For the mapping f2 , one has to differentiate relation 16 , and for f4 — use the left-hand side of 4. It is easy to see that 19 together with 20 yield exactly the required symmetry. The examples below in Sections 6 and 7 show that there are many enough interesting cases where complex 15 turns out to be acyclic see Definition 2. However, at this time we cannot make this statement more precise. Convention 3. From now on, we assume that we are working with an acyclic complex. Definition 3 and Remark 5 after it.
We write this minor as det f3 B , where f3 B is the submatrix of f3 whose rows and columns correspond to the edges in B. The set B is the complement of B in the set of all edges except those depicted in Figs. Martyushev Remark This sign is changed when we change the order of basis vectors in any of the vector spaces. In our case, however, this is not a problem: due to the symmetry proved in Theorem 3, we can choose our torsion in the form 21 where the numerator is a square and the denominator is a diagonal minor.
Both thus do not depend on the order of basis vectors. Theorem 4. If complex 15 is acyclic for T1 , then it is also acyclic for T2 , and the value I M given by 22 is the same for T1 and T2.
By Theorem 1, it is enough to show the invariance of I M under relative Pachner moves. Thus, a new edge QR appears in the triangulation. We are going to express the new matrix f3 in terms of the old one.
Essentially, we follow [5, Section 4]. The fact that no minors considered in our proof vanished obviously implies that the acyclicity of our complex 15 is preserved under the Pachner moves.
Theorem 5. Let P be any vertex in the triangulation, not belonging to the chain of two distinguished tetrahedra. First, we are going to prove that there exists a sequence of relative Pachner moves removing P from the triangulation. We begin with subdividing the initial triangulation so that it becomes combinatorial. Denote by star P and lk P the star and the link of P respectively in this combinatorial triangulation.
Continuing this way for other pairs of adjacent tetrahedra from star P containing M as a vertex, we can reduce to three the number of edges starting from M and belonging to lk P. Continuing this way, we reduce to four the number of vertices in lk P. Finally, we return to the initial triangulation inverting the whole sequence of Pachner moves.
It remains to say that, due to the invariance under Pachner moves, the value I M is unchanged at every step. However, it turns out that with the multiplier 6VABCD 4 introduced in 22 the invariant is just a number at least in the examples considered below in Sections 6 and 7.
A topological field theory dealing with triangulated manifolds must answer the question what happens with an invariant like I M under shellings. We also show what happens with matrices f3 and f2 from complex 15 under these shellings. These results will be used in calculations in Sections 6 and 7. It is enough to show how to change the knot framing by one-half of a revolution.
Of course, the framing can be changed in other direction similarly. In this case, we should first draw the left-hand-side tetrahedron in Fig. Return to Fig. So, we glue the same tetrahedron as drawn in the left-hand side of Fig. How will the invariant I M change? Thus, f3 is a submatrix of F3. Just as f3 , matrices F3 and G3 are symmetric. Now we describe what happens with G3 when we change the framing.
Recall that we have chosen a consistent orientation for all tetrahedra in the triangulation, which means, for every tetrahedron, a proper ordering of its vertices up to even permutations. The initial tetrahedron in the left-hand side of Fig. Let G G3 after the change of framing which adds two new edges g and CD AB g to the triangulation in the way described above.
The normalization 26 of matrix G3 has been chosen keeping in mind formulas of type 6. It remains to show that the other new matrix elements vanish, e. A similar explanation works for the second equality in 29 as well. As for the matrix f2 in complex 15 , it will just acquire two new rows whose elements, like all elements in f2 , are obtained by differentiating relations of type Martyushev 6 Calculation for unknot in three-sphere We now turn to applications of our ideas to examples — manifolds with framed knots in them.
In this section we do calculations for the simplest case — an unknot in a three-sphere. Recall that MK is a three-manifold with the boundary consisting of a single torus. Then, M is a filled torus. We glue M out of six tetrahedra in the following way. First, we take two identically oriented tetrahedra ABCD and glue them together in much the same way as in Fig.
