D IAGRAMMES

M ARCO G RANDIS
ROBERT PARÉ
From cubical to globular higher categories
Diagrammes, tome S67-68 (2012), p. 117-148
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DIAGRAMMES , SUPPLEMENT AUX VOLUMES 67 + 68 , 2012, PARIS, pp. 117-148

FROM CUBICAL TO GLOBULAR
HIGHER CATEGORIES

M. GRANDIS and R. PARE

EDITE PAR L. COPPEY , F. FOLTZ , R. GUITART, C. HENRY , C. LAIR

119

From cubical to globular higher categories (*)
Marco Grandis - Robert Paré
dedié à Andrée Ehresmann, en amitié et admiration
Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146-Genova, Italy
Department of Mathematics and Statistics, Dalhousie University, Halifax NS, Canada B3H 3J5

Abstract. We show that a strict symmetric (infinite-dimensional) cubical category A has an
associated !-category Glb(A), consisting of its 'globular cubes'. The procedure of globularisation
generally destroys important features of A, like the existence of limits and colimits or the presence of
symmetries. Then we examine the much more complex weak case, up to constructing the tricategory
associated to a weak symmetric 3-cubical category.
Mathematics Subject Classifications (2000): 18D05, 55U10, 20B30
Key words: higher category, cubical category, !-category, double category, bicategory, cubical set,
symmetry, span, cospan, profunctor.

Introduction
This paper takes on a study of the theory of weak cubical categories, begun in [G1-G5] with the
aim of extending to higher dimension the study of weak double categories developed in [GP1-GP4].
Here we prove that a strict symmetric (infinite-dimensional) cubical category A has an associated
!-category Glb(A), consisting of its 'globular cubes'. Then we examine in low dimension the much

more complex weak case, up to constructing the tricategory associated to a weak symmetric 3-cubical
category. As a general fact, the procedure of globularisation tends to destroy important features of
cubical categories, like the existence of limits and colimits or the presence of symmetries; the main
motivation of this analysis is that, presently, higher categories are mostly studied in the globular form with some exceptions like those cited above.
Let us recall that a weak cubical category [G1-G5] has a cubical structure, with faces and
degeneracies; moreover, there are weak compositions in countably many directions, which we call
cubical (or geometric), and one strict composition, in the transversal (or structural) direction.
As a leading example, linked to higher dimensional cobordism (see [G1-G3]), one can think of the
weak cubical category !C osp(X) of cubical cospans in a category with pushouts X. An nX, where " is the 'formal cospan' category
dimensional object is a functor x: "n

(*) Work partially supported by grants of MIUR (Italy), Università di Genova (Italy) and NSERC (Canada).

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2

(1)

–1

0

1

"

(-1,-1)

(0,-1)

(1,-1)

(-1, 0)

(0, 0)

(1, 0)

(-1, 1)

(0, 1)

(1, 1)

"2.

An n-dimensional transversal map, or structural map, is a natural transformation f: x
X of such functors. Their composition is also called transversal, or structural.

