is, the sorts of R X are all the 1-cells of rank 1 in X .The 2-cells of type ( a , . . . , a n ) ⇒ ( b , . . . , b m ) in R X are the 2-cells of type a . . . a n ⇒ b . . . b m in X .Compositions are induced by those of X in the obvious way, and a functor f : X → Y of 2-categories induces a morphism R f : R X → R Y of bicolouredpros by ϕ : ( a , . . . , a n ) ⇒ ( b , . . . , b m ) f ( ϕ ) : f ( a ) . . . f ( a n ) ⇒ f ( b ) . . . f ( b m ) . amar hadzihasanovic For each 2-cell ϕ of type ( a , . . . , a n ) ⇒ ( b , . . . , b m ) in a bicoloured pro( T, T ), there is a 2-cell a . . . a n ⇒ b . . . b m in U ( T, T ), whichin turn induces a 2-cell of type ( a , . . . , a n ) ⇒ ( b , . . . , b m ) in RU ( T, T ); theunit of the adjunction sends ϕ to this 2-cell.Conversely, for each 2-category X and 2-cell ϕ : ( a , . . . , a n ) ⇒ ( b , . . . , b m )in UR X , the counit sends ϕ to the cell of type a . . . a n ⇒ b . . . b m in X from which it was induced. It is an exercise to show that the unit andcounit satisfy the required equations and determine an adjunction. Lemma 2.14 —
The category
Pro bi has equalisers.Proof. Let f, g : ( T, T ) → ( S, S ) be parallel morphisms of bicoloured pros.Define T ′ to be the restriction of T to the cells x that satisfy f ( x ) = g ( x ) in S . Then T ′ is a 2-category.A 1-cell ( a , . . . , a n ) in T belongs to T ′ if and only if f ( a i ) = g ( a i ) for all i ∈ { , . . . , n } . It follows that T ′ := { a ∈ T | f ( a ) = g ( a ) } gives σ ≤ T ′ thestructure of a 1-polygraph, so that the inclusion of T ′ into T is a morphism ofbicoloured pros. By a routine argument, it is the equaliser of f and g . (cid:4) Proposition 2.15 —
The functor U : Pro bi → Cat is comonadic.Proof.
We have shown that U has a right adjoint. Moreover, equalisers asconstructed in the proof of Lemma 2.14 are evidently created by U .In order to apply the dual of Beck’s monadicity theorem [ML71, §VI.7], itsuffices to show that U reflects isomorphisms. Let f : ( T, T ) → ( S, S ) bea morphism of bicoloured pros and suppose that U f is an isomorphism of2-categories with inverse g . Then both U f and g must preserve the rank of allcells.Let a ∈ S . Then g ( a ) can be written uniquely as a finite path ( a , . . . , a n )with a i ∈ T for all i ∈ { , . . . , n } , and a = U f ( g ( a )) = ( f ( a ) , . . . , f ( a n ))where n > f ( a i ) ∈ S for all i ∈ { , . . . , n } . Because σ ≤ S is free, thisis only possible if n = 1 and f ( a ) = a . It follows that g sends generators togenerators, hence it determines a morphism of bicoloured pros, inverse to f in Pro bi . (cid:4) Corollary 2.16 —
The categories
Pro bi and Pro have all small limits andcolimits.Proof.
By the dual of [ML71, Exercise 2, §VI.2], a comonadic functor createsall colimits in its codomain; since 2
Cat has all small colimits, so does
Pro bi . he smash product of monoidal theories Moreover, 2
Cat has all small limits and
Pro bi has equalisers, so Pro bi has allsmall limits by the dual of [Lin69, Corollary 2].Since Pro is defined equationally, it is a reflective subcategory of
Pro bi . Itfollows from [Rie17, Proposition 4.5.15] that it also has all small limits andcolimits. (cid:4) (Braided strict monoidal category) . A braided strict monoidal category is a strict monoidal category X together with a family of 2-cells σ x,y : x y ⇒ y x called braidings , indexed by 1-cells x, y , satisfying the following axioms:1. the braidings are invertible, that is, there are unique 2-cells σ − x,y , called inverse braidings , such that σ x,y σ − x,y and σ − x,y σ x,y are units;2. they are natural in their parameters, that is, for all 2-cells ϕ : x ⇒ x ′ and ψ : y ⇒ y ′ , ( ϕ y ) σ x ′ ,y = σ x,y ( y ϕ ) , ( x ψ ) σ x,y ′ = σ x,y ( ψ x );3. they are compatible with 0-composition and units, that is, σ x x ′ ,y = ( x σ x ′ ,y ) ( σ x,y x ′ ) ,σ x,y y ′ = ( σ x,y y ′ ) ( y σ x,y ′ ) ,σ ε • ,y = εy, σ x,ε • = εx, whenever the left-hand side is defined.A functor f : X → Y of braided strict monoidal categories is a functor of theunderlying 2-categories that preserves braidings, that is, f ( σ x,y ) = σ f ( x ) ,f ( y ) for all 1-cells x, y in X . With their functors, braided strict monoidal categoriesform a category BrMonCat str . (Prob) . A prob is a pro together with a structure of braided strict mon-oidal category on its underlying strict monoidal category.A morphism of probs is a morphism of pros that preserves the braidings.Probs and their morphisms form a category Prob . Comment 2.19.
Models of probs live in braided monoidal categories M , notnecessarily strict. A model of a prob ( T, T ) in M is a braided strong monoidalfunctor from T to M . amar hadzihasanovic Remark 2.20.
To determine a unique structure of braided strict monoidal cat-egory on a pro it is, in fact, sufficient to give braidings σ a,b for all pairs ofgenerating 1-cells a, b ; a morphism of pros that preserves these braidings auto-matically preserves all braidings. This is a consequence of axiom 3 of braidedstrict monoidal categories, since every 1-cell in a pro can be decomposed as acomposite of generating 1-cells. (Dual braided structure) . Let X be a braided strict monoidal categorywith braidings { σ x,y } . The family of 2-cells σ ∗ x,y := σ − y,x defines a second structure X ∗ of braided strict monoidal category on the un-derlying strict monoidal category of X .If f : X → Y is a functor of braided strict monoidal categories, the sameunderlying functor of 2-categories determines a functor f ∗ : X ∗ → Y ∗ . Thisdefines an involution − ∗ on BrMonCat str , which also induces a duality on
Prob . (Symmetric strict monoidal category) . A symmetric strict monoidal cat-egory is a braided strict monoidal category X satisfying X = X ∗ . (Prop) . A prop is a prob whose underlying braided strict monoidalcategory is symmetric. We let Prop denote the full subcategory of
Prob onprops. . There is an obvious forgetful functor U : Prob → Pro and an inclusionof subcategories
Prop ֒ → Prob . Both of these have left adjoints: • the left adjoint F : Pro → Prob of U freely adds braidings σ x,y and inversebraidings σ − x,y for all pairs of 1-cells x, y of a pro (or just the generatingones, see Remark 2.20), then quotients by the axioms of braided strictmonoidal categories; • the reflector r : Prob → Prop quotients by the equation σ x,y = σ ∗ x,y forall pairs of 1-cells x, y of a prob.Since we have not yet shown that Prob has coequalisers, for the moment wecan interpret the latter as a coequaliser in
Pro , then observe that the imagesof the σ x,y still form a family of braidings in the quotient. Example 2.25.
The free prob B := F N is the theory of braids . With 1-compos-ition, 2-cells of type ( n ) ⇒ ( n ) in B form the braid group B n on n strands. he smash product of monoidal theories Example 2.26.
The prop reflection S := r B of the theory of braids is the theoryof permutations . With 1-composition, 2-cells of type ( n ) ⇒ ( n ) in S form thesymmetric group S n on n elements. Example 2.27.
Let
CMon be defined as
Mon , but 2-cells of type ( n ) ⇒ ( m )are all functions from ( n ) to ( m ), not just the order-preserving ones. This isa one-sorted prop with braidings generated by σ , : (2) ⇒ (2) , , . It corresponds to the theory of commutative monoids in symmetric monoidalcategories. Similarly, models of
CMon co are commutative comonoids . Example 2.28.
There is a one-sorted pro
Mat Z whose 2-cells A : ( n ) ⇒ ( m )are ( m × n )-matrices of integers, the 1-composite A B is the product BA of matrices, and A B := A B ! . This is a prop with braidings generated by σ , := ! . There is a morphism of props
CMon → Mat Z sending ϕ : ( n ) ⇒ ( m ) to the( m × n )-matrix A ϕ with entries A ϕ ( j, i ) := j = ϕ ( i ) , , and a morphism CMon co → Mat Z sending ϕ co : ( m ) ⇒ ( n ) to the transposeof A ϕ . As a symmetric monoidal theory, Mat Z corresponds to the theory of commutative and cocommutative Hopf algebras [BSZ17, Section 7]. (Gray-category) . A Gray-category is a 3-precategory G together with afamily of 3-cells χ x,y : ( x ∂ − y ) ( ∂ + x y ) ⇒ ( ∂ − x y ) ( x ∂ + y )called interchangers , indexed by 2-cells x, y with ∂ +0 x = ∂ − y , satisfying thefollowing axioms:1. the interchangers are invertible, that is, there are unique 3-cells χ − x,y : ( ∂ − x y ) ( x ∂ + y ) ⇒ ( x ∂ − y ) ( ∂ + x y )called inverse interchangers , such that χ x,y χ − x,y and χ − x,y χ x,y areunits; amar hadzihasanovic
2. the interchangers are natural in their parameters, that is, for all 3-cells ϕ : x ⇒ x ′ and ψ : y ⇒ y ′ with ∂ +0 ϕ = ∂ − ψ ,(( ϕ ∂ − y ) ( ∂ +1 ϕ y )) χ x ′ ,y = χ x,y (( ∂ − ϕ y ) ( ϕ ∂ + y )) , (( x ∂ − ψ ) ( ∂ + x ψ )) χ x,y ′ = χ x,y (( ∂ − x ψ ) ( x ∂ +1 ψ ));3. the interchangers are compatible with 1-compositions and units, that is, χ x x ′ ,y = (( x ∂ − y ) χ x ′ ,y ) ( χ x,y ( x ′ ∂ + y )) ,χ x,y y ′ = ( χ x,y ( ∂ + x y ′ )) (( ∂ − x y ) χ x,y ′ )) ,χ εx,y = ε ( εx y ) , χ x,εy = ε ( x εy ) , whenever the left-hand side is defined;4. for all pairs of 3-cells ϕ, ψ with ∂ +1 ϕ = ∂ − ψ , the equation( ϕ ∂ − ψ ) ( ∂ +2 ϕ ψ ) = ( ∂ − ϕ ψ ) ( ϕ ∂ +2 ψ )holds in G .A functor f : G → H of Gray-categories is a functor of the underlying 3-prec-ategories that preserves the interchangers, that is, f ( χ x,y ) = χ f ( x ) ,f ( y ) for allsuitable 2-cells x, y in G . With their functors, Gray-categories form a category GrayCat . Remark 2.30.
Axiom 4 is an instance of (5), allowing us to univocally definethe 1-composition ϕ ψ of 3-cells with ∂ +1 ϕ = ∂ − ψ in a Gray-category. Comment 2.31.
A more concise definition is that a Gray-category is a smallcategory enriched over 2
Cat with the “pseudo” Gray tensor product [GPS95,Chapter 5]. As in [Lac11, §1.4], one derives that
GrayCat is locally finitelypresentable, and in particular has all small limits and colimits. . By Proposition 1.36, every 3-category seen as a 3-precategory admitsa natural structure of Gray-category with units as interchangers. This definesan embedding 3
Cat ֒ → GrayCat , which makes 3
Cat a reflective subcategoryof
GrayCat : the reflector universally turns the interchangers into units. . Given a braided strict monoidal category X , we define a Gray-category B X as follows. For all n ∈ N , we let B X n +1 := X n , with the same boundaryand unit operators as X between B X n +2 and B X n +1 . We let B X := {•} ,with the only possible unit and boundary operators relating it to B X . Thisdefines the underlying reflexive ω -graph of B X .To make B X a 3-precategory, it suffices to define the principal compositions.Because B X has no rank-1 cells, the principal compositions are of the form x k y where he smash product of monoidal theories k = 1 and min { rk( x ) , rk( y ) } = 2, or • k = 2 and rk( x ) = rk( y ) = 3.In either case, x k − y is defined in X , and we let x k y be equal to it in B X .Finally, given 2-cells x, y in B X , we let the interchanger χ x,y correspondto the braiding σ x,y in X . It is an exercise to check that this gives B X thestructure of a Gray-category.This assignment extends to a functor B : BrMonCat str → GrayCat in theobvious way. By [CG11, Theorem 2.16], this functor is full and faithful, andits essential image consists exactly of those Gray-categories that have a single0-cell and a single 1-cell. This can be seen as an alternative characterisationof
BrMonCat str as a full subcategory of
GrayCat . Remark 2.34.
Through B , the duality − ∗ on BrMonCat str is the restrictionof the duality on
GrayCat that reverses the orientation of 1-cells. . If X is a braided strict monoidal category, the structure of a 1-polygraphon σ ≤ X determines a unique structure of 2-prepolygraph on σ ≤ B X , and viceversa. A functor f of braided strict monoidal categories sends generators togenerators if and only if B f does.Thus, a prob is equivalently defined as a Gray-category T with a single0-cell, a single 1-cell, and the structure of a 2-prepolygraph ( σ ≤ T, T ) onits 2-skeleton. A morphism f : ( T, T ) → ( S, S ) of probs is a functor ofGray-categories such that f ( a ) ∈ { ε •} ∪ S for all a ∈ T .We conclude that there is a triangle of functors BrMonCat str
Prob GrayCat U U B commuting up to natural isomorphism, where U and U are the forgetfulfunctors associated to the two alternative definitions of prob. Proposition 2.36 —
The categories
BrMonCat str , Prob , and
Prop haveall small limits and colimits.Sketch of the proof.