We are going to remove this difficulty by gluing some more tetrahedra to the chain. Beginning of the construction of the triangulated solid torus: a chain of two tetrahedra ABCD. To distinguish edges of the same name, we introduce the following notations. Then, we think of one of the tetrahedra in Fig. It remains to denote four edges, of which two lie inside the filled torus, and two — in the boundary.
The obtained triangulated filled torus is depicted in Fig. Now we can glue the tetrahedron chain from Fig. Thus, sphere S 3 appears with a framed unknot in it determined by the tetrahedra from Fig. Triangulated filled torus. A simple calculation using the explicit form of matrix blocks concludes the proof of the theorem. We think there is no need to present here all details of this calculation, especially because in Section 7 we give details for a similar calculation in a more complicated case of unknots in lens spaces.
Now we describe triangulations of L p, q which will be used in our calculations. Consider the bipyramid of Fig. The connection of Fig. A chain of two tetrahedra in a lens space. A generator of the fundamental group can be represented, e. The boldface lines solid and dashed in Fig.
In this paper, we call a knot in L p, q determined by a tetrahedron chain of the kind of Fig. We are going to calculate our invariant for such unknots with different framings. As we have explained in Section 5, it makes sense to consider matrix F3 which consists, by definition, of the partial derivatives of all deficit angles with respect to all edge lengths in the triangulation of the lens space and of which f3 is a submatrix.
Matrix G3 has many zero entries. We will denote the triangulation edges by indicating the origin and end vertices of a given edge. As different edges may have the same origin and end vertices, we will assign indices to them, as indicated in Fig. For example, as one can see from this figure, there exist p different edges AB, and each of them is equipped with an index from 1 to p. So, we denote by AB n the edge AB equipped with index n.
To the explanation of the structure of matrix G3. To describe the structure of matrix G3 , we introduce the following ordering on the set of all edges in triangulation: AB 1 ,.
Here and below the empty spaces in matrices are of course occupied by zeroes. We now describe the structure of S1 , S2 and S3. It is convenient to choose the orientation of the four tetrahedra in Figs. In this section we consider the simplest case when a framed knot is determined directly by a tetrahedron chain of the type depicted in Fig. Let Se1 resp. Proof of Theorem 9. Equation 42 is directly deduced from the block structure in equa- tion 36 of matrix G3 and equation So, it remains only to prove the formulas 43 and 44 for values sn and tn.
We first prove We use the factorization of matrix S1 given by 38 in order to simplify the matrix Se1 with the help of certain sequence of elementary transformations preserving the determinant.
So, here are our elementary transformations. Quite similarly, we obtain formula We assume that we do the first half-revolution exactly as described in Section 5, and the second half-revolution goes in a similar way but with the pair of edges AB, CD replaced by the pair AC, BD, the third half-revolution involves again the pair AB, CD and so on.
We keep the notations used in Section 5. The changes made in matrices S1 and S2 follow from formula Unitarity of functors is essentially different from unitarity of weak double categories. On the other hand, the importance of unitarity for lax or colax functors is structural, and will clearly appear when dealing with limits and their relationship with adjunctions see 4. The main notions of fullness and faithfulness we are interested in are of a horizontal kind, related to the notions of equivalence that we shall see in 3.
Of course in the transpositive case the vertical analogues are of the same interest. If F is unitary it is easy to see that the second condition implies the first: August 6, bk-9x6 main page 3. Again, if F is unitary the second condition implies the first. X is then determined by any set of vertical arrows of A closed with respect to vertical identities of domains and codomains and vertical composition.
The naturality condition comprises its level of degree 0 hX 0. We have now the 2-category LxDbl of weak double categories, lax functors and horizontal transformations. The adjoint case will be considered in 4. We speak of a special equivalence when both h and k can be chosen to be special isomorphisms, so that F and G give inverse isomorphisms between Hor0 X and Hor0 A.