y: " n

The ordinary categories Cospn(X) = Cat(" n, X) form a cubical object in Cat, with obvious
faces and degeneracies. Moreover, n-dimensional objects (and maps) have cubical, or geometric,
composition laws x +i y in each direction i = 1,..., n, which are constructed with pushouts; these
compositions are consistent with faces and degeneracies, but only behave well up to suitable
transversal maps, which yield invertible comparisons for their associativity, unitarity and interchange.
Actually, as already stressed in the papers mentioned above, !C osp(X) is a symmetric weak
cubical category, when equipped with the obvious action of the symmetric group Sn on Cat("n, X);
namely, the action of permuting the factors of " n, i.e. the directions of n-cubical cospans in X.
These symmetries allow one to only consider the faces, degeneracies and cubical compositions in a
single direction (see 1.3), which greatly simplifies the coherence conditions. Notice also that cubical 1truncation, keeping one weak direction and the strict transversal one, yields the weak double category
C osp(X) of ordinary cospans and their transversal maps, studied in [GP1]; here, symmetries
'disappear', since the groups S0 and S1 are trivial.
It should be noted that the importance of considering the transversal maps goes beyond the fact of
containing the comparisons for the associativity, unitarity and interchange of the cubical compositions:
this point only requires invertible transversal maps and disappears in the strict case. In fact, cubical
limits (or colimits) have been dealt with in [G5], extending the theory of double limits developed in
[GP1]: their projections (or injections) are transversal maps. Thus, if our previous category X. is
(co)complete, then !C osp(X) has all cubical (co)limits, in a way that is consistent with faces,
degeneracies and transposition, and (co)lax functorial with respect to cubical composition. (The last
fact is a general, straightforward consequence of the universal property; but here, where cubical
compositions are based on pushouts, the cubical colimits are actually pseudo functorial.)
Outline. In the first two sections we begin by considering a (strict) symmetric cubical category A
and its globularisation GlbA, a strict !-category consisting of the 'globes', or globular cubes of A. As
an example, the symmetric cubical category !Rel of cubical relations of sets (1.8) yields an !category (2.7); the transversal maps of A do not intervene in this procedure of globularisation (but,
again, are essential for the cubical (co)limits of !Rel).
Then, in Sections 3 and 4, we recall the definition of a weak symmetric cubical category and some
main examples, like cubical cospans on a category with pushouts. We also give a construction of the
weak symmetric cubical category !Cat of cubical profunctors, modifying a similar construction given
in [G4] with less adequate transversal maps.
Finally, in the last section, we examine globularisation in the weak case, up to constructing the
tricategory (see Gordon, Power and Street [GPS]) associated to a weak symmetric 3-cubical category;
here, the transversal maps contain the comparisons for associativity, unitarity and interchange, and

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intervene in the globularisation by their companion cubes (see 5.1). The problem of globularisation
becomes very complicated in the (higher or) infinite-dimensional case, and should be based on one of
the many - more or less equivalent - definitions of weak !-category that have been proposed.
References to the rich literature on higher globular categories can be found in two recent books, by
T. Leinster [Le] and E. Cheng - A. Lauda [CL]. Strict cubical categories with connections (and without
transversal maps) have been studied in [ABS], and proved to be equivalent to (globular) !-categories.
As a matter of notation, the indices #, $ take the values 0, 1, that are more often written as –, +.

1. Strict symmetric cubical categories
We begin by considering (strict) symmetric cubical categories; their globularisation will be defined
in Section 2.
Some simpler notions will be used as preliminary steps. We use the term 'basic' when transversal
maps have not yet been inserted in the structure, so that a cubical category can be defined as a category
object in the category of basic cubical categories (cf. 1.5).
Moreover, as a preliminary step for the weak case dealt with in the sequel, we use the prefix 'pre'
when we are not assuming the axioms of associativity, unitarity and interchange of the cubical
compositions (and comparisons have not yet been introduced).
We end by constructing, as an example, the symmetric cubical category !Rel of cubical relations
(1.8).
1.1. Cubical sets. A cubical set X = ((Xn), (%#i ), (ei)), in the usual sense [K1, K2, BH1, BH2], has
faces (%#i ) and degeneracies (ei)
(1) %#i : Xn

X n–1 :ei

(i = 1,..., n; # = ±),

satisfying the cubical relations :
(2) %#i .%$j = %$j .%#i +1 (j & i),
%#i .ej = ej.%#i –1

(j < i),

ej.ei = ei+1.ej
or

id (j = i),

(j & i),
or

ej–1.%#i (j > i).

Elements of Xn are called n-cubes; vertices and edges for n = 0 or 1, respectively. Every n-cube
x ' Xn has 2n vertices: %1#%2$%3( (x) = %1( %$1%1#(x) for n = 3 and #, $, ( = ±.
A morphism f = (fn): X
and degeneracies.

Y is a sequence of mappings fn: Xn

Yn commuting with faces

Small cubical sets and their morphisms form a category Cub, which has all limits and colimits
Set, where I is the
and is cartesian closed. In fact, it is the presheaf category of functors X: I op
n
n
subcategory of Set consisting of the elementary cubes 2 = {0, 1} , together with the maps {0, 1}m
{0, 1}n which delete some coordinates and insert some 0's and 1's, without modifying the order of
the remaining coordinates [GM].