First of all, the essential image of B is defined equationallyin GrayCat , so
BrMonCat str is, up to equivalence, a reflective subcategory.Since
GrayCat has all small limits and colimits, by [Rie17, Proposition 4.5.15]so does
BrMonCat str .Then, we can mimic the proofs of Lemma 2.14 and Proposition 2.15 to showthat
Prob has equalisers and that the functor U : Prob → BrMonCat str is amar hadzihasanovic comonadic. As in the proof of Corollary 2.16, we deduce that Prob has allsmall limits and colimits, and so does its reflective subcategory
Prop . (cid:4) Remark 2.37.
The functors U : Pro bi → Cat , U : Prob → BrMonCat str are pseudomonic : that is, in addition to being faithful, they reflect and are fullon isomorphisms. This captures the fact that a 2-category admits at most onestructure of bicoloured pro, a consequence of the general statement, proved byMichael Makkai [Mak05, Section 4, Proposition 8], that an ω -category admitsat most one structure of polygraph.Because the composite of a pseudomonic with a full and faithful functor ispseudomonic, it follows that U : Prob → GrayCat is also pseudomonic.
3. Combinatorial results
In this section, we use results of [Had20a, Section 6]. These results are statedrelative to the restricted class of constructible directed complexes, but theproofs do not involve any properties that are not satisfied by all regular direc-ted complexes. Thus all cited statements hold with constructible replaced by regular . . Let U be a closed subset of a regular directed complex. For each n ≥ − M n U has { x ∈ U | dim( x ) ≤ n } + { x ∈ U | x is maximal and dim( x ) > n } as set of vertices, and an edge y → x if and only if • dim( y ) ≤ n , dim( x ) > n , and y ∈ ∂ − n x \ ∂ n − x , or • dim( y ) > n , dim( x ) ≤ n , and x ∈ ∂ + n y \ ∂ n − y . (Frame dimension) . Let U be a closed subset of a regular directed com-plex. The frame dimension of U is the integerfrdim( U ) := max { dim(cl { x } ∩ cl { y } ) | x, y maximal in U , x = y } . Remark 3.3.
If frdim( U ) = −
1, then U is a disjoint union of atoms. (Frame acyclicity) . A regular directed complex P is frame-acyclic if, forall molecules U in P , if frdim( U ) = k , then M k U is acyclic. Lemma 3.5 —
Let P be a frame-acyclic regular directed complex. Then he smash product of monoidal theories
1. for all molecules U in P , if U = U ∪ U for some closed subsets U , U such that U ∩ U = ∂ + k U = ∂ − k U , then U and U are molecules;2. ( M oℓP ∗ , { cl { x }} x ∈ P ) is a polygraph.Proof. A corollary of [Had20a, Proposition 26]. (cid:4) ( k -Order) . Let U be a regular n -molecule. For k < n , a k -order on U isa linear ordering ( x , . . . , x m ) of the set { x ∈ U | x is maximal and dim( x ) > k } with the property that, if there is a path from x i to x j in M k U , then i ≤ j . Proposition 3.7 —
Let U be a regular n -molecule, k < n . If M k U is acyclic,then U admits a k -order.Proof. Every directed acyclic graph admits a topological sorting, that is, alinear order (cid:22) on its vertices with the property that if there is a path from x to y , then x (cid:22) y . The restriction of a topological sorting of M k U to themaximal elements of dimension greater than k is a k -order on U . (cid:4) Proposition 3.8 —
Let U be a regular n -molecule. Then M n − U is acyclicand U admits an ( n − -order.Proof. Follows from Proposition 3.7 and [Had20a, Proposition 20]. (cid:4)
Lemma 3.9 —
Let U be a frame-acyclic regular molecule, k ≥ frdim( U ) , andlet ( x , . . . , x m ) be a k -order on U . There exist molecules V , . . . , V m , such that U = V k . . . k V m and x i ∈ V j if and only if i = j for all i, j ∈ { , . . . m } .Proof. If m = 1 then V := U satisfies the statement. Suppose that m > k = frdim( U ). Then we proceed as in the proof of [Had20a, Proposition26] to produce i ∈ { , . . . , m − } and a decomposition U = U ∪ U such that1. U contains x , . . . , x i and U contains x i +1 , . . . , x m ,2. U ∩ U = ∂ + k U = ∂ − k U .By Lemma 3.5, both U and U are molecules. Moreover ( x , . . . , x i ) and( x i +1 , . . . , x m ) are k -orders on U and U , respectively. We conclude by theinductive hypothesis applied to U and U . amar hadzihasanovic Finally, suppose that k > ℓ := frdim( U ). By frame acyclicity, we can fix an ℓ -order ( y , . . . , y p ) on U , and by the first part of the proof we can decompose U as W ℓ . . . ℓ W p with y i ∈ W j if and only if i = j . Now for each i ∈ { , . . . , m } there is aunique j ( i ) ∈ { , . . . , p } such that x i = y j ( i ) . Let V i := ∂ α ( i, k W ℓ . . . ℓ W j ( i ) ℓ . . . ℓ ∂ α ( i,p ) k W p , where α ( i, j ) := + if j = j ( i ′ ) for some i ′ < i , − otherwise . Then U = V k . . . k V m is the required decomposition. (cid:4) (Substitution) . Let V and W be regular n -molecules with sphericalboundary, let U be a regular n -molecule, and suppose V ⊑ U . Then U \ ( V \ ∂V )is a closed subset of U .Suppose that ∂ α V is isomorphic to ∂ α W for all α ∈ { + , −} . From [Had20b,Lemma 2.2] we obtain a unique isomorphism ı : ∂U ∼ ֒ → ∂V . We define U [ W/V ]to be the pushout ∂V U [ W/V ] W U \ ( V \ ∂V )in DCpx R , and call it the substitution of W for V ⊑ U . By [Proposition 2.4, ibid. ] this is an n -molecule with boundaries isomorphic to those of U , andsuch that W ⊑ U [ W/V ]. Remark 3.11.
As shown in [Had20b, Lemma 2.5], if U is an n -molecule witha decomposition U = V n − . . . n − V m as in Lemma 3.9, then ∂ α V i isisomorphic to ∂ − α V i [ ∂ α x i /∂ − α x i ] for all α ∈ { + , −} and i ∈ { , . . . , m } . (Totally loop-free molecule) . Given a regular molecule U , let H o U bethe directed graph obtained from H U by reversing all the edges labelled − .We say that U is totally loop-free if H o U is acyclic as a directed graph.If U is totally loop-free, for all x, y ∈ U , we let x (cid:22) y if and only if there isa path from x to y in U . Proposition 3.13 —
Let U be a regular molecule. If dim( U ) ≤ , then U istotally loop-free and (cid:22) is a linear order on U . he smash product of monoidal theories Proof.
If dim( U ) ∈ { , } or if U is a 2-dimensional atom, this is easy. Oth-erwise, decompose U as V . . . V m so that each V i contains a unique2-dimensional element x i , and let V := ∂ − U and U ′ := V . . . V m − .Then U = U ′ ∪ cl { x m } , and we may assume inductively that the statementholds for U ′ .Suppose that there is a cycle in H o U . Because both U ′ and cl { x m } aretotally loop-free, such a cycle must leave U ′ , enter cl { x m } \ ∂ − x m , then returnto U ′ . Such a path either • enters x m via ∆ − x m , then enters ∂ + x m and leaves via ∂ +0 x m , or • enters ∆ + x m directly via ∂ − x m , stays in ∂ + x m and leaves via ∂ +0 x m .In both cases, the path through cl { x m } \ ∂ − x m can be replaced with theunique path to ∂ +0 x m that stays in ∂ − x m ⊆ U ′ . In this way, we create a cyclein H o U ′ , contradicting the inductive hypothesis.This proves that U is totally loop-free. To show that (cid:22) is a linear order, itsuffices to compare elements of U ′ and of cl { x m } \ ∂ − x m . Let x ∈ U ′ . Thereare two possible cases: • ∂ +0 x m (cid:22) x in U ′ . Then z (cid:22) x for all elements z ∈ cl { x m } . • x ≺ ∂ +0 x m in U ′ . Let y be the unique 1-dimensional element of ∆ − x m thatcovers ∂ +0 x m . Suppose that y ≺ x in U ′ , that is, there is a non-trivial pathfrom y to x in H o U ′ . Such a path cannot pass through ∂ + y = ∂ +0 x m , forotherwise ∂ +0 x m (cid:22) x ; nor it can enter a 2-dimensional element, because y is not covered by any element of U ′ with orientation − . Therefore x (cid:22) y ,so x ≺ x m and x ≺ z for all elements z ∈ ∂ + x m \ ∂ x m .This proves that (cid:22) is a linear order on U . (cid:4) Remark 3.14. If U is a regular molecule with dim( U ) ≤
2, by [Ste93, Theorem2.17] combined with Proposition 3.13, M oℓU ∗ is equal to M oℓU . Proposition 3.15 —
Let U be a regular 2-molecule, k ∈ { , } , and let x, y ∈ U be maximal elements of dimension > k . If there is a path from x to y in M k U , then x (cid:22) y .Proof. Suppose k = 1. A path x = x → w → . . . → w m − → x m = y in M U is a concatenation of two-step paths x i → w i → x i +1 where dim( x i ) = 2for all i ∈ { , . . . , m } and w i ∈ ( ∂ + x i \ ∂ x i ) ∩ ( ∂ − x i +1 \ ∂ x i +1 ) . If dim( w i ) = 1 then w i ∈ ∆ + x i ∩ ∆ − x i +1 , so x i ≺ w i ≺ x i +1 . Supposedim( w i ) = 0. Because ∂ + x i is pure and 1-dimensional, w i is covered by some amar hadzihasanovic element of ∆ + x i , and because w i / ∈ ∂ x i = ∆( ∂ + x i ), by [Had20b, Lemma1.16] it is in fact covered by two elements of ∆ + x i with opposite orientations.If w i ∈ ∆ + x i covers w i with orientation +, we have x i ≺ w ′ i ≺ w i .Dually, we find w ′′ i ∈ ∆ − x i +1 that covers w i with orientation − , so that w i ≺ w ′′ i ≺ x i +1 . It follows that x i ≺ x i +1 for all i ∈ { , . . . , m − } , and weconclude that x (cid:22) y .Now suppose that k = 0. A path from x to y in M U is a concatenationof two-step paths x i → w i → x i +1 where dim( x i ) ∈ { , } and w i is theonly element of ∂ +0 x i = ∂ − x i +1 . If dim( x i ) = 1 then immediately x i ≺ w i ,otherwise there is exactly one element w ′ i ∈ ∆ + x i such that ∂ +0 x i = ∂ + w ′ i , so x i ≺ w ′ i ≺ w i . Similarly we find that w i ≺ x i +1 . (cid:4) Corollary 3.16 — If U is a regular 2-molecule, the restriction of (cid:22) to 2-di-mensional elements determines a 1-order on U . (Normal 1-order) . Let U be a regular 2-molecule. The normal 1-order on U is the 1-order determined by Corollary 3.16. Example 3.18.
In the shape of the 2-diagrams , the normal 1-order is indicated by the labels of 2-cells.In general, a rule-of-thumb for reconstructing the normal 1-order from astring diagram is:1. if there is an upward path between two 2-cells, then the lowermost precedesthe uppermost ;2. if there is no such path, then the leftmost precedes the rightmost .Due to a certain flexibility in the depiction of string diagrams, the second rulemay not always strictly hold (but it will hold up to a harmless deformation ofthe picture). Corollary 3.19 —
Let P be a regular directed complex with dim( P ) ≤ .Then P is frame-acyclic. Lemma 3.20 —
Let U be a regular molecule with dim( U ) ≤ and let I ⊆ U be a 1-molecule with ∂ − I = ∂ − U and ∂ + I = ∂ +0 U . Then(a) there is a unique decomposition U = U + U − with ∂ +1 U + = ∂ − U − = I ; he smash product of monoidal theories (b) for all α ∈ { + , −} , if V ⊑ U is a 2-molecule with spherical boundary and V ∩ I = ∂ α V , then V ⊑ U α .Proof. By induction on the number m of 2-dimensional elements of U : if m = 0, then necessarily U = I and U = I I is the unique decomposition. If m >
0, we can write U = U ′ U x where U x contains a single 2-dimensionalelement x . Now either • I ⊆ U ′ , in which case we have a unique decomposition U ′ = U ′ + U ′− andwe can set U + := U ′ + and U − := U ′− U x , or • ∂ + x ⊆ I , since I traces a path in H o U through 0-dimensional and 1-di-mensional elements, and given that ∂ + x ⊆ ∂ + U , such a path can onlyenter ∂ + x through ∂ − x , traverse the entire ∂ + x , and leave through ∂ +0 x .Then I ′ := I [ ∂ − x/∂ + x ] is well-defined and a 1-molecule in U ′ ; by the in-ductive hypothesis, we have a decomposition U ′ = U ′ + U ′− relative to I ′ .Then setting U + := U ′ + ∪ cl { x } and U − := U ′− produces a decompositionof U relative to I .Uniqueness is straightforward since the removal of { x } ∪ ( ∂ + x \ ∂ x ) from adecomposition of U produces a decomposition of U ′ either relative to I or to I ′ .Let V ⊑ U be a 2-molecule with spherical boundary and V ∩ I = ∂ α V . If V is an atom, then clearly V ⊑ U α . Otherwise, observe that I is not affected bythe substitution U [ h V i /V ] and ∂ α h V i = ∂ α V ⊆ I . Decomposing U [ h V i /V ] as U ′ + U ′− , by the atom case we have h V i ⊑ U ′ α . Now U ′ α [ V / h V i ] and U ′− α arefactors of a decomposition of U relative to I , so by uniqueness U α = U ′ α [ V / h V i ]and V ⊑ U α . (cid:4) Proposition 3.21 —
Let
U, V, W be regular 2-molecules. Suppose V and W have spherical boundary, V, W ⊑ U , and V ∩ W ⊆ ∂V ∪ ∂W . Then W ⊑ U [ h V i /V ] and V ⊑ U [ h W i /W ] .Proof. Fix α ∈ { + , −} and let U ′ := U [ h V i /V ]; by assumption, as a closedsubset W is unaffected by this substitution.We construct a sequence of 1-molecules I , . . . , I n as follows. Let I := ∂ α W .For i ≥
0, if ∂ − I i = ∂ − U , then let k := i and move to the next cycle, otherwisepick a 1-dimensional element x with ∂ + x = ∂ − I i and let I i +1 := cl { x } I i .For i ≥ k , if ∂ + I i = ∂ +0 U , then let n := i and stop, otherwise pick a1-dimensional element x with ∂ − x = ∂ + I i and let I i +1 := I i cl { x } . Thisprocess terminates by finiteness of U ′ and acyclicity of H o U .Now I := I n is unaffected by the reverse substitution U = U ′ [ V / h V i ], has ∂ − I = ∂ − U and ∂ + I = ∂ +0 U , and W ∩ I = ∂ α W . Consider the unique amar hadzihasanovic decomposition U ′ + U ′− of U ′ relative to I given by Lemma 3.20. Clearly h V i ⊑ U ′ β for some β ∈ { + , −} , so U β := U ′ β [ V / h V i ] and U − β := U ′− β producesthe unique decomposition of U relative to I .Now observe that if we decompose relative to I ′ := I [ ∂ − α W/∂ α W ] insteadof I , only the 2-dimensional elements of W “switch sides” in the factorisation,so we can vary α without affecting β . Choosing α := − β , we have V ⊑ U β , W ⊑ U − β and the substitution of h V i for V , or of h W i for W , only affects one factor. (cid:4) Comment 3.22.