We shall see that the lax functor S is produced by a limit, the tabulator of a relation in 3. Here a weak double category is not assumed to be unitary. Proof Let us recall that the weak double category X has a vertical bicategory VerX see 3.
We shall repeatedly use the Coherence Theorem for bicategories see 2. First we replace VerX with the free category V on the graph of the old vertical arrows. The coherence theorem for the bicategory VerX says that we get the same result, no matter how this composition is done, and that 3.
It coincides with X at the level Hor0 , i. The vertical arrows of A are the previous strings in V. Horizontal composition of double cells in A is like in X, and forms a category.
These cells do coincide. August 6, bk-9x6 main page Double categories In this way the central prism commutes by definition of weak double category. The identity laws are left to the reader.
Let us note that this procedure needs invertible comparisons S, and cannot be applied to lax or colax functors. Here we begin by introducing their 1-dimensional universal property. This topic is based on the paper [GP11]. Tabulators and cotabulators often describe important, well-known constructions and clarify the structure we are examining — as we have already seen while introducing the weak double category Cat of categories, functors and profunctors, in 3.
The 1-dimensional tabulator in a strict double category A was introduced in [BaE], p. Explicitly, this means that tab. Moreover, we shall try to represent A as a weak double category of spans, over the category Hor0 A.
The 2-dimensional property of tabulators and cotabulators will be considered in Section 5. In this chapter the terms tabulator and cotabulator always refer to the 1-dimensional aspect. The present computations are clearer without assuming that A is unitary. The lax functor S will be called the span representation of A. Related results can be found in the paper [Ni]. The last condition is trivially satisfied when A is flat.
This property will be studied in Sections 3. In these hypotheses, we say that A is cospan representable if the colax functor C is horizontally faithful. In an obvious comparison with homotopy theory, the category 2 is playing the role of the standard topological interval.
As of now, the reader will only consider the 1dimensional universal property of the latter, while the 2-dimensional one is deferred to Exercise 5. Prove that, if C has all cell corepresenters, then the ordinary limit of any 2-functor with values in C is automatically a 2-limit.
Most of the strict or weak double categories we are considering have them, and are representable by spans or cospans. On the other hand, the weak double category Rng of rings, homomorphisms and bimodules lacks tabulators — a fact related to an anomalous behaviour of colimits, as we shall see in 5.
After computing some cases, many others are presented as exercises, with solution or hints in Appendix C. We compute now their tabulators and cotabulators, showing that they exist and how they are related. Since these double categories are flat, they are automatically span and cospan representable. RelSet is closed in pOrd and therefore in Mtr under tabulators. How can one extend this fact? It may be convenient to prove things in the following order.
We fix two unitary weak double categories I and A, of which the former is small. Strict horizontal transformations and pseudo vertical transformations form a weak double subcategory Lx I, A. Then these cells become the redundant naturality condition 3. August 27, bk-9x6 main page 3.
We also have a weak double subcategory Lx I, A by restricting the horizontal arrows to the strict horizontal transformations, but keeping all the vertical arrows and all modifications whose horizontal arrows are strict.
Proof By straightforward verification. Their functors, transformations and modifications are particular cases of the corresponding items for double categories. A 2-natural resp. Their universal property is based on the main composition, the horizontal one. It will be defined as an orthogonal adjunction see Section 4. One should use a version of Dbl based on a universe V to which the ground universe U belongs; but also here we do not insist on problems of size. It is important to note that these adjunctions cannot be viewed in a 2category: a globular structure, of any dimension, can only have one sort of arrows.
The same holds for general adjunctions between bicategories, which still are of the colax-lax type. The last section of this chapter briefly studies pseudo algebras for a 2-monad, and shows that weak double categories can be seen as normal pseudo algebras for an obvious 2-monad on the 2-category of graphs of categories.