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4

The terminal object
is freely generated by one vertex * and will also be written {*}; but
notice that each of its components is a singleton. The initial object is empty, i.e. all its components are;
the other cubical sets have a non-empty component in each degree.
1.2. Symmetric cubical sets. As in [G1], a symmetric cubical set is a cubical set which is further
equipped with mappings, called transpositions
(1) si: Xn

Xn

(i = 1,..., n–1; n ) 2).

These have to satisfy the Moore relations
(2) si.si = 1,

si.sj.si = sj.si.sj

(i = j–1),

si.sj = sj.si

(i < j–1),

and the following equations of coherence with faces and degeneracies:

(3)

j<i

j=i

j = i+1

j > i+1

%#j .si =

si–1.%#j

%#i +1

%#i

si.%#j ,

si.ej

e j.si–1

ei+1

ei

ej.si.

=

Assigning the mappings (1) under conditions (2) amounts to letting the symmetric group Sn
operate on Xn. Indeed, it is well known that Sn is generated, under the Moore relations, by the
'ordinary' transpositions s1,..., sn–1, where si, acting on the set {1,..., n}, exchanges i with i+1
(see Coxeter-Moser [CM], 6.2; or Johnson [Jo], Section 5, Thm. 3).
Y is a sequence of mappings fn: Xn
Yn commuting with faces,
A morphism f = (fn): X
degeneracies and transpositions. The resulting category sCub (of small symmetric cubical sets and
Set, for the symmetric cubical site I s.
their morphisms) is again a category of presheaves X: I sop
The latter can be defined as the subcategory of Set consisting of the elementary cubes 2n = {0, 1}n
together with the maps 2m
2n which delete some coordinates, permute the remaining ones and
insert some 0's and 1's. It is a subcategory of the extended cubical site K of [GM], which also
contains the 'connections' (higher degeneracies).
The truncated cases will also be of interest. A symmetric n-cubical set X = ((Xk), (%#i ), (ei), (si))
Xk–1, degeneracies ei:
has components indexed by k = 0,..., n. Of course, also its faces %#i : Xk
Xk and transpositions si: Xk
Xk undergo the restriction k & n, and satisfy the
X k-1
symmetric cubical relations as far as appropriate.
A symmetric n-cubical set is a presheaf on the truncated site TrnI s, namely the full subcategory of
I s with objects 2k for k & n. We write as TrnsCub the category of symmetric n-cubical set and
their (obvious) morphisms.
1.3. A simpler presentation of symmetric cubical sets. In a symmetric cubical set, the presence of
transpositions makes all faces and degeneracies determined by the 1-directed ones, %1–, %1+ and e1 . In
fact, from %#i +1 = %#i .si and ei+1 = si.ei, we deduce that:
(1) %#i = %1#.s'i,

ei = si.e1

where we are using the inverse 'permutations' si and s'i
(2) si = si–1. ... .s1,

s'i = s1. ... .si–1.

(i = 2,..., n; # = ±),

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5

This leads to a more economical presentation of our structure, as proved in [G3]. Namely, a
symmetric cubical set can be equivalently defined as a system
(3) X = ((Xn), %1–, %1+, e1, (si)),
under the Moore relations for transpositions (1.2.2) and the axioms:
(4) %1#.%1$ = %$1.%1#.s1,

e1.e1 = s1.e1.e1,

si.%1# = %1#.si+1,

e1.si = si+1.e1.

%1#.e1 = id,

In other words, X can be presented as a system ((Xn), %1–, %+1, e1) where each Xn is an Sn-set
(equipped with an action of the symmetric group Sn) and the axioms (4) are satisfied.
1.4. Basic cubical categories. As a further step towards cubical categories, we define now the notion
of a basic cubical category (called a 'reduced' cubical category in [G1, G4]), as a cubical set equipped
with cubical compositions in all directions; these are assumed to be strictly categorical (i.e. strictly
associative and unital, units being given by degeneracies) and to satisfy the interchange property.
More explicitly, our notion is defined as follows.
(cub.1) A basic cubical category A is, first of all, a cubical set (1.1):
(1) A = ((An), (%#i ), (ei)).
(cub.2) Moreover, for 1 & i & n, the i-concatenation x +i y (or i-composition) of two n-cubes x, y is
defined when the latter are i-consecutive, i.e. %+i(x) = %–i(y); the following 'geometric' interactions with
faces and degeneracies are required:
(2)

%–i(x +i y) = %–i(x),
%#j (x +i y) =

(3)

ej(x +i y) =

%+i(x +i y) = %+i(y),

%#j (x) +i–1 %#j (y),

if j < i,

%#j (x) +i %#j (y),

if j > i,

ej(x) +i+1 ej(y),

if j & i & n,

ej(x) +i ej(y),

if i < j & n+1

(nullary interchange).