As a consequence of Proposition 3.21, if V and W are sub-molecules with spherical boundary of a regular 2-molecule U that only overlapon their boundaries, then they can both be substituted in U : if U [ W ′ /W ] and U [ V ′ /V ] are both defined as 2-molecules, then so are U [ W ′ /W ][ V ′ /V ] and U [ V ′ /V ][ W ′ /W ], which are in fact equal. This generalises to an arbitrarynumber V , . . . , V n ⊑ U of 2-molecules such that V i ∩ V j ⊆ ∂V i ∪ ∂V j for all i, j ∈ { , . . . , n } , i = j .Dimension 2 is, in fact, the largest dimension in which this result holds.The following is an example of a regular 3-molecule for which the analogousstatement fails; it is a simplified version of [Ste93, Section 8], itself based on[Pow91, Example 3.11].The point in our proof that fails to generalise to higher dimensions is theseemingly innocuous fact that ∂ + W can always be extended to a 1-molecule I with ∂I = ∂ U . In the example below, ∂ + W cannot be extended to any2-molecule in U [ h V i /V ] whose boundary is equal to ∂ U . Example 3.23.
Let U be the shape of the 3-diagram t b l r λ t b l x r ρ t b l x yr β t b l yr τ t b l r where we use the labels of cells to refer to the corresponding atoms of U . Thenboth V := λ ∪ τ and W := ρ ∪ β are submolecules of U , they have sphericalboundary, and they do not share any 3-atoms, so they only intersect in theboundary.However, W is not a submolecule of U [ h V i /V ], and V is not a submoleculeof U [ h W i /W ]. Indeed, there are paths ρ → y → h V i → x → β, λ → x → h W i → y → τ he smash product of monoidal theories in M U [ h V i /V ] and M U [ h W i /W ], respectively; note that we are confusingan atom with its greatest element. If W ⊑ U [ h V i /V ] or V ⊑ U [ h W i /W ],then it would be possible to substitute h W i for W in U [ h V i /V ], or h V i for V in U [ h W i /W ], to obtain a regular 3-molecule U ′ . These paths would thenbecome cycles in M U ′ , contradicting Proposition 3.8. Theorem 3.24 —
Let P be a regular directed complex with dim( P ) ≤ .Then P is frame-acyclic.Proof. It suffices to show that for all regular 3-molecules U , if frdim( U ) = k ,then M k U is acyclic. The case k = 2 is handled by Proposition 3.8, so suppose k ∈ { , } .By [Had20b, Lemma 2.5] we can decompose U as V . . . V m , where V i contains a unique 3-dimensional element x i and ∂ α V i = ∂ − α V i [ ∂ α x i /∂ − α x i ]for all α ∈ { + , −} and i ∈ { , . . . , m } .Since cl { x i } ∩ cl { x j } has dimension at most 1 when i = j , we have that1. ∂ − x i ⊆ ∂ − U for all i ∈ { , . . . , m } , and2. by [Had20a, Lemma 18], cl { x i } ∩ cl { x j } = ∂ − x i ∩ ∂ − x j ⊆ ∂ x i ∪ ∂ x j when i = j .Since for all i ∈ { , . . . , m } we have ∂ − x i ⊑ ∂ − V i and ∂ − U = ∂ − V i [ ∂ − x i − /∂ + x i − ] . . . [ ∂ − x /∂ + x ] , applying Proposition 3.21 repeatedly we find that ∂ − x i ⊑ ∂ − U and the sim-ultaneous substitution U ′ := ∂ − U [ h ∂ − x i /∂ − x ] . . . [ h ∂ − x m i /∂ − x m ] (6)is defined as a regular 2-molecule with the same frame dimension as U .Now from every path in M k U , we construct a path in M k U ′ as follows. Thepath in M k U is a concatenation of two-step paths y − → x → y + , where x ismaximal in U , y − ∈ ∂ − x and y + ∈ ∂ +1 x .If dim( x ) <
3, then this path stays inside ∂ − U , and x, y − , y + are unaffectedby the substitution (6). If dim( x ) = 3, then this path can be replaced by apath y − → ˜ x → y + in M k U ′ , where ˜ x is the greatest element of h ∂ − x i .Assuming there is a cycle in M k U , with this procedure we construct a cyclein M k U ′ , which contradicts Corollary 3.19. Thus M k U is acyclic. (cid:4) amar hadzihasanovic
4. Pros and diagrammatic sets . Given a regular directed complex P and n ∈ N , let σ ≤ n P ⊆ P be theclosed subset of elements x ∈ P with dim( x ) ≤ n . Then M oℓ ( σ ≤ n P ) ∗ and σ ≤ n M oℓP ∗ are isomorphic n -categories.By Lemma 3.5 combined with Theorem 3.24, for n ≤ n -category σ ≤ n M oℓP ∗ admits the structure of a polygraph with { cl { x } | dim( x ) ≤ n } asgenerating cells. Because, in general, for an ω -category X , σ ≤ k X = σ ≤ k ( τ ≤ n X ) when k < n, the 2-category τ ≤ M oℓP ∗ has the structure of a bicoloured pro with generators { cl { x } | dim( x ) ≤ } .Moreover, if f : P → Q is a morphism in DCpx R , then τ ≤ M oℓf ∗ sendseach generator cl { x } to a generator cl { f ( x ) } , so it is compatible with thisstructure. This defines a functor P : DCpx R → Pro bi that fits into a com-mutative square DCpx R Pro bi ω Cat Cat . P M oℓ − ∗ τ ≤ U Because
DCpx R is small, Set is locally small, and by Corollary 2.16
Pro bi has all small colimits, by [Rie17, Corollary 6.2.6] the left Kan extension of P : DCpx R → Pro bi along the embedding DCpx R ֒ → Set exists. Thisproduces a functor P : Set → Pro bi . Remark 4.2.
We may reason as in [Had20b, Proposition 7.10] to show that P : DCpx R → Pro bi preserves the colimits that are already in DCpx R , anddeduce from [Corollary 1.34, ibid. ] that the left Kan extension of P along DCpx R ֒ → Set is the left Kan extension of its restriction to along theYoneda embedding. (Diagrammatic nerve of bicoloured pros) . The diagrammatic nerve ofbicoloured pros is the right adjoint N : Pro bi → Set to the functor P : Set → Pro bi . . In each bicoloured pro ( T, T ), he smash product of monoidal theories morphisms P I n → ( T, T ) classify 1-cells in T of the form a . . . a n ,where a i ∈ T (including T ) for all i ∈ { , . . . , n } , and • morphisms P U n,m → ( T, T ) in Pro bi classify 2-cells of type a . . . a n ⇒ b . . . b m in T where a i , b j ∈ T .These correspond to morphisms I n → N ( T, T ) and U n,m → N ( T, T ), respect-ively, in Set , that is, 1-diagrams and 2-cells in N ( T, T ).If U is a 3-atom, a morphism e : U → N ( T, T ) restricts, for each α ∈ { + , −} ,to a 2-diagram ∂ α e of shape ∂ α U in N ( T, T ), whose transpose d ∂ α e : P ∂ α U → T is a diagram of 2-cells in T . Because T is a 2-category,1. the morphism b e : P U → ( T, T ) exhibits an equation between the compos-ites of the diagrams d ∂ + e and d ∂ − e in T , and2. if e ′ : U → N ( T, T ) is another 3-cell with ∂ α e ′ = ∂ α e for all α ∈ { + , −} ,then e = e ′ .More in general, if U is an atom, then a cell U → N ( T, T ) is uniquely determ-ined by its restriction σ ≤ e to σ ≤ U ⊆ U . Lemma 4.5 —
Let X be a diagrammatic set, ( T, T ) a bicoloured pro, andlet f, g : X → N ( T, T ) be morphisms of diagrammatic sets. If f ( x ) = g ( x ) forall 2-cells x in X , then f = g .Proof. Let x : U → X be a cell in X with dim( U ) >
2. Then f ( x ) and g ( x )are uniquely determined by their restrictions σ ≤ ( f ( x )) = ( σ ≤ x ); f, σ ≤ ( g ( x )) = ( σ ≤ x ); g to the directed complex σ ≤ U . If f and g agree on 2-cells, these are equal. (cid:4) Proposition 4.6 —
The functor N is full and faithful.Proof. Suppose N f = N g for two morphisms f, g : ( T, T ) → ( S, S ) in Pro bi .Given a 2-cell ϕ : ( a , . . . , a n ) ⇒ ( b , . . . , b m ) in T , classified by a morphism ϕ : P U n,m → ( T, T ) with transpose b ϕ : U n,m → N ( T, T ), we have b ϕ ; N f = b ϕ ; N g in Set . It follows that ϕ ; f = ϕ ; g in Pro bi , that is, f ( ϕ ) = g ( ϕ ). Because f and g agree on all 2-cells, they are equal. This proves that N is faithful.Let f ′ : N ( T, T ) → N ( S, S ) be a morphism of diagrammatic sets. Givena 2-cell ϕ : ( a , . . . , a n ) ⇒ ( b , . . . , b m ) in T , classified by ϕ : P U n,m → ( T, T ) amar hadzihasanovic with transpose b ϕ : U n,m → N ( T, T ), we define f ( ϕ ) to be the unique 2-cellin S whose classifying morphism f ( ϕ ) : P U n,m → ( S, S ) is the transpose of b ϕ ; f ′ : U n,m → N ( S, S ).We want to show that f determines a morphism of bicoloured pros. It isstraightforward to verify that f is compatible with all boundaries and withcomposition and units for 1-cells.Let x be a 1-cell in T , classified by x : P I n → ( T, T ) with transpose b x ; theunit εx is classified by εx : P U n,n → ( T, T ) with transpose c εx . Let U := O ( I n ) ⇒ U n,n , where O ( − ) is the construction of [Had20b, §2.21]. This is well-defined as aregular 3-atom. There is a unique cell e : U → N ( T, T ) such that1. e is equal to ε b x on ∂ − U , see [§4.16, ibid. ], and2. e is equal to c εx on ∂ + U .Then e ; f ′ : U → N ( S, S ) is a 3-cell of type εf ′ ( b x ) ⇒ f ′ ( c εx ), whose transposeexhibits the equation εf ( x ) = f ( εx ) in S .Next, let ϕ, ψ be 2-cells in T , classified by morphisms ϕ : P U n,m → ( T, T ) , ψ : P U p,ℓ → ( T, T )with transposes b ϕ , b ψ . Suppose that ϕ ψ is defined; then we may assume p = m , and the composite is classified by ϕ ψ : P U n,ℓ → ( T, T )with transpose \ ϕ ψ . Let U := ( U n,m U m,ℓ ) ⇒ U n,ℓ ;this is a regular 3-atom. There is a unique cell e : U → N ( T, T ) such that1. e is equal to b ϕ on U n,m ֒ → ∂ − U and to b ψ on U m,ℓ ֒ → ∂ − U , and2. e is equal to \ ϕ ψ on ∂ + U .Then e ; f ′ : U → N ( S, S ) is a cell of type f ′ ( b ϕ ) f ′ ( b ψ ) ⇒ f ′ ( \ ϕ ψ ) , whose transpose exhibits an equation f ( ϕ ) f ( ψ ) = f ( ϕ ψ ) in S .Finally, suppose that ϕ ψ is defined; this composite is classified by ϕ ψ : P U n + p,m + ℓ → ( T, T ) he smash product of monoidal theories with transpose \ ϕ ψ . Let U := (( U n,m U p,ℓ ) U m + ℓ,m + ℓ ) ⇒ U n + p,m + ℓ . This is a regular 3-atom and there is a unique cell e : U → N ( T, T ) such that1. e is equal to b ϕ on U n,m ֒ → ∂ − U and to b ψ on U p,ℓ ֒ → ∂ − U ,2. e is equal to the transpose of ε ( ∂ + ϕ ∂ + ψ ) on U m + ℓ,m + ℓ ֒ → ∂ − U , and3. e is equal to \ ϕ ψ on ∂ + U .Then e ; f ′ : U → N ( S, S ) is a cell of type( f ′ ( b ϕ ) f ′ ( b ψ )) f ′ ( \ ε ( ∂ + ϕ ∂ + ψ )) ⇒ f ′ ( \ ϕ ψ ) . whose transpose exhibits the equation( f ( ϕ ) f ( ψ )) f ( ε ( ∂ + ϕ ∂ + ψ )) = f ( ϕ ψ )in S . Because we already know that f is compatible with units, we deducethat f ( ϕ ) f ( ψ ) = f ( ϕ ψ ).This proves that f : ( T, T ) → ( S, S ) is a morphism of bicoloured pros. Now N f and f ′ are morphisms N ( T, T ) → N ( S, S ) that, by construction, agree onall 2-cells of N ( T, T ). It follows from Lemma 4.5 that N f = f ′ . This provesthat N is full. (cid:4) Comment 4.7.