Kan extensions in or for weak double categories form a complex topic, that will not be covered here. Also because of this, the study of monads in weak double categories is deferred to the infinite dimensional case, in Chapter 8. Such phenomena, introduced in [GP1, GP2], are interesting in themselves and typical of double categories.
Later, orthogonal adjoints will be used to define adjunctions between weak double categories. A is always a unitary weak double category. VerA denotes its vertical bicategory of objects, vertical arrows and special cells u 11 v , introduced in 3.
We say that A has vertical companions if every horizontal arrow has a vertical companion. Let us recall that A is assumed to be unitary. Here f is called the horizontal adjoint and v the vertical one. We say that A has vertical adjoints if every horizontal arrow has a vertical adjoint. August 6, bk-9x6 main page 4. The converse is obvious. Proof By 4. Proof First d implies b , by 4.
By horizontal duality, c also is equivalent to d. If it holds, two objects A, B horizontally isomorphic are always sesqui-isomorphic, hence vertically equivalent i. In fact, we prove more: F and G are sesqui-isomorphic objects in the weak double category of lax double functors Lx I, A introduced in Section 3. It consists of weak double categories, with lax and colax double functors and suitable cells.
It contains a double category Mnc of monoidal categories, with monoidal and comonoidal functors. Limits and span representation in Dbl will be studied in 5. August 6, bk-9x6 main 4. But they can be organised in a strict double category Dbl, where orthogonal adjunctions will provide our general notion of double adjunction in the next section , while companion pairs will amount to pseudo double functors Theorem 4.
The objects of Dbl are the small weak double categories A, B, Its horizontal arrows are the lax double functors F, G Proof a To show that the formulas 4. August 6, bk-9x6 main page Double adjunctions 4. Then the arrow mark of a cell, in 4. In particular, restricting Dbl to trivial vertical arrows and globular cells 1 F G 1 , as in 3. This agrees with the 2-category considered by Carboni and Rosebrugh to define lax monads of bicategories [CaR], Prop. Note, on the other hand, that lax functors and lax transformations of bicategories or 2-categories do not form a bicategory.
This construction can be extended from Dbl to any double category, see Exercise 4. Moreover, L is pseudo if and only if both of G and H are. Proof It is sufficient to prove a , since Dbl is transpositive by Exercise 4. Uniqueness is obvious. As we have seen in 3.
A vertical arrow is a comonoidal functor, which is colax. One should now define the new morphisms and their composition, and prove that they do form a category. Corollary 4. Most of the proofs of this section are deferred to their infinite-dimensional extension, in Section 7.
All this will be analysed below. Also here the arrow of a colax double functor is marked with a dot when displayed vertically, in a diagram of Dbl. We speak of a pseudo-lax resp. From general properties of adjoint arrows in 4. A general colax-lax adjunction cannot be presented as an internal adjunction in some 2-category, but we shall see in the next section that this is possible in the pseudo-lax or the colax-pseudo case.
Loosely speaking, we are saying that a colax double functor can only have a lax right adjoint. Proof See 7. Again the proofs are deferred to their infinite-dimensional extension, in Section 7. The comparison cells of F are then horizontally invertible and the composites GF and F G are lax double functors, while the unit and counit are horizontal transformations of such functors as remarked in 4.
Therefore, a pseudo-lax adjunction gives an adjunction in the 2-category LxDbl of weak double categories, lax double functors and horizontal transformations; we shall prove that these two facts are actually equivalent Theorem 4. Applying Proposition 4. Adjoint equivalences are characterised below.
We recall, from 3. It follows that Hor0 F is also essentially surjective on objects because F is pseudo. Proof Conditions i to iv are proved to be equivalent in Theorem 7.