(cub.3) For 1 & i & n, we have a category Ani = (An–1, An, %–i, %+i, ei, +i), where faces give domains
and codomains, and degeneracy yields the identities. In other words, we have the following equations
for i-consecutive n-cubes x, y, z:
(4) (x +i y) +i z = x +i (y +i z),

ei%–ix +i x = x = x +i ei%+ix.

(cub.4) For 1 & i < j & n, and n-cubes x, y, z, u, we have
(5) (x +i y) +j (z +i u) = (x +j z) +i (y +j u)

(middle-four interchange),

whenever these compositions make sense:
(6) %+i(x) = %–i(y), %+i(z) = %–i(u),

x

y

z

u

i
j

%+j(x)

=

%–j(z),

%+j(y)

=

%–j(u),

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6

A cubical functor F: A
B between basic cubical categories is a morphism of cubical sets which
preserves all composition laws.
1.5. Cubical categories. Now, a cubical category A [G1] is a category object in the category of
basic cubical categories (and their cubical functors)
%0#
(1)

A(0)

e0

c0
A(1)

A(2)

(# = ±),

or, equivalently, a basic cubical category in the category of categories
(2) A = ((tvnA), (%#i ), (ei), (+i)),

tvnA = (An, Mn, (%0#), e0, c0),

where An and Mn denote the set of objects and morphisms of the category tvnA, respectively.
Explicitly, this statement means that A is a basic cubical category where each component tvnA is
a category (namely, the category of n-cubes of A and their transversal n-maps, called the transverse
category of A of degree n), while the cubical faces, degeneracies and concatenations are functors
(3) %#i : tvnA

tvn–1A : ei,

+i: tvnA !i tvnA

tvnA.

(The pullback tvnA !i tvnA is the category of pairs of i-consecutive n-cubes.) We distinguish
between the cubical compositions x +i y or f +i g (of i-consecutive n-cubes or n-maps), and the
y, g: y
z.
transversal composition gf = c0(f, g) of transversal maps f: x
A basic cubical category amounts to a cubical category all of whose transversal maps are identities.
A cubical category A has an associated basic cubical category A(0), without transversal maps, that will
be called the basic form of A.
A cubical functor F: A
B between cubical categories strictly preserves the whole structure; in
other words, F is an internal functor between category objects, as specified above.
A transversal (or structural) transformation h: F
G: A
B between cubical functors is an
internal transformation between internal functors. Concretely, it assigns, to every n-cube x of A, a
transversal map in B
(4) h(x): F(x)

G(x),

consistently with faces, degeneracies, concatenations, and satisfying the naturality condition
(nat) hy.Ff = Gf.hx,

for every n-map f: x

y in A.

In a cubical category, as well as in all the weaker cases considered below, a transversal n-map f:
x
x' is said to be special if its 2n vertices are identities
(5) %#f: %#x

%#x'

%# = %1#1 %2#2 ... %n#n

(# i = ±).

In degree 0, this just means an identity.
1.6. Remarks. (a) In a cubical category, a k-map between k-cubes should be viewed as a (k+1)dimensional cell. Therefore an n-truncated cubical category will be called an (n+1)-cubical category.