String diagrams are commonly used to depict cells in a pro,usually after an appeal to the Joyal–Street soundness result [JS91]. The dia-grammatic nerve construction offers an alternative justification, where dia-grams are attributed a combinatorial, rather than topological interpretation.Unless otherwise stated, our string diagrams will represent diagrams in adiagrammatic set. A caveat is that, contrary to custom, we are not allowedto have nodes with no input or output wires; instead, we need to explicitlyintroduce units and unitors [Had20b, §4.17] where necessary.To distinguish them visually, we draw unit 1-cells as dotted wires, andunitor 2-cells as “dotless nodes”: for example, a 2-cell of type (0) ⇒ (1) in aone-sorted pro will be depicted asas opposed to , while a left unitor 2-cell will be depicted asas opposed to . This may seem like unnecessary trouble in dimension 2; the pay-off is thatdiagrammatic sets provide sound diagrammatic reasoning in all dimensions. amar hadzihasanovic . Our next goal is to construct a functor G : DCpx R → GrayCat , differ-ent from the “obvious” one obtained by composing M oℓ − ∗ : DCpx R → ω Cat with τ ≤ : ω Cat → Cat and then including 3
Cat in GrayCat . In particu-lar, G P will in general have non-trivial interchangers, so it will not be a strict3-category.Every regular directed complex is the colimit of the diagram of inclusions ofits atoms [Had20b, Corollary 1.34]. We impose that G preserve these colimitdiagrams. Then it suffices to define G on atoms of increasing dimension. Foreach n ∈ N + {− } , let n be the full subcategory of on the atoms ofdimension ≤ n . ( G in dimension ≤ . On regular atoms of dimension ≤
2, we define G tobe M oℓ − : → Cat followed by the embedding 3
Cat ֒ → GrayCat . Weextend G along colimits to all regular directed complexes of dimension ≤ . Let P be a 2-dimensional regular directed complex. Then G ( σ ≤ P ) isequal to (the image under the embedding 3 Cat ֒ → GrayCat of) M oℓσ ≤ P ∗ and has the structure of a 1-(pre)polygraph with the 1-atoms of P as gener-ators. Now, for all 2-atoms x ∈ P , ∂O M oℓ (cl { x } ) M oℓ ( ∂x ) O cl { x } is a pushout both in ω PreCat and
GrayCat . By the dual of the pullbacklemma, the pushout of the span a dim( x )= n M oℓ ( ∂x ) ֒ → a dim( x )= n M oℓ (cl { x } ) , a dim( x )= n M oℓ ( ∂x ) ֒ → M oℓσ ≤ P ∗ in ω PreCat determines a 2-prepolygraph ( G P ) , while in GrayCat it is equi-valent to the construction of G P . The results of [FM18, Section 1.6] implythat1. G P is obtained from ( G P ) by freely attaching some 3-cells (interchangegenerators) indexed by generating cells of ( G P ) , and imposing some equa-tions of 3-cells, so in particular2. ( G P ) is the 2-skeleton of G P . he smash product of monoidal theories In the terminology of Forest and Mimram, P determines a presentation of the2-precategory ( G P ) , which can be completed to a Gray presentation of G P by freely adding the necessary structural generators . Lemma 4.11 —
Let U be a regular 2-molecule. There is a bijective corres-pondence between1. cells of rank 2 in G U , and2. 2-molecules V ⊆ U together with a 1-order.Proof. By the discussion in §4.10, the 2-cells in G U are the same as the 2-cellsin the 2-prepolygraph ( G U ) , so they are freely generated by the atoms of U under principal compositions (§1.33), subject to the axioms of ω -precategories.Let V be a 2-molecule with a 1-order ( x , . . . , x m ). By Lemma 3.9, weobtain a decomposition V = V . . . V m . (7)Now each V i has frame dimension 0 or -1, so it has a (clearly unique) decom-positioncl { y i, } . . . cl { y i,k } cl { x i } cl { y i,k +1 } . . . cl { y i,p } (8)where dim( y i,j ) = 1 for all j ∈ { , . . . , p } . Replacing the (8) into (7), weobtain a decomposition of U into atoms using only principal compositions,which determines a cell of rank 2 in ( G U ) .Conversely, by [FM18, Proposition 2], every cell y of rank 2 in ( G U ) hasa unique expression of the form y . . . y m where y i is an expression ofthe form (8). Now the expression of y is also a valid expression for a 2-cell in M oℓU ∗ , which by Remark 3.14 is equal to M oℓU , so it determines a 2-molecule V ⊆ U together with a decomposition into atoms. From this decompositionwe recover uniquely a 1-order ( x , . . . , x m ) on V . The two constructions areclearly inverse to each other. (cid:4) Remark 4.12.
By Lemma 4.11, every cell of rank 2 in G U is identified uniquelyby a pair ( V, ( x i ) mi =1 ) of a 2-molecule and a 1-order.More in general, if P is a 2-dimensional regular directed complex, a pair( V, ( x i ) mi =1 ) of a 2-molecule in P and a 1-order on it determines a unique cellof rank 2 in G P , although these may not exhaust all cells of rank 2 when P isnot totally loop-free. Proposition 4.13 —
Let U and V ⊆ U be regular 2-molecules and let ( x , . . . , x m ) and ( x ′ , . . . , x ′ m ) be two 1-orders on V . Then in G U there isa unique 3-cell from ( V, ( x i ) mi =1 ) to ( V, ( x ′ i ) mi =1 ) . amar hadzihasanovic Proof.
For each 1-order ( x , . . . , x m ) on V , let w (( x i ) mi =1 ) be equal to thenumber of pairs ( i, j ) such that i < j but x j ≺ x i in the total order on U . Then • ≤ w (( x i ) mi =1 ) ≤ (cid:0) m (cid:1) , • w (( x i ) mi =1 ) = 0 if and only if ( x i ) mi =1 is the normal 1-order, and • for all non-trivial interchangers χ x,y : ( V, ( x i ) mi =1 ) ⇒ ( V, ( x ′ i ) mi =1 ), we have w (( x i ) mi =1 ) < w (( x ′ i ) mi =1 ).It follows that w ( − ) induces a termination order on the 2-cells of G U in theterminology of [FM18, Section 2.2]. Because the Gray presentation of G U determined by U as in §4.10 has no non-structural 3-generators, it is alwayslocally confluent, so [Theorem 11, ibid. ] applies and G U has at most one 3-cellbetween any parallel pair of 2-cells. This proves uniqueness.For existence, it suffices to observe that, if w (( x i ) mi =1 ) >
0, then there is anon-trivial inverse interchanger with input ( V, ( x i ) mi =1 ); we leave the proof asan exercise. Applying inverse interchangers repeatedly, we obtain invertible3-cells of type( V, ( x i ) mi =1 ) ⇒ ( V, normal 1-order) , ( V, ( x ′ i ) mi =1 ) ⇒ ( V, normal 1-order) . Composing the first with the inverse of the second produces a 3-cell of type( V, ( x i ) mi =1 ) ⇒ ( V, ( x ′ i ) mi =1 ). (cid:4) ( G in dimension 3) . Let U be a regular 3-atom. We define G U to be thepushout ∂O G U G ( ∂U ) O f in GrayCat , where f sends 2 α to ( ∂ α U, normal 1-order) for each α ∈ { + , −} .Now every map f : U → V in determines an assignment of generators of G V to generators of G U which is compatible with boundaries, hence extendsuniquely to a functor G f : G U → G V . This defines G : → GrayCat . Weextend G along colimits to all regular directed complexes of dimension ≤ . By construction, if P is a regular directed complex of dimension 3, wecan associate to each 3-atom U of P a 3-cell J U K : ( ∂ − U, normal 1-order) ⇒ ( ∂ + U, normal 1-order)in G P . We want to extend this assignment to all 3-molecules in P . he smash product of monoidal theories . Suppose that U contains a single 3-dimensional element x . Then ∂ α x ⊑ ∂ α U for all α ∈ { + , −} , the substitution ∂ α U [ h ∂ α x i /∂ α x ] is well-defined, and ∂ − U [ h ∂ − x i /∂ − x ] = ∂ + U [ h ∂ + x i /∂ + x ] . Pick a 1-order ( x i ) mi =1 on ∂ α U [ h ∂ α x i /∂ α x ]; then cl { x k } = h ∂ α x i for a unique k ∈ { , . . . , m } . Let y , . . . , y p be the normal 1-order on ∂ − x and let z , . . . , z q be the normal 1-order on ∂ + x . Then( x − i ) m + p − i =1 := ( x , . . . , x k − , y , . . . , y p , x k +1 , . . . , x m ) , ( x + i ) m + q − i =1 := ( x , . . . , x k − , z , . . . , z q , x k +1 , . . . , x m )are 1-orders on ∂ − U and ∂ + U , respectively.Now substituting J cl { x } K for cl { x k } in the decomposition of ∂ α U [ h ∂ α x i /∂ α x ]corresponding to the 1-order ( x i ) mi =1 yields a valid expression for a 3-cell c [ x ] : ( ∂ − U, ( x − i ) m + p − i =1 ) ⇒ ( ∂ + U, ( x + i ) m + q − i =1 ) (9)in G P . By Proposition 4.13, there are unique 3-cells χ − : ( ∂ − U, normal 1-order) ⇒ ( ∂ − U, ( x − i ) m + p − i =1 ) ,χ + : ( ∂ + U, ( x + i ) m + q − i =1 ) ⇒ ( ∂ + U, normal 1-order)obtained as composites of interchangers and inverse interchangers, respect-ively. We define J U K to be the composite χ − c [ x ] χ + : ( ∂ − U, normal 1-order) ⇒ ( ∂ + U, normal 1-order) . We need to show that this is independent of our choice of 1-order ( x i ) mi =1 .Suppose ( x ′ i ) mi =1 is another 1-order on ∂ α U [ h ∂ α x i /∂ α x ], leading to a potentiallydifferent interpretation χ ′− c ′ [ x ] χ ′ + . There are unique 3-cells ψ − : ( ∂ − U, ( x − i ) m + p − i =1 ) ⇒ ( ∂ − U, ( x ′− i ) m + p − i =1 ) ,ψ + : ( ∂ + U, ( x + i ) m + q − i =1 ) ⇒ ( ∂ + U, ( x ′ + i ) m + q − i =1 )obtained as composites of interchangers and inverse interchangers, and sincethey “fix” ∂ − x and ∂ + x , by naturality of interchangers we have ψ − c ′ [ x ] = c [ x ] ψ + hence c ′ [ x ] = ( ψ − ) − c [ x ] ψ + and χ ′− c ′ [ x ] χ ′ + = χ ′− ( ψ − ) − c [ x ] ψ + χ ′ + . Finally, by Proposition 4.13, χ ′− ( ψ − ) − = χ − and ψ + χ ′ + = χ + . amar hadzihasanovic . Let U be any regular 3-molecule in P and fix a 2-order ( x , . . . , x m )on U . Then Lemma 3.9 combined with Theorem 3.24 gives a decomposition U = V . . . V m where x i is the only 3-dimensional element of V i for each i ∈ { , . . . , m } . We let J U K in G P be the composite J V K . . . J V m K of the3-cells J V i K : ( ∂ − V i , normal 1-order) ⇒ ( ∂ + V i , normal 1-order)defined in §4.16.We need to show that this interpretation is independent of the 2-order chosenon U . Observe that any pair of 2-orders on U is related by a sequence ofelementary transpositions of consecutive elements that are not connected bya path in M U . Thus it suffices to show that if W W = W ′ W ′ as 3-molecules, where x is the only 3-dimensional element in W and W ′ , while y is the only 3-dimensional element in W and W ′ , then J W K J W K = J W ′ K J W ′ K . The interpretation of W involves a choice of 1-order on ∂ α W [ h ∂ α x i /∂ α x ]but it is independent of this choice. Now ∂ − y ⊑ ∂ − W = ∂ + W , and since x and y are not connected by a path in M U , necessarily ∂ + x ∩ ∂ − y ⊆ cl { x } ∩ cl { y } ⊆ ∂ x ∪ ∂ y, and by Proposition 3.21 ∂ − y ⊑ ∂ + W [ h ∂ + x i /∂ + x ].Applying the known equalities between the boundaries of W , W , W ′ , W ′ ,we deduce that the double substitutions ∂ + W [ h ∂ + x i /∂ + x ][ h ∂ − y i /∂ − y ] ,∂ − W [ h ∂ − y i /∂ − y ][ h ∂ + x i /∂ + x ] ,∂ + W ′ [ h ∂ + y i /∂ + y ][ h ∂ − x i /∂ − x ] ,∂ − W ′ [ h ∂ − x i /∂ − x ][ h ∂ + y i /∂ + y ]are all well-defined and equal to the same regular 2-molecule. Fix a 1-order( z i ) pi =1 on it. Then h ∂ α x i = cl { z k } and h ∂ β y i = cl { z ℓ } for a unique pair k, ℓ ∈ { , . . . , p } . Now • to interpret W , choose the 1-order on ∂ + W [ h ∂ + x i /∂ + x ] obtained byreplacing z ℓ with the normal 1-order on ∂ − y in ( z i ) pi =1 , • to interpret W , choose the 1-order on ∂ − W [ h ∂ − y i /∂ − y ] obtained byreplacing z k with the normal 1-order on ∂ + x in ( z i ) pi =1 , he smash product of monoidal theories and similarly for W ′ and W ′ . With the construction of §4.16, any choices of1-orders lead to expressions χ − c [ x ] χ +1 χ − c [ y ] χ +2 ,χ ′− c ′ [ y ] χ ′ +1 χ ′− c ′ [ x ] χ ′ +2 , and with the particular choice that we made, χ − = χ ′− , χ +2 = χ ′ +2 , while χ +1 χ − and χ ′ +1 χ ′− are units, so they can be eliminated. Finally, χ − c [ x ] c [ y ] χ +2 = χ − c ′ [ y ] c ′ [ x ] χ +2 is a consequence of axiom 4 of Gray-categories. This proves that J W K J W K is equal to J W ′ K J W ′ K , and we conclude that J U K is independent of the choiceof 2-order. Example 4.18.