Conditions iv and v are equivalent by 1. Many of them are presented here as exercises. The non-obvious solutions can be found in Appendix C, or here when they are to be used in the next chapters. Prove that this embedding is reflective and lax coreflective, i. Show that the lax functor F is not pseudo unitary. We shall see, in Theorem 5. In fact, Su is the span associated to the tabulator of u, so that the lax functor S is the span representation of RelSet see 3.
R is the obvious strict functor taking a span to the associated relation. The solution will be used in the next chapters, and is given here.
We just specify its action on the objects. Now let I be an arbitrary weak double category. The previous construction of the graph S step A of 5.
Note that Guv , puv , quv is the pullback of qu , pv in Hor0 A. Solutions or hints are given in Appendix C. Lax functoriality of limits and colax functoriality of colimits are obviously important, also because they imply the 2-dimensional property. In some cases we also examine their pseudo functoriality, but this property is not of much interest here; more complete results on this point can be found in [GP1], Section 6.
For instance it is the case in the monoidal category Rng of rings, with the usual tensor product and tensor unit Z, examined in Exercises 2. As remarked there, this structure is not cocartesian, unless we restrict to commutative rings. It is important to note that these limits can be transposed to vertical limits. This embedding can be generalised see 6. Colimits of rings give 1-dimensional level colimits in Rng, which are not 2-dimensional colimits, generally. August 6, bk-9x6 main page Double limits d Give a similar study of the weak double category of monoids, homomorphisms and bimodules, introduced in 3.
When F is not unitary, this can fail even for a strict G. But it is still true that G must preserve level limits, see 5. We begin by reformulating the definition of cones and limit of T in terms of cells in Dbl, whose vertical arrows are strict functors; such a cell is inhabited by a horizontal transformation of lax functors as remarked in 4.
First, we have already seen in 4. The general case will only be dealt with at the end of this section, in 5. In particular, 5. As defined in 5. Here mod. The remaining points are straightforward. Marking the cells of Dbl with an arrow will help us to manage this drawback. In a , taking F strictly unitary gives a less clear argument in the proof. There is August 6, bk-9x6 main page 5. Since F X is invertible, k is uniquely determined as well.
In diagram 5. August 27, bk-9x6 main page Double limits 5. Using the construction theorem of double limits, it follows that weighted limits in C amount to double limits in C, viewed as a horizontal double category. We write as [I, Cat] resp. The 2-category C is said to be 2-complete if it has all weighted limits. Then it also has all weighted pseudo limits, as proved in [BKPS]. The reader can write its 1-dimensional and 2-dimensional universal properties and prove that all conical limits in C can be constructed from 2products and 2-equalisers.
The reader can write the universal properties of S t X, p , for a small category S, and verify that 2 t X coincides with the cell representer defined in 3. The 1-dimensional aspect has already been considered, in Exercise 3. It is a cone when all the comparison cells h a, X are vertical identities. Speaking of a W -pair i, X , or i, x , or a, X , below, we mean a pair as above.
Horizontal morphisms compose, forming a category. August 27, bk-9x6 main 5. Point b is well known in the theory of weighted limits: see [St2], Theorem The converse follows from Theorem 5. Point b is proved in the same way. The terminal objects of these double categories give the W -weighted limit of F , strict or pseudo, respectively.
Note that one cannot express the 2-dimensional universal property of weighted strict or pseudo limits by terminality in a 2-category. This is probably why weighted cones are rarely considered in the theory of 2categories.
In this case the double category QC is span representable. August 6, bk-9x6 main 5. I is small weak double category.
Here we follow a shortcut, using a characterisation proved in [GP13], Theorem 6. If this happens for a given weak double category A, we say that I-limits are persistent in A. Namely, we say that the weak double category I is grounded if every connected component of the ordinary category Hor0 I has a natural weak initial object.
August 6, bk-9x6 main page Double limits 5. Proof See [GP13], Theorems 6. Proof See [GP14], Theorem 3. As proved in Theorem 2. In this chapter we start from a simple example, a triple category built on the double category Dbl of weak double categories Section 6. In Sections 6. Again, solutions and hints to the exercises can be found in Apendix C. An explicit definition of multiple categories of any dimension will be given in Section 6.