Let U be the shape of diagram (3). We introduce names forsome atoms of U as follows: a deb c w zx y ϕ a deb z ′ w ′ x y ψ a e z ′ w ′ y ′ x ′ .The 3-atoms ϕ and ψ are interpreted in G U as 3-cells J ϕ K : ( a z ) ( w c ) ⇒ z ′ w ′ , J ψ K : ( x d ) ( b y ) ⇒ x ′ y ′ ;notice that in this case both ∂ α ϕ and ∂ α ψ admit a single 1-order, which isnecessarily the normal 1-order.We pick the 2-order ( ϕ, ψ ) on U , which determines the decomposition U = V V , V := ϕ ∪ ∂ − ψ, V := ψ ∪ ∂ + ϕ. To interpret V in G U , first we need to consider V [ h ∂ − ϕ i /∂ − ϕ ]. This is theshape of the diagram zwx y amar hadzihasanovic on which we pick the 1-order ( x, y, zw ). The 3-cell c [ ϕ ] corresponding to this1-order, defined as in (9), is( a x d ) ( a b y ) ( J ϕ K e )of type ( ∂ − V , ( x, y, z, w )) ⇒ ( ∂ + V , ( x, y, z ′ , w ′ )) in G U .The normal 1-order on ∂ − V is in fact ( x, z, w, y ). Applying a pair of inter-changers to first move w after y , then z after y , we obtain a 3-cell χ − : ( ∂ − V , ( x, z, w, y )) ⇒ ( ∂ − V , ( x, y, z, w )) . Similarly, the normal 1-order on ∂ + V is ( x, z ′ , w ′ , y ), so we apply a pair ofinverse interchangers to move y after z ′ , then y after w ′ , producing a 3-cell χ +1 : ( ∂ + V , ( x, y, z ′ , w ′ )) ⇒ ( ∂ + V , ( x, z ′ , w ′ , y )) . Then J V K is defined to be χ − c [ ϕ ] χ +1 .Next, consider V [ h ∂ − ψ i /∂ − ψ ]. This is the shape of the diagram z ′ w ′ xy which admits only the 1-order ( xy, z ′ , w ′ ). Correspondingly, we construct the3-cell c [ ψ ] := ( a J ψ K ) ( z ′ e ) ( w ′ e )which is of type ( ∂ − V , ( x, y, z ′ , w ′ )) ⇒ ( ∂ + V , ( x ′ , y ′ , z ′ , w ′ )). The normal1-order on ∂ − V is ( x, z ′ , w ′ , y ), and we have a composite of interchangers χ − : ( ∂ − V , ( x, z ′ , w ′ , y )) ⇒ ( ∂ − V , ( x, y, z ′ , w ′ )) , which is in fact the inverse of χ +1 . On the other hand, ( x ′ , y ′ , z ′ , w ′ ) is alreadythe normal 1-order on ∂ + V , so J V K is just χ − c [ ψ ]. Overall, J U K is χ − c [ ϕ ] c [ ψ ] : ( ∂ − U, ( x, z, w, y )) ⇒ ( ∂ + U, ( x ′ , y ′ , z ′ , w ′ )) . If we had picked the 2-order ( ψ, ϕ ), we would have instead ended up withthe expression χ − c ′ [ ψ ] c ′ [ ϕ ] where c ′ [ ψ ] := ( a J ψ K ) ( a z e ) ( w c e ) ,c ′ [ ϕ ] := ( a x ′ ) ( a y ′ ) ( J ϕ K e ) . It follows from axiom 4 of Gray-categories that c [ ϕ ] c [ ψ ] = c ′ [ ψ ] c ′ [ ϕ ],confirming that J U K is independent of the 2-order on U . he smash product of monoidal theories ( G in dimension ≥ . Let U be a regular 4-atom. We define G U to bethe quotient of G ( ∂U ) by the equation J ∂ − U K = J ∂ + U K , where the 3-molecules ∂ α U are interpreted in G ( ∂U ) as by §4.17.If f : U → V is a map in , its restriction to ∂U determines a functor G ( ∂f ) : G ( ∂U ) → G V . If U is a 4-atom, then either dim( f ( U )) < f ( ∂ − U ) = f ( ∂ + U ), or dim( f ( U )) = 4, f ( U ) = V and f ( ∂ α U ) = ∂ α V foreach α ∈ { + , −} . In either case, G ( ∂f ) is compatible with the equation J ∂ − U K = J ∂ + U K , so it factors uniquely through a functor G f : G U → G V .This defines G : → GrayCat , and we extend it along colimits to allregular directed complexes of dimension ≤ f : P → Q is a map of regular directed complexes of arbitrarydimension, it restricts to a map σ ≤ f : σ ≤ P → σ ≤ Q , and we let G f be equalto G ( σ ≤ f ). This defines G : DCpx R → GrayCat . Comment 4.20.
By construction, G ignores any elements of dimension > . Because
GrayCat has all small colimits, we are in the conditions of[Rie17, Corollary 6.2.6] and we can define a functor G : Set → GrayCat as the left Kan extension of G : DCpx R → GrayCat along the embedding
DCpx R ֒ → Set . Remark 4.22.
Since we made sure at every step that G preserve the colimitsin DCpx R , this is in fact equal to the left Kan extension of the restriction of G to along the Yoneda embedding. Remark 4.23.
For the usual reasons, G has a right adjoint, of which we will notmake use. Unlike the diagrammatic nerve of pros, it is not full; see [Had20b,Remark 7.20] for a counterexample that also applies to the present case. Comment 4.24.
The following (generally non-commutative) diagram of func- amar hadzihasanovic tors recaps the adjunctions that we have established: SetPro bi Pro Prob Prop . GrayCat BrMonCat str G ⊥ ⊥ B ⊥ P ⊣ N F ⊥ U r ⊥ U ⊣ U
5. The smash product
We reconstruct the tensor product of props, as defined by Hackney and Robertson,as a reflection of an “external” tensor product of pros producing a prob, whosecombinatorics are only slightly more involved.
Lemma 5.1 —
Let s be a permutation on the set { , . . . , n } . Then s is eitherthe identity or admits a unique decomposition s = s ; . . . ; s p with the following properties. For each i ∈ { , . . . , p } , let s ( i ) := s i ; . . . ; s p .Then1. s i is an elementary transposition ( k k + 1) of two consecutive elements,and2. k is the least element of { , . . . , n } such that s ( i ) ( k + 1) < s ( i ) ( k ) .Proof. We construct, step by step, decompositions s = s ; . . . ; s i − ; s ( i ) . For i = 1, we let s = s (1) trivially. For each i ≥
1, if s ( i ) is the identity, we let p := i − k such that s ( i ) ( k + 1) < s ( i ) ( k ). We let s i := ( k k + 1) and s ( i +1) := s − i ; s ( i ) . Then s = s ; . . . ; s i ; s ( i +1) .At each step, the number of pairs j, j ′ ∈ { , . . . , n } such that j < j ′ but s ( i ) ( j ′ ) < s ( i ) ( j ) strictly decreases, and it is equal to 0 if and only if s ( i ) isthe identity. It follows that the algorithm terminates after a finite number ofsteps, producing a decomposition with the desired properties.Uniqueness is clear, since the conditions determine the factor s i uniquely ateach step. (cid:4) he smash product of monoidal theories . Let s be a permutation on the set { , . . . , n } . For all 1-cells ( a , . . . , a n )in a prob ( T, T ), we define an invertible 2-cell σ ( s ) : ( a , . . . , a n ) ⇒ ( a s (1) , . . . , a s ( n ) )in T ; the dependence of σ ( s ) on ( a , . . . , a n ) is left implicit. • If s is the identity, we let σ ( s ) be the unit on ( a , . . . , a n ). • If s is an elementary transposition ( k k + 1) of two consecutive elements,we let σ ( s ) := a . . . a k − σ a k ,a k +1 a k +2 . . . a n . • In general, if s = s ; . . . ; s p is the decomposition of s given by Lemma 5.1,we let σ ( s ) := σ ( s ) . . . σ ( s p ) . We also define a second invertible 2-cell σ ∗ ( s ) : ( a , . . . , a n ) ⇒ ( a s (1) , . . . , a s ( n ) )by σ ∗ ( s ) := ( σ ( s − )) − . Remark 5.3.
If ( T, T ) is a prop, then σ ( s ) = σ ∗ ( s ) for all permutations s . Example 5.4.
Let s be the permutation (1 , , , , (3 , , , , s given by Lemma 5.1 is s = (2 3); (1 2); (3 4); (4 5); (3 4) . We use the graphical notation a bba ,for the braiding σ a,b and the inverse braiding σ ∗ a,b , respectively, in a prob. The2-cells σ ( s ) and σ ∗ ( s ) of type ( a , a , a , a , a ) ⇒ ( a , a , a , a , a ) can bepictured as , ,respectively. In a prop, these are identical and may both be pictured as their“shadow” . amar hadzihasanovic . Let ( T, T ) be a prob and let { a i,j | ≤ i ≤ n, ≤ j ≤ m } be a doublyindexed collection of 1-cells in T . We denote by (( a i,j ) ni =1 ) mj =1 the 1-cell( a , , . . . , a n, , a , , . . . , a n, , . . . . . . , a ,m , . . . , a n,m ) (10)and by (( a i,j ) mj =1 ) ni =1 the 1-cell( a , , . . . , a ,m , a , , . . . , a ,m , . . . . . . , a n, , . . . , a n,m ) . (11)We let σ : (( a i,j ) mj =1 ) ni =1 ⇒ (( a i,j ) ni =1 ) mj =1 ,σ ∗ : (( a i,j ) ni =1 ) mj =1 ⇒ (( a i,j ) mj =1 ) ni =1 be equal to σ ( s − ) and its inverse σ ∗ ( s ), respectively, for the permutation s implied by the reordering of (10) into (11). Example 5.6.
The 2-cells σ : ( a , , a , , a , , a , , a , , a , ) ⇒ ( a , , a , , a , , a , , a , , a , ) ,σ ∗ : ( a , , a , , a , , a , , a , , a , ) ⇒ ( a , , a , , a , , a , , a , , a , )can be pictured as , ,respectively. (Tensor product of pros) . The tensor product ( T, T ) ⊗ ( S, S ) of two pros( T, T ) and ( S, S ) is the prob ( T ⊗ S, T ⊗ S ) constructed as follows.1. Let ( T ⊗ S ) := {•} and ( T ⊗ S ) := { a ⊗ c : • ⇒ • | a ∈ T , c ∈ S } .This determines σ ≤ ( T ⊗ S ) together with its 1-polygraph structure, whichmakes it a pro.2. Construct the coproducts a c ∈ S ( T, T ) , a a ∈ T ( S, S ) (12)in Pro . Denote by − ⊗ d : ( T, T ) ֒ → a c ∈ S ( T, T ) , b ⊗ − : ( S, S ) ֒ → a a ∈ T ( S, S ) he smash product of monoidal theories the inclusions into the d -indexed and b -indexed summand, respectively.There are morphisms σ ≤ ( T ⊗ S ) → a c ∈ S ( T, T ) , σ ≤ ( T ⊗ S ) → a a ∈ T ( S, S )uniquely determined by the “tautologous” assignments a ⊗ c a ⊗ c .Construct the pushout σ ≤ ( T ⊗ S ) T (cid:3) S ` a ∈ T ( S, S ) ` c ∈ S ( T, T ) (13)in Pro .3. Construct the free prob F ( T (cid:3) S ) and quotient it by the following equations:for all 2-cells ϕ : ( a , . . . , a n ) ⇒ ( b , . . . , b m ) in T , ψ : ( c , . . . , c p ) ⇒ ( d , . . . , d q ) in S ,the 1-composite of( a ⊗ ψ ) . . . ( a n ⊗ ψ ) : (( a i ⊗ c k ) pk =1 ) ni =1 ⇒ (( a i ⊗ d ℓ ) qℓ =1 ) ni =1 ,σ : (( a i ⊗ d ℓ ) qℓ =1 ) ni =1 ⇒ (( a i ⊗ d ℓ ) ni =1 ) qℓ =1 , ( ϕ ⊗ d ) . . . ( ϕ ⊗ d q ) : (( a i ⊗ d ℓ ) ni =1 ) qℓ =1 ⇒ (( b j ⊗ d ℓ ) mj =1 ) qℓ =1 ,σ ∗ : (( b j ⊗ d ℓ ) mj =1 ) qℓ =1 ⇒ (( b j ⊗ d ℓ ) qℓ =1 ) mj =1 is equal to the 1-composite of σ : (( a i ⊗ c k ) pk =1 ) ni =1 ⇒ (( a i ⊗ c k ) ni =1 ) pk =1 , ( ϕ ⊗ c ) . . . ( ϕ ⊗ c p ) : (( a i ⊗ c k ) ni =1 ) pk =1 ⇒ (( b j ⊗ c k ) mj =1 ) pk =1 ,σ ∗ : (( b j ⊗ c k ) mj =1 ) pk =1 ⇒ (( b j ⊗ c k ) pk =1 ) mj =1 , ( b ⊗ ψ ) . . . ( b m ⊗ ψ ) : (( b j ⊗ c k ) pk =1 ) mj =1 ⇒ (( b j ⊗ d ℓ ) qℓ =1 ) mj =1 . We label this equation ϕ ⊗ ψ .Note that any composite indexed by an empty list must be interpreted as aunit on • of the appropriate dimension.If f : ( T, T ) → ( T ′ , T ′ ) and g : ( S, S ) → ( S ′ , S ′ ) are morphisms of pros,we can define morphisms a c ∈ S ( T, T ) → U ( T ′ ⊗ S ′ , T ′ ⊗ S ′ ) , a a ∈ T ( S, S ) → U ( T ′ ⊗ S ′ , T ′ ⊗ S ′ ) ,x ⊗ c f ( x ) ⊗ g ( c ) , a ⊗ y f ( a ) ⊗ g ( y ) . amar hadzihasanovic Taking the transpose morphisms in
Prob , and using the universal propertyof the pushout (13) which is preserved by F , we obtain a unique morphism F ( T (cid:3) S ) → ( T ′ ⊗ S ′ , T ′ ⊗ S ′ ) of probs which is compatible with the ϕ ⊗ ψ equations, hence factors uniquely through a morphism f ⊗ g : ( T ⊗ S, T ⊗ S ) → ( T ′ ⊗ S ′ , T ′ ⊗ S ′ ) . This defines a functor − ⊗ − : Pro × Pro → Prob . Remark 5.8.
When either ϕ or ψ is a unit, the equation ϕ ⊗ ψ holds automat-ically by the axioms of braidings. So ϕ ⊗ ψ is only non-trivial when both cellshave rank 2.One can derive, as a consequence, that the monoid N is a “relative unit”for the tensor product, in the sense that the functors N ⊗ − and − ⊗ N arenaturally isomorphic to F : Pro → Prob . Example 5.9.
We compute the tensor product
Bialg := Mon ⊗ Mon co of thetheories of monoids and comonoids. Both Mon and
Mon co are one-sorted, so Bialg is also one-sorted.In fact, the indexed coproducts (12) are equal to
Mon and
Mon co , respect-ively, while σ ≤ ( Bialg ) is isomorphic to N , so the pushout (13) can be computedas N Mon ⊎ Mon co Mon co Mon in Pro . The 2-cells in
Mon ⊎ Mon co are freely generated by those of Mon and
Mon co , modulo any equations that hold in the two factors separately: a modelof Mon ⊎ Mon co is a pair of a monoid and a comonoid structure on the sameobject.Finally, to obtain Bialg , we quotient F ( Mon ⊎ Mon co ) by the ϕ ⊗ ψ equations.It suffices to let ϕ and ψ range over 2-cells that generate Mon and
Mon co ,respectively, under composition.An obvious choice is to take the unique maps µ : (2) ⇒ (1) and η : (0) ⇒ (1)as generators of Mon , and their duals δ : (1) ⇒ (2) and ε : (1) ⇒ (0) asgenerators of Mon co . We may picture these as µ , η , δ , ε . he smash product of monoidal theories The four corresponding equations are= µ ⊗ δ , = η ⊗ δ ,= µ ⊗ ε , = η ⊗ ε .In a symmetric monoidal category, a pair of a monoid and a comonoid satisfy-ing these equations is a bialgebra [Pir02]. In a braided monoidal category, thisnotion forks into two variants, distinguished by the use of braidings or inversebraidings, classified by Bialg and by
Bialg ∗ = Mon co ⊗ Mon , respectively.
Example 5.10.
We compute the tensor product
BrCMon := Mon ⊗ Mon .We proceed as in Example 5.9 to derive that
BrCMon is the quotient of F ( Mon ⊎ Mon ) by the equations ϕ ⊗ ψ where ϕ, ψ range over { µ, η } .Using different colours to distinguish cells from each copy of Mon , these canbe pictured as = µ ⊗ µ , = η ⊗ µ ,= µ ⊗ η , = η ⊗ η .A model of Mon ⊎ Mon is a pair of monoid structures on the same object.It is a consequence of the Eckmann–Hilton argument, valid in every braidedmonoidal category, that a pair of monoid structures satisfying the equations µ ⊗ µ and η ⊗ η coincide with a single commutative monoid structure. Theequations η ⊗ µ and µ ⊗ η are derivable from the rest.We conclude that BrCMon is the braided monoidal theory of commutat-ive monoids, whose reflection r ( BrCMon ) is isomorphic to
CMon . Dually,
Mon co ⊗ Mon co is the braided monoidal theory of commutative comonoids. (Tensor product of props) . The tensor product ( T, T ) ⊗ S ( S, S ) of twoprops ( T, T ) and ( S, S ) is the quotient of r ( U ( T, T ) ⊗ U ( S, S )) by the equa-tions σ a,b ⊗ c = σ a ⊗ c,b ⊗ c , a ⊗ σ c,d = σ a ⊗ c,a ⊗ d (14) amar hadzihasanovic for all a, b ∈ T and c, d ∈ S , where σ a,b and σ c,d are the original braidingsof T and S .As shown in [HR15, Section 3], the tensor product of props is part of asymmetric monoidal closed structure on Prop , whose unit is the theory ofpermutations S . Example 5.12.
Given a prop ( T, T ), the tensor product ( T, T ) ⊗ S CM on co is a cartesian prop, also known as a Lawvere theory. It is in fact the freecartesian prop on ( T, T ) [Bae06]. Comment 5.13.
The tensor product of props is compatible with the tensorproduct of pros in the sense that the diagram of functors
Pro × Pro PropProp × Prop Prob ⊗ rF × rF ⊗ S r (15)commutes up to natural isomorphism. The reason why this works is that,when ϕ or ψ is a braiding σ a,b , the equation ϕ ⊗ ψ combined with (14) holdsautomatically in a prop. It follows that, while UrF ( T, T ) ⊗ UrF ( S, S ) hasadditional generators and equations compared to ( T, T ) ⊗ ( S, S ), these areall trivialised by the combined action of r and (14).This fact is specific to props and does not generalise to probs: the quo-tient of UF ( T, T ) ⊗ UF ( S, S ) by (14) in Prob is not in general isomorphicto ( T, T ) ⊗ ( S, S ). For example, the quotient of U B ⊗ U B by (14) is notisomorphic to N ⊗ N ≃ B . Indeed, if σ , : (2) ⇒ (2) is a braiding in B , theequation σ , ⊗ σ , becomes = ,which does not hold in the braid group on 4 strands. This can be checked byconsidering the link diagrams , , he smash product of monoidal theories and observing that the first is an unlink while the second is not. On the otherhand, the reflected equation =is valid in the theory of permutations.As a consequence, there does not seem to be an interesting monoidal struc-ture on Prob that generalises the one on
Prop . Following the interpretationof the tensor product as a smash product, we believe that symmetric monoidaltheories being closed under the tensor product is a consequence of symmetricmonoidal structures being stable under smash products in the sense of stablehomotopy theory.
Remark 5.14.
As shown in [HR15, Proposition 40], the tensor product of propsextends the Boardman-Vogt product of symmetric operads [BV06], in the sensethat there is an embedding of the category of symmetric operads into thecategory of props which is strong monoidal with respect to the two monoidalstructures.
Remark 5.15.
The tensor product of pros is not symmetric. Up to the defini-tion of T (cid:3) S as the pushout (13), the construction of ( T, T ) ⊗ ( S, S ) and of( S, S ) ⊗ ( T, T ) is, indeed, identical up to a change of notation. However, inthe final quotient, the roles of σ and σ ∗ , or braidings and inverse braidings,are reversed.Nevertheless, this argument reveals a natural isomorphism between( T, T ) ⊗ ( S, S ) and (( S, S ) ⊗ ( T, T )) ∗ , where − ∗ is the duality defined in §2.21. From this we can recover a symmetryfor the tensor product of props. (Gray product) . Let
P, Q be regular directed complexes. The
Grayproduct P ⊗ Q of P and Q is the cartesian product P × Q of their underlyingposets with the following orientation. Write x ⊗ y for a generic element of P ⊗ Q . For all x ′ covered by x in P and all y ′ covered by y in Q , o ( x ⊗ y → x ′ ⊗ y ) := o P ( x → x ′ ) ,o ( x ⊗ y → x ⊗ y ′ ) := ( − ) dim( x ) o Q ( y → y ′ ) , where o P and o Q are the orientations of P and Q , respectively. amar hadzihasanovic As shown in [Had20b, Section 2.2], P ⊗ Q is a regular directed complex.If f : P → P ′ and g : Q → Q ′ are maps of regular directed complexes, let f ⊗ g : P ⊗ Q → P ′ ⊗ Q ′ have the cartesian product of f and g as underlyingfunction. Then f ⊗ g is a map of regular directed complexes.Gray products determine a monoidal structure on DCpx R whose unit isthe terminal object 1. . The monoidal structure on
DCpx R restricts to a monoidal structureon , which, by Day’s theory [Day70], extends along the Yoneda embeddingto a monoidal biclosed structure on Set .Explicitly, let X and Y be diagrammatic sets. The Gray product X ⊗ Y of X and Y is the colimit in Set of the diagram /X × /Y × Set , dom × dom ⊗ (16)where /X is the category whose objects are cells x : U → X and morphismsfrom x : U → X to y : V → X are commutative triangles U X V fx y ,while dom sends such a triangle to the map f : U → V in .In particular, for each pair of cells x : U → X and y : V → Y , the image ofthe pair ( x, y ) through the diagram (16) is U ⊗ V , so we obtain a morphism U ⊗ V → X ⊗ Y to the colimit, that is, a cell of shape U ⊗ V in X ⊗ Y . Thisis the cell x ⊗ y obtained as the Gray product of x and y in Set . Remark 5.18.
The dimensions of cells add under the Gray product, that is, if x is an n -cell and y is an m -cell, then x ⊗ y is an ( n + m )-cell. Remark 5.19.
The Gray product is not the cartesian product in
Set . How-ever, the monoidal unit is the terminal object, which gives us “projection”morphisms X ⊗ Y → X and X ⊗ Y → Y . These send a cell x ⊗ y of shape U ⊗ V to p ; x and p ; y , respectively, where p : U ⊗ V ։ U and p : U ⊗ V ։ V are projections in . Comment 5.20.
We use string diagrams to give some intuition about cells x ⊗ y of shape U ⊗ V in low dimension; in the pictures, we write xy for x ⊗ y . Firstof all, if x or y is a 0-cell, then U ⊗ V is isomorphic to V or U , respectively,and x ⊗ y has the same dimension and shape as y or x . he smash product of monoidal theories Let a : x − ⇒ x + be a 1-cell in X and c : y − ⇒ y + a 1-cell in Y . Then a ⊗ c is a 2-cell of the form ay − x + cx − c ay + ac ,that is, it is of type ( x − ⊗ c ) ( a ⊗ y + ) ⇒ ( a ⊗ y − ) ( x + ⊗ c ) in X ⊗ Y .Next, let ϕ : a . . . a n ⇒ b . . . b m be a 2-cell in X , and let ψ : c . . . c p ⇒ d . . . d q be a 2-cell in Y . Then ϕ ⊗ c is a 3-cell ofthe form a c a n cϕy − ϕc b c b m cϕy + , (17)while a ⊗ ψ is a 3-cell of the form ad ad q x − ψ aψ ac ac p x + ψ (18)in X ⊗ Y . It is useful to think of these as sliding moves : ϕ ⊗ c slides a 2-cell inthe fibre of ϕ left-to-right, top-to-bottom past a 1-cell in the fibre of c , while a ⊗ ψ slides a 2-cell in the fibre of ψ left-to-right, bottom-to-top past a 1-cellin the fibre of a .Next, we consider the 4-cell ϕ ⊗ ψ ; to simplify, we depict ϕ and ψ as if theyhad only 2 inputs and 2 outputs each. Then ∂ − ( ϕ ⊗ ψ ) is the 3-diagram a ψ a n ψ ϕc ϕc p where the sequence of sliding moves a ⊗ ψ, . . . , a n ⊗ ψ is followed by the se-quence ϕ ⊗ c , . . . , ϕ ⊗ c p , while ∂ + ( ϕ ⊗ ψ ) is the 3-diagram ϕd ϕd q b ψ b m ψ where the sequence of sliding moves ϕ ⊗ d , . . . , ϕ ⊗ d q is followed by the se-quence b ⊗ ψ, . . . , b m ⊗ ψ . In the case n, m, p, q = 2, one can recognise thetwo sides of the Zamolodchikov tetrahedron equation [KV94]. amar hadzihasanovic Next, let ρ be a 3-cell in X and consider the 4-cell ρ ⊗ c . To simplify, wedepict ρ as if it were of type ϕ ⇒ ϕ ′ where ϕ and ϕ ′ are both 2-cells. Then ∂ − ( ρ ⊗ c ) has the form ϕy − ϕc ϕy + ρy + ϕ ′ y + while ∂ + ( ρ ⊗ c ) has the form ϕy − ρy − ϕ ′ y − ϕ ′ c ϕ ′ y + .Dually, if τ : ψ ⇒ ψ ′ is a 3-cell in Y , ∂ − ( a ⊗ τ ) has the form x − ψ x − τ x − ψ ′ aψ ′ x + ψ ′ while ∂ + ( a ⊗ τ ) has the form x − ψ aψ x + ψ x + τ x + ψ ′ . (Pointed diagrammatic set) . A pointed diagrammatic set is a diagram-matic set X together with a distinguished 0-cell • : 1 → X , the basepoint .A morphism f : ( X, • X ) → ( Y, • Y ) of pointed diagrammatic sets is a morph-ism f : X → Y such that f ( • X ) = • Y . With their morphisms, pointed dia-grammatic sets form a category Set • . . The obvious forgetful functor
Set • → Set has a left adjoint sendinga diagrammatic set X to the coproduct X + 1, pointed with the inclusion of1 into the coproduct.The terminal object 1 of Set , pointed with its only 0-cell, is a zero objectin
Set • , both terminal and initial. (Wedge sum) . The wedge sum of two pointed diagrammatic sets ( X, • X )and ( Y, • Y ) is the pointed diagrammatic set ( X ∨ Y, • ) where1. X ∨ Y is the quotient of X + Y by the equation • X = • Y , and he smash product of monoidal theories • is the result of the identification of • X and • Y . (Smash product) . Let ( X, • X ) and ( Y, • Y ) be pointed diagrammatic sets.There is an inclusion X ∨ Y ֒ → X ⊗ Y defined by x x ⊗ • Y , y
7→ • X ⊗ y on cells in X and Y , respectively.The smash product of ( X, • X ) and ( Y, • Y ) is the pointed diagrammatic set( X ? Y, • ) obtained from the pushout diagram X ∨ Y X ? Y X ⊗ Y • in Set (the “quotient of X ⊗ Y by the subspace X ∨ Y ”).The smash product is part of a monoidal structure on Set • , whose unit isthe diagrammatic set 1 + 1, pointed with one of the coproduct inclusions, andall structural isomorphisms are derived from those of the Gray product. Comment 5.25.