This introductory section constructs some simple triple categories, and gives a first motivation for studying them and their limits. Therefore all these laws are strictly associative and unitary, and each pair of them satisfies the interchange law. The fact that any triple cell of StcDbl is determined by its boundary i.
August 6, bk-9x6 main 6. Since P and Q are strict double functors, this construction also gives the tabulator, or e2 -tabulator, of the 2-arrow U of StcDbl: it will be defined in 6. Again, P and Q are strict double functors, and this construction also gives the tabulator, or e1 -tabulator, of the 1-arrow F of StcDbl: it will be defined in 6.
In other words p, as a horizontal arrow, is companion to itself as a vertical one see 4. Therefore all these laws are strictly associative, unitary and satisfy interchange.
Again the triple category StgD is box-like. Another possible choice is the pseudo functors: this would give a larger triple category.
Applying our procedure to the double category LCCat resp. LRAc defined in 3. The objects are those of A. The transversal arrows f, g, Composition in direction 0 in dimension 1, 2, or 3 works with the horizontal composition of A of arrows or double cells ; composition in direction 1 or 2 works with the vertical composition of A. All these laws are strictly associative, unitary and satisfy interchange. This is not the case of Dbl, for which we used a different procedure. The objects and arrows are those of C and QC.
The 2-dimensional cells are quintets, i. The triple category Q3 C is also box-like. Let us note that the triple category Cub3 QC is a substructure of Q3 C , where all cells are commutative quintets.
It could be written in a more complete form, e. The same can be said of degeneracies. August 6, bk-9x6 main page Weak and lax multiple categories 6. Precisely, X has to satisfy the following relations, where the multi-index August 6, bk-9x6 main page 6. This notion is equivalent to the classical notion of a cubical set, by a rewriting of multi-indices. Here we prefer to avoid such rewritings and stay within multiple sets.
Besides N, we shall frequently use the ordinal n and the ordered set Z of all integers. As above i is any multi-index, i.
Let us note that the lower interchanges are already expressed above: August 6, bk-9x6 main page Weak and lax multiple categories the last condition of 6. Again, we can more generally consider N -indexed multiple categories, where N is a totally ordered set, pointed at 0.
The 0-, 1- and 2-dimensional versions amount — respectively — to a set, a category or a double category. We have already studied some triple categories in Section 6. Infinite dimensional examples of cubical type will be seen in Section 6. In Section 7. It will be used for the transformations of multiple functors and for the structural arrows of limits and colimits; its composition will stay strict, in all the weak or lax extensions we shall consider. A transversal isomorphism is an isomorphism in a transversal category.
All these form the 2-category Mlt, in an obvious way. Multiple categories have dualities, generated by reversing each direction i and permuting directions; they form an infinite-dimensional hyperoctahedral group. This corresponds to ordinary duality in dimension 1 for categories and horizontal duality in dimension 2 for double categories. The 0-, 1- and 2-dimensional versions amount — respectively — to a set, or a category, or a double category.
We are particularly interested in the 3-dimensional notion, called a triple category, already explored in Section 6. Other kinds of truncation, by degree or dimension, will be mentioned in 6.
An independent definition will be given in Appendix B. The notion of cubical category which we use here was defined in [G2, G4]: it includes transversal maps, which can be of a different sort from the other arrows, and are crucial for the weak and lax extensions. A multiple category of cubical type is equivalent to a cubical category, as defined in [G2, G4] and Appendix B. Generally, we shall make no difference between these two notions. Again, symmetric cubical categories, in their own right, will be dealt in Appendix B; here they can be viewed as follows.
A multiple category of symmetric cubical type is a multiple category of cubical type A as defined above with an assigned action of the symmetric group Sn on each transversal category tv[n] A , generated by transposition August 27, bk-9x6 main 6. This will be made precise in Appendix B.