The smash product of pointed diagrammatic sets is a “direc-ted” counterpart to the smash product of pointed topological spaces, with theGray product playing the rôle of the cartesian product of spaces.The formal correspondence between definitions is made concrete throughthe geometric realisation of diagrammatic sets [Had20b, §8.38]. This functor |−| : Set → cgHaus sends 0-cells in a diagrammatic set to points in a space,so it lifts to a functor | − | : Set • → cgHaus • to the category of pointed compactly generated Hausdorff spaces and pointedcontinuous maps.We claim that this functor sends smash products in Set • to smash productsin cgHaus • , that is, it is strong monoidal with respect to the two monoidalstructures. Proof.
On regular atoms, | − | is defined as the forgetful functor from to thecategory of posets and order-preserving maps, followed by the simplicial nerveof posets, followed by the geometric realisation of simplicial sets. The firstsends Gray products to cartesian products and the other two preserve finiteproducts. Thus | U ⊗ V | ≃ | U | × | V | naturally in U and V .Both Gray products in Set and products in cgHaus are part of a biclosedmonoidal structure, so they preserve colimits separately in each variable. Since amar hadzihasanovic | − | , a left adjoint functor, also preserves colimits, we can extend to an iso-morphism | X ⊗ Y | ≃ | X | × | Y | natural in the diagrammatic sets X and Y .Finally, | − | also preserves the terminal object, so it sends the colimit dia-grams that define wedge sums and smash products in Set • to the colimitdiagrams that define them in cgHaus • . (cid:4) . Like the smash product of pointed spaces, the smash product of pointeddiagrammatic sets is part of a biclosed structure on
Set • .Left homs and right homs can be computed by a formal argument. If( X, • X ) ⊸ ( Y, • Y ) is a right hom in ( Set • , ? , U in its underlying diagrammatic set correspond to pointed morphisms from U + 1 to ( X, • X ) ⊸ ( Y, • Y ), which correspond to pointed morphisms from( X, • X ) ? ( U + 1) to ( Y, • Y ).Now X ? ( U + 1) is isomorphic to the quotient of X ⊗ U by the subspace {• X } ⊗ U . By the universal property of this quotient, we conclude that thereis a bijection between1. cells of shape U in ( X, • X ) ⊸ ( Y, • Y ) and2. morphisms X ⊗ U → Y which send {• X } ⊗ U to {• Y } .Similarly, cells of shape U in the left hom ( Y, • Y ) › ( X, • X ) correspond biject-ively to morphisms U ⊗ X → Y sending U ⊗ {• X } to {• Y } .In particular, the 0-cells in both the left and the right hom are the pointedmorphisms from ( X, • X ) to ( Y, • Y ). The basepoint, classified by the onlymorphism from the zero object, is the constant morphism X
7→ • Y . Comment 5.27.
From the string diagram of a cell in X ⊗ Y , it is easy to obtaina picture of the same cell in X ? Y : we simply need to identify every cell ofthe form x ⊗ • Y or • X ⊗ y and shape U with the cell !; • of the same shape,where ! : U ։ X and Y have a single 0-cell, then any 3-cellof the form ϕ ⊗ c as in (17) or a ⊗ ψ as in (18) becomes a ca c a n c ϕc b cb c b m c , ad ad ad q aψ ac ac ac p ,respectively, in X ? Y . he smash product of monoidal theories We are ready to state our main theorem. . Let ( T, T ) be a pro. Its diagrammatic nerve N ( T, T ) has a single 0-cell,so it is canonically pointed, and every morphism in the image of N triviallypreserves the basepoint. Thus N , restricted to Pro , lifts uniquely to a functor N : Pro → Set • . Theorem 5.29 —
The diagram of functors
Pro × Pro ProbSet • × Set • Set • GrayCat N × N − ⊗ −− ? ( − ) ◦ U G commutes up to natural isomorphism.Comment 5.30. The form of Theorem 5.29 does not suggest, at first sight, thatthe smash product of pointed diagrammatic sets subsumes and generalises thetensor product of props. Nevertheless, we argue that this is essentially thecase.First of all, since U is pseudomonic by Remark 2.37, if G X is isomorphic to U ( T, T ) for some prob ( T, T ), then this prob is essentially unique. It followsthat, on the image of N − ? ( N − ) ◦ , we can lift G to a functor with codomain Prob , and compute the tensor product of two pros through the lower leg ofthe diagram. In this sense, the smash product on
Set • strictly subsumes the“external” tensor product of pros.From the tensor product of pros, we can recover the tensor product of propsvia a universal characterisation in Prop , independent of the specific construc-tion. If ( T, T ) and ( S, S ) are two props, we have families of morphismsid T ⊗ c : FU ( T, T ) → U ( T, T ) ⊗ U ( S, S ) ,a ⊗ id S : FU ( S, S ) → U ( T, T ) ⊗ U ( S, S )in Prob indexed by c ∈ S and a ∈ T , where c : N → U ( S, S ) and a : N → U ( T, T ) send the generating 1-cell of N to c and a , respectively;here we use the fact that − ⊗ N and N ⊗ − are naturally isomorphic to F .Then ( T, T ) ⊗ S ( S, S ) is the pushout ` c ∈ S rFU ( T, T ) + ` a ∈ T rFU ( S, S ) ( T, T ) ⊗ S ( S, S ) ` c ∈ S ( T, T ) + ` a ∈ T ( S, S ) r ( U ( T, T ) ⊗ U ( S, S )) amar hadzihasanovic in Prop , where the top leg is obtained universally from the family of morph-isms { r (id T ⊗ c ) , r ( a ⊗ id S ) | c ∈ S , a ∈ T } and the left leg from the counitof the adjunction between rF and U . (Skeleta of diagrammatic sets) . For each n ∈ N + {− } , the restrictionfunctor Set → PSh( n ) has a left adjoint; let σ ≤ n be the comonad inducedby this adjunction.The n -skeleton of a diagrammatic set X is the counit σ ≤ n X → X . For all k ≤ n , the k -skeleton factors uniquely through the n -skeleton of X . By astandard argument X is the colimit of the sequence of its skeleta. Proof of Theorem 5.29.
Let ( T, T ) and ( S, S ) be two pros and let ( X, • X )and ( Y, • Y ) be equal to N ( T, T ) and ( N ( S, S )) ◦ , respectively.As seen in §5.17, X ⊗ Y is the colimit of a diagram of atoms U ⊗ V indexedby pairs of cells x : U → X and y : V → Y , which are transposes of morphisms x : P U → ( T, T ) and y : P V ◦ → ( S, S ) in Pro . The smash product X ? Y isthen the colimit of this diagram extended with a morphism U ⊗ V ։ U ⊗ V indexed by ( x, • Y ) or by ( • X , y ).Each diagrammatic set is the colimit of the sequence of its skeleta, and thiscolimit is preserved by smash products separately in each variable. Becausecolimits commute with colimits, we can compute X ? Y in steps, increasing i and j separately in σ ≤ i X ? σ ≤ j Y . This corresponds to restricting the indexingcategory to pairs ( x, y ) with dim( x ) ≤ i and dim( y ) ≤ j .The functor G : Set → GrayCat , a left adjoint, preserves colimits, andwe know how to explicitly compute G on atoms. We will use this to compute G ( X ? Y ). • Let i = 0 or j = 0. Since the only 0-cell in X and Y is their basepoint, both σ ≤ X ? Y and X ? σ ≤ Y are isomorphic to the terminal diagrammaticset. Their image through G is the terminal Gray-category with one 0-celland no cells of higher rank. • Let i = j = 1. The 1-cells in X that do not factor through σ ≤ X corres-pond bijectively to generating 1-cells a ∈ T , and the 1-cells in Y that donot factor through σ ≤ Y to generating 1-cells c ∈ S .The boundary of a ⊗ c contains only 1-cells of the form !; • , which G sendsto units on • . Through G , then, a ⊗ c becomes a 2-cell of type ε • ⇒ ε • .Thus G ( σ ≤ X ? σ ≤ Y ) has a single 0-cell, a single 1-cell, and its 2-cells arefreely generated by the a ⊗ c : this makes it a prob in the sense of §2.35,isomorphic to F ( σ ≤ ( T ⊗ S )). This structure of prob will be inherited by G ( σ ≤ i X ? σ ≤ j Y ) for all higher i, j . he smash product of monoidal theories We fix j = 1 and increase i ; observe that we can stop at i = 3, since for i > >
4, whose contribution through G is trivial.Let c ∈ S . Each 2-cell ϕ : ( a , . . . , a n ) ⇒ ( b , . . . , b m ) in T contributesa 3-cell ϕ ⊗ c in σ ≤ X ? σ ≤ Y ; any other 2-cell in X factors through oneof this form, so it does not give a contribution. We can read the form of ϕ ⊗ c from Comment 5.27: unravelling the definition of G on 3-atoms, wesee that it sends ϕ ⊗ c to a 3-cell of type( a ⊗ c ) . . . ( a n ⊗ c ) ⇒ ( b ⊗ c ) . . . ( b m ⊗ c ) . Extending along colimits, a 2-diagram x k y in X induces a diagram( x ⊗ c ) k +1 ( y ⊗ c ) in G ( σ ≤ X ? σ ≤ Y ) for each k ∈ { , } .From Proposition 4.6, we know that 3-cells ρ in X exhibit all and only theequations of diagrams x = y that hold in T . Reading the form of ρ ⊗ c from Comment 5.20, we see that the only part surviving both the smashproduct quotient and G is an equation between the composites of x ⊗ c and y ⊗ c . For each c ∈ S , then, we can define a morphism of probs − ⊗ c : F ( T, T ) → G ( X ? σ ≤ Y )by x x ⊗ c on cells in T , extending universally to the free prob, andprove that it is injective. Moreover, the family of the − ⊗ c is jointly sur-jective and only overlaps on • . We conclude that there is an isomorphismbetween G ( X ? σ ≤ Y ) and ` c ∈ S F ( T, T ) ≃ F ( ` c ∈ S ( T, T )) , the coproducts being in Prob and
Pro , respectively. • The case where we fix i = 1 and increase j is dual, with a subtlety dueto the way Gray products change orientations in their second factor de-pending on the dimension of the first factor. Since we defined Y to be the dual of N ( S, S ), each 2-cell ψ : ( c , . . . , c p ) ⇒ ( d , . . . , d q ) corresponds toa 2-cell ψ : d q . . . d ⇒ c p . . . c in Y , which for each a ∈ T contributes a 3-cell a ⊗ ψ in σ ≤ X ? σ ≤ Y .By inspection of the shape of a ⊗ ψ in Comment 5.27, we see that G sendsit to a 3-cell of type( a ⊗ c ) . . . ( a ⊗ c p ) ⇒ ( a ⊗ d ) . . . ( a ⊗ d q ) , amar hadzihasanovic which matches the original orientation of ψ in S . Proceeding as before,then, we construct an isomorphism between G ( σ ≤ X ? Y ) and ` a ∈ T F ( S, S ) ≃ F ( ` a ∈ T ( S, S )) . It follows that G (( X ? σ ≤ Y ) ∪ ( σ ≤ X ? Y )) can be computed as thepushout F ( σ ≤ ( T ⊗ S )) G (( X ? σ ≤ Y ) ∪ ( σ ≤ X ? Y )) F ( ` a ∈ T ( S, S )) F ( ` c ∈ S ( T, T ))in Prob , isomorphic to F ( T (cid:3) S ). • Finally, let i = j = 2; increasing either i or j beyond 2 only includes cellsof dimension >
4, whose contribution is trivial.Each pair of a 2-cell ϕ : ( a , . . . , a n ) ⇒ ( b , . . . , b m ) in T and a 2-cell ψ : ( c , . . . , c p ) ⇒ ( d , . . . , d q ) in S contributes a 4-cell ϕ ⊗ ψ to X ? Y ,and any other pair factors through one of this form.Remember that the orientation of ψ is reversed in Y . Reading the form of ϕ ⊗ ψ from Comment 5.20 and unravelling the definition of G on 4-atoms,we find that the boundaries of ϕ ⊗ ψ are mapped by G to the two sides ofthe ϕ ⊗ ψ equation in F ( T (cid:3) S ).We conclude that G ( X ? Y ) with its unique prob structure is isomorphic to( T, T ) ⊗ ( S, S ). It is straightforward to check that the isomorphism is naturalin ( T, T ) and ( S, S ). (cid:4) Example 5.32.