In the truncated case the symmetric structure, that only works on positive indices, is trivial up to dimension 2: for sets, categories and double categories as well. One should not confuse this notion with the transpositive property of strict double categories, that works on the indices 0, 1; see 3. It is of symmetric cubical type, and gives the previous Cub C when applied to the double category QC of commutative squares of the category C.
It is of symmetric cubical type, in a strong sense because here everything is invariant under renaming all indices, in the same order; the transversal direction plays no special role.
They form the category cati M , underlying the 2-category C and independent of i. We have thus a triple category trpijk M. The former approach was followed in [GP8]; here we follow the latter, that fits better with the theory of adjunctions and monads developed below. The prime examples are based on cubical spans and cospans, in 6.
We also have a multiple set Ai indexed by the finite subsets of N. The last condition in 6. We say that f is ij-special if it is special in both directions i, j. Every i-special map is of this kind. In the diagrams below a line segment represents a cell and a double one represents a cell degenerate in the direction of the line itself. August 6, bk-9x6 main page Weak and lax multiple categories wmc. August 6, bk-9x6 main page Weak and lax multiple categories The transversal dual Atr of a weak multiple category A reverses the transversal faces and compositions as in 6.
As in 6. The truncated case is considered below, in 6. In particular, from 3. Now the formulas 6. We speak of symmetric cubical type when we further have a left action of the symmetric groups Sn on A1, A detailed construction will be given in Section B4, taking advantage of the cubical machinery developed in Appendix B, but the interested reader can begin to work it out now, from the following inputs.
The comparisons come from the universal property of the latter. Concatenation can be given a formal definition. Span C is transversally dual to Cosp Cop , with positive compositions computed by pullbacks.
The 2-dimensional truncation Bisp2 C is the weak double category Bisp C described in 4. The 0- and 1-dimensional versions still amount to a set or a category, but the 2-dimensional notion is now a weak double category. We are particularly interested in the 3-dimensional case, a weak triple category. This notion is no longer transversally selfdual and has two instances. A right chiral multiple category has the same structure and satisfies the same axioms considered above in the weak multiple case, except for the fact that the ij-interchanger, for 0 6.
By transversal duality, a left chiral multiple category has an ij-interchange comparison directed the other way round, for 0 6. Both structures still have the strict degenerate interchanges mentioned in 6.
In the infinite dimensional case this also works if we replace the natural indices with the integral ones, or with any infinite selfdual totally ordered pointed set.
In Section 6. August 6, bk-9x6 main page 6. Each coherence condition of 6. More generally, we are also interested in the intermediate cases, dealt with in 6. The opposite arrangement makes no sense. Given two chiral multiple categories A and B we have thus the category StCmc A, B of their strict multiple functors and transversal transformations. All these form the 2-category StCmc. As in the theory of double categories, this unitarity condition is at a more basic level than unitarity of weak or August 27, bk-9x6 main page Weak and lax multiple categories chiral multiple categories: indeed only in this case F commutes with all degeneracies and is a morphism of multiple sets.
For instance: cmf. Full and faithful lax or colax multiple functors are defined in 7. A lax functor which is transversally isomorphic to a pseudo functor is pseudo as well. Similarly one defines the 2-category CxCmc, for the colax case; the last axiom above has now a transversally dual form trt. It will be called a pseudo isomorphism of chiral multiple categories. F is made colax by inverting its comparisons; this gives a general invertible arrow in CxCmc. In fact, for every positive multi-index i, Gi is inverse to Fi.
Moreover, composing their unit comparisons, as in 6. The same holds for the composition comparisons. The same holds for colax functors, pseudo functors, unitary co lax functors. We have now a family of 2-categories Mxp Cmc of chiral multiple categories, p-mixed functors and transversal transformations. It has been studied in [GP7], Subsection 6. Interchanging the positive directions one gets the left chiral triple category CS C of cospans and spans of C.