We compute an equation of the prob
Bialg = Mon ⊗ Mon co through Set • , to illustrate how it arises from a 4-cell in the smash productof X := N ( Mon ) and Y := N ( Mon co ) ◦ .The two generating 1-cells of Mon and
Mon co produce a 2-cell 1 ⊗ X ? Y . From the generator µ : (2) ⇒ (1) of Mon , we obtain a 3-cell µ ⊗ X ? Y . Because 1 ⊗ µ ⊗
1, we may informally picture this 3-cell as a string diagramin 3-dimensional space, “tracing the history” of the various copies of 1 ⊗ µ .:= he smash product of monoidal theories Similarly, from the generator δ : (1) ⇒ (2) of Mon co , we obtain a 3-cell 1 ⊗ δ of the form δ .:=Now ∂ − ( µ ⊗ δ ) has the form δ δ µ µ while ∂ + ( µ ⊗ δ ) has the form µ δ .In 3-dimensional string diagrams, we may picture µ ⊗ δ as µδ . (19)If we also “trace the history” of the dotted wires to produce surface diagrams in the style of [VD19], we recover, up to a deformation, the “intersectingsurfaces” picture (2).The picture of the equation µ ⊗ δ in Example 5.9 can be interpreted as aplanar projection of (19). Technically, the single instance of a braiding in thisequation arises, by definition of G , from the fact that the input 2-cells of thefirst instance of µ ⊗ ∂ − ( µ ⊗ δ ) are not consecutive in the normal 1-orderon the overall 2-diagram.
6. Higher-dimensional cells
Having established that the smash product of pointed diagrammatic sets gen-eralises the tensor product of pros, we briefly explore the potential of thisgeneralisation in higher-dimensional universal algebra and rewriting. amar hadzihasanovic (Diagrammatic complex) . A diagrammatic complex is a diagrammaticset X together with a set X = P n ∈ N X n of generating cells such that, for all n ∈ N , ` x ∈ X n ∂U ( x ) σ ≤ n Xσ ≤ n − X ` x ∈ X n U ( x ) ( x ) x ∈ X n (20)is a pushout in Set . Here U ( x ) denotes the shape of the cell x . Remark 6.2.
It follows from the results of [Had20b, Section 8.3] that the geo-metric realisation of diagrammatic sets sends a diagrammatic complex ( X, X )to a CW complex with one cell for each generating cell in X . Proposition 6.3 —
Let ( X, X ) and ( Y, Y ) be diagrammatic complexes.Then X ⊗ Y is a diagrammatic complex with ( X ⊗ Y ) n := n X k =0 { x ⊗ y | x ∈ X k , y ∈ Y n − k } . Proof.
Essentially the same as [Had17, Theorem 1.35], replacing “polygraph”with “diagrammatic complex” and “globe” with “regular atom”. (cid:4)
Remark 6.4.
A straightforward consequence: the smash product X ? Y ofpointed diagrammatic complexes ( X, X , • X ) and ( Y, Y , • Y ) is a pointed dia-grammatic complex whose generating cells are • and the pairs x ⊗ y with x ∈ X \ {• X } and y ∈ Y \ {• Y } . (Diagrammatic presentation) . Let ( T, T ) be a bicoloured pro. A present-ation of ( T, T ) is a diagrammatic complex ( X, X ) such that P X is isomorphicto ( T, T ).Similarly let ( T, T ) be a prob. A presentation of ( T, T ) is a diagrammaticcomplex ( X, X ) such that G X is isomorphic to U ( T, T ). Example 6.6.
The pro of monoids
Mon admits the following presentation( X, X ). To begin, X contains a single 0-cell • and X a single 1-cell 1 : • ⇒ • .Next, X contains a 2-cell µ : 1 ⇒ η : ε • ⇒
1, which wepicture as , .Finally, X contains 3-cells α, λ, ρ of the form α , λ , ρ . he smash product of monoidal theories . Unless a pro ( T, T ) embodies a free monoidal theory, a presentation( X, X ) contains some generating 3-cells, exhibiting equations in P X . Higher-dimensional generators have no effect on the presented pro, as P turns theminto trivial “equations of equations”.For presentations of a prob , the same statement applies shifted by one dimen-sion: generating 4-cells exhibit equations, while higher-dimensional generatorsare trivialised by G .For the purpose of computing a tensor product of pros, we can replace thenerves of pros ( T, T ), ( S, S ) with any pair of presentations ( X, X ), ( Y, Y ),pointed with their unique 0-cell, to obtain a presentation X ? Y ◦ of the prob( T, T ) ⊗ ( S, S ).Even if ( X, X ) and ( Y, Y ) contain no generating cells in dimension higherthan 3, X ? Y ◦ contains generating cells up to dimension 6, two more thanthe threshold of significance for G . Thus the smash product X ? Y ◦ containsstrictly more data than the tensor product ( T, T ) ⊗ ( S, S ).We suggest that these data can be interpreted through the lens of higher-di-mensional rewriting, and in particular the concepts of syzygies and coherence ;we refer to Yves Guiraud’s thèse d’habilitation [Gui19] for an introduction.Rewriting theory is concerned with computational properties of presenta-tions, in particular the properties of confluence and termination. When apresentation is embodied by a polygraph, confluence at a critical branchingis exhibited by a pair of parallel cells. In a coherent presentation, this isstrengthened to the requirement that the parallel pair be filled by a higher-di-mensional cell, sometimes called a syzygy [Lod00]. This requirement can beextended by asking that higher-dimensional parallel pairs also have fillers.As the following example shows, it appears that the higher-dimensional cellsproduced by the smash product of two presentations of pros are syzygies forthe presentation of their tensor product. Example 6.8.
Let ( X, X ) be the presentation of Mon from Example 6.6. Then( X ◦ , X ◦ ) is a presentation of Mon co , so the smash product X ? X is a present-ation of the prob Bialg of bialgebras.Let us compute this smash product. To simplify, we employ the followingabuse of notation: we represent a 3-diagram x in X ? X as a 2-diagram in Bialg whose image through U has the same composite as G x . This allows usto depict n -cells in X ? X as if they were ( n − amar hadzihasanovic To begin, X ? X has a single generating 0-cell and no generating 1-cells.The only generating 2-cell is 1 ⊗ µ ⊗ η ⊗
1, 1 ⊗ µ , and 1 ⊗ η . These are thestandard generators of Bialg as in Example 5.9, and with our abuse of notation,we use the same depiction: µ , η , µ , η .There are 10 generating 4-cells which can be subdivided into three groups.Those of the form x ⊗ x ∈ X have the same representation as x , that is, α , λ , ρ .Those of the form 1 ⊗ x for x ∈ X have the same representation as x ◦ : α , λ , ρ .Finally, those of the form x ⊗ y for x, y ∈ X present the additional equationsof Example 5.9 with the orientation µµ , ηµ , µη , ηη .This presentation of Bialg contains new critical branchings that do not cor-respond to critical branchings in the presentations of
Mon or Mon co . Forexample, we have the following critical branching involving α ⊗ µ ⊗ µ : µµα . (21)There are 12 generating 5-cells of X ? X , of the form x ⊗ y where either x ∈ X and y ∈ X or x ∈ X and y ∈ X . We observe that these are syzygiesexhibiting confluence at these critical branchings. he smash product of monoidal theories For example, ∂ − ( α ⊗ µ ) is µµ µµ α ,while ∂ + ( α ⊗ µ ) is α α µµ µµ ,which exhibits confluence at the critical branching (21).As another example, ∂ − ( η ⊗ λ ) is ηη ηµ ,while ∂ + ( η ⊗ λ ) is λ .Here the unlabelled 4-cell is a degenerate 4-cell [Had20b, §4.14] of the form η ⊗ x where x is a unitor 2-cell. It is mapped by G to a diagram in Bialg whose image is the unit ε ( η ⊗ ∂ + ( η ⊗ λ ) as asingle rewrite step. Thus η ⊗ λ exhibits confluence at a critical branchinginvolving η ⊗ η and 1 ⊗ λ .These syzygies are oriented, so they can be interpreted as higher-dimen-sional rewrites creating critical branchings one dimension up. The 9 generat-ing 6-cells of X ? X , of the form x ⊗ y where x ∈ X and y ∈ X , are highersyzygies exhibiting confluence at these higher branchings. Comment 6.9.
Diagrammatic complexes are closely related to polygraphs, sothe definitions of confluent, terminating, and coherent presentations shouldadmit sufficiently straightforward translations to our framework.The only difficulty is the treatment of degenerate cells. This can mostlikely be circumvented by considering finite sub-presheaves of the underlying combinatorial polygraph of a diagrammatic complex [Had20b, Section 6.2]. We amar hadzihasanovic note that, in fact, a combinatorial polygraph is equivalent to a polygraph if[Conjecture 7.7, ibid. ] holds, which for now is proven up to dimension 3.We would like to make the informal conjecture that, if ( X, X ) and ( Y, Y )are two presentations of pros that are coherent in a suitable sense, then X ? Y ◦ is a coherent diagrammatic presentation of their tensor product; or, at least,a coherent presentation can be extracted from it. We leave the formal devel-opment of this problem to future work. . To conclude, we may want to leave behind the interpretation of dia-grammatic complexes as presentations of pros or probs and consider themdirectly as embodiments of higher-dimensional theories, such as homotopicalalgebraic theories. ( n -Tuply monoidal diagrammatic set) . For each n >
0, a pointed dia-grammatic set ( X, • ) is n -tuply monoidal if • : 1 → σ ≤ n − X is an isomorphism.We say monoidal instead of 1-tuply monoidal and doubly monoidal insteadof 2-tuply monoidal. Example 6.12.
A presentation of a pro is monoidal, while a presentation of aprob is doubly monoidal. . We propose the following basic setup for higher-dimensional universalalgebra in diagrammatic sets: • a presentation of a k -tuply monoidal higher-dimensional theory is embod-ied by a k -tuply monoidal diagrammatic complex ( X, X , • X ); • a “semantic universe” for such a theory is a k -tuply monoidal diagram-matic set with weak composites ( M, • M ) [Had20b, §6.1], a form of weakhigher category; • both the right and the left hom ( X, • X ) ⊸ ( M, • M ), ( M, • M ) › ( X, • X )are spaces of models of the theory in M . These coincide on 0-cells, whichare pointed morphisms from ( X, • X ) to ( M, • M ), but have different (“lax”or “oplax”) higher transformations.Diagrammatic sets with weak composites encompass strict ω -categories viathe diagrammatic nerve construction of [Section 7.2, ibid. ] but also homotopytypes via the right adjoint of geometric realisation, leading to a strictly moregeneral class of semantic universes compared to the theory of polygraphs. Example 6.14.
We extend the presentation of Example 6.6 to a presentationof the 2-dimensional theory of pseudomonoids [SD97]. he smash product of monoidal theories First we add generating 4-cells π, τ where ∂ − π := α α α , ∂ + π := α α ,while ∂ − τ := ρ , ∂ + τ := α λ .Here the unlabelled 3-cell in ∂ − τ is a degenerate 3-cell of the form p ; µ for anappropriate surjective map of atoms p .We let PsMon be the localisation of this diagrammatic complex at the set { α, λ, ρ } [Had20b, §6.4]. This operation weakly inverts the generating 3-cells.We note that a localisation of a diagrammatic complex is always a diagram-matic complex. Example 6.15.
The paradigmatic 2-category has small categories as 0-cells,functors as 1-cells, and natural transformations as 2-cells. This can be givena cartesian monoidal structure, and this monoidal structure can be strictified,producing a strict monoidal 2-category.This is equivalent to a 3-category with a single 0-cell. If we restrict tosuitably small categories, we can make sure that this defines an object of 3
Cat ,which through the diagrammatic nerve produces a pointed diagrammatic set
Cat × that has weak composites and is monoidal.A model of PsMon in Cat × , that is, a pointed morphism PsMon → Cat × in Set • , is then precisely a small monoidal category. Remark 6.16.
Let ( X, • X ) be an n -tuply monoidal and ( Y, • Y ) an m -tuplymonoidal diagrammatic set. Then X ? Y is ( n + m )-tuply monoidal. As aspecial case, we recover the fact that the smash product of two presentationsof pros is doubly monoidal, so it presents a prob.Following Baez and Dolan’s stabilisation hypothesis [BD95], we expect a k -tuply monoidal diagrammatic complex to present a prop when k >
2. We amar hadzihasanovic have not defined a realisation functor that would make this statement precise.Nevertheless, this gives us an idea of what iterated smash products of monoidaltheories produce: a single smash product yields a braided monoidal theory,any number above that a symmetric monoidal theory.For higher-dimensional theories, this ought to generalise as follows: thesmash product of m different k -tuply monoidal diagrammatic sets, interpretedas presentations of n -dimensional theories, presents a symmetric monoidal n -dimensional theory as soon as mk > n + 1. Acknowledgements
This work was supported by the ESF funded Estonian IT Academy researchmeasure (project 2014-2020.4.05.19-0001).I am grateful to Simon Forest, Fosco Loregian, François Métayer, SamuelMimram, Paweł Sobociński, and Sophie Turner for various forms of help.
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