Higher dimensional examples are considered in 6. For the sake of simplicity, we assume that, in our choices, the pullback or pushout of an identity along any map is an identity. In the second part of this section we introduce some multiple categories of partial maps, and a premultiple category RelSet of cubical relations.
The basic part can be found below, in 6. This composition will be written as qp, or q. It is also easy to see that the 0-directed composition has a strict interchange with all the other compositions.
Because of our assumption on the choice of pushouts and pullbacks, all 1- or 2-directed composition laws are strictly unitary. To complete the construction of the chiral triple category SC C , there are invertible comparisons for the associativity of 1- and 2-directed composition, and a directed comparison for their interchange.
All these comparisons are constructed in [GP7], Section 6, where their coherence is proved. The comparisons for the 1-directed resp. August 27, bk-9x6 main page Weak and lax multiple categories 6.
Similarly we have a left chiral n-tuple category Cp Sq C see 6. It is transversally dual to Sp Cq Cop. The category C is supposed to have a choice of subobjects, stable under composition; counterimages of subobjects are determined by this choice.
This construction has an obvious extension to a strict multiple category Pmap C of cubical partial map. The n-cubes are the n-cubical spans formed by the spans of C whose first arrow is a subobject, as shown below August 6, bk-9x6 main page 6. This corrects an error in [G2]. The definition of degeneracies extends 6. There is no interchanger, as we show below: already in dimension three, the truncated structure Rel3 Set of 2-cubical relations and their maps is neither left-chiral nor right-chiral.
We study here these important multiple limits, which often clarify the structure we are examining, and can lead to representing it by cubical spans. This extends the span representation of weak double categories, in Section 3. We work in a chiral multiple category A.
The extension to intercategories will be briefly considered in Section 6. We say that A has multiple tabulators if it has tabulators of all degrees, preserved by faces and degeneracies. In this case, if A is transversally invariant, one can always make a choice August 6, bk-9x6 main page Weak and lax multiple categories of multiple tabulators such that this preservation is strict see a similar proof in 7. Note that the conditions 6.
Proof Obvious, composing universal arrows of ordinary functors. Similarly, all the n-cubical spans T x are well defined. The comparisons of T are then constructed by an inductive procedure that extends what we have seen in Theorem 3. In this situation we say that A is cospan representable if this colax functor is transversally faithful.
August 6, bk-9x6 main page Weak and lax multiple categories b Same question for the cubical categories Pmap C and Piso C of 6. In A the only non-trivial tabulators are the total tabulators of 1-cubes, i. Therefore the weak multiple category A has multiple tabulators if and only if A has 1-dimensional tabulators. We are not considering in A any higherdimensional universal property. The difference can be better appreciated noting that a 3-dimensional intercategory is a pseudo category in the 2-category of weak double categories, lax double functors and horizontal transformations see [GP6], Section 2 , while a chiral triple category is a unitary pseudo category in the 2-category of weak double categories, unitary lax double functors and horizontal transformations.
The general framework, besides providing an effective unification of these three-dimensional structures, makes also possible to consider morphisms between them and study how they relate to each other.
Here we present infinite-dimensional intercategories, introduced in [GP8], and give a short synopsis of the results recalled above. Other examples of 3-dimensional intercategories can be found in Section 6. The coherence axioms stated in 6. Furthermore there are coherence conditions for the interchangers, stated below in 6.
The transversally dual notion of a left intercategory has interchangers in the opposite direction. Full details and proofs for this part can be found in [GP7]. The interchangers express the fact that the second tensor is a lax functor with respect to the first, or equivalently that the first tensor is colax with respect to the second as in the structure of a bialgebra.
These interchangers must be natural and satisfy the coherence axioms 6. This is the same as a three-dimensional intercategory on a single object written as a dot, and called the vertex , with trivial arrows in all directions and trivial and cells. A used soccer ball over white background.
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