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Thus, a 1-cubical category is just a category, with objects and 0-maps. A 2-cubical category
amounts to a (strict) double category: its double cells are given by 1-maps, and their boundary consists
of 0-maps and 1-cubes. One level up, a 3-cubical category amounts to a (strict) triple category of a
particular kind, with:
- objects (of one type);
- arrows in directions 0, 1 and 2, where the last two types coincide;
- 2-dimensional cells in directions 01, 02, 12, where the first two types coincide;
- and 3-dimensional cells (of one type).
(b) For fixed positive integers i & n, the cubical category A has an associated double category Ani,
whose double cells are the transversal n-maps of A; the two composition laws are the transversal
composition and i-concatenation of A . More explicitly: objects are the (n–1)-cubes, horizontal arrows
are the transversal (n–1)-maps (with their composition) and vertical arrows are the n-cubes (with iconcatenation); the faces of a double cell are given by %0# and %#i .
(c) We speak of a precubical category when we do not want to assume the axioms of associativity,
unitarity and interchange of the cubical compositions (cf. 1.4). This notion will be of interest later, for
the weak case introduced in Section 3, where we will replace such axioms with 'comparisons', realised
as transversal maps (invertible and special).
(d) It should be noted that transversal maps, apart from being essential in the weak case (as mentioned
above) are already important in the present strict case. For instance, they allow us to define (and
construct) cubical (co)products, and more generally cubical (co)limits, as shown in [G5] (or in [GP1]
for the truncated case of double categories).
1.7. Symmetric cubical categories. A symmetric cubical category
(1) A = ((tvnA ), (%#i ), (ei), (+i), (si)),
is a cubical category (1.5) equipped with cubical functors si: tvnA
tvnA (1 = 1,..., n–1) called
transpositions, which make it a symmetric cubical set. Furthermore, concatenations and transpositions
must be consistent, in the following sense
(2) si–1(x +i y) = si–1(x) +i–1 si–1(y),
sj(x +i y) = sj(x) +i sj(y)

si(x +i y) = si(x) +i+1 si(y),
(j * i–1, i),

where the variables x, y can denote cubes or transversal maps.
As with symmetric cubical sets, all faces, degeneracies and concatenations are now determined by
the 1-directed ones (i.e. %1#, e1, +1), together with transpositions (see 1.3).
The involutive case, further equipped with reversions under axioms which can be easily deduced
from [GM], is also of interest - e.g. for higher relations, higher (co)spans and singular cubes of a
space; however, we will not go here into such details.
A symmetric cubical functor is a cubical functor which also preserves transpositions. A symmetric
transversal transformation (or structural transformation) h: F
G: A
B between such
functors is defined as above (1.5), by further requiring that the transversal maps h(x): F(x)
G(x)
commute with all transpositions.

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8

1.8. Cubical relations. As a simple, non-trivial example we recall here the symmetric cubical category
!Rel of cubical relations (of sets), introduced in [G4] (Sections 4.1, 4.2). We follow here the direct
construction of cubical relations as subsets of suitable cartesian products, but that paper also gives
another realisation of !Rel, as a quotient of the weak symmetric cubical category of spans of sets; the
latter can more easily be extended to other domains.
Items will be indexed by the three-element set {0, u, 1} and its powers. A 1-cubical relation is an
a1 of sets, viewed as a subset au + a0 ! a1, and will be written with a dotordinary relation a: a0
marked arrow; their composition will be written in additive notation. The cubical structure so far is
obvious:
%1#(a: a0

a1) = a#,

e1(x) = ,(x): x

x

(for x a set),

where ,(x) is the diagonal of x ! x.
A 2-cubical relation a consists of:
- four vertices (aij): 2!2

Set (where 2!2 = {0, 1}2 is a discrete category on four objects),

- four (binary) relations on the sides of a square, written auj and aiu (see the diagram below, where no
condition of commutativity is assumed),
- and one quaternary relation auu + - a ij whose projection on each side is contained in the
corresponding binary relation
au0

a00
(1)

a0u

a01

a10
a uu
a u1

auu + a00 ! a01 ! a10 ! a11,

1

a1u

(p0j, p1j)(auu) + auj,

2

a11

(pi0, pi1)(auu) + aiu.

(We write pij: a00 ! a01 ! a10 ! a11

aij the four cartesian projections.)

The 1-concatenation c = a +1 b is defined when the 2-cubes a, b are consecutive in direction 1,
i.e. a1u = b0u, and is shown below, at the right
a u0

a00
(2)

a0u

a01

bu0

a10
a uu
a u1

a1u

a11

=
=
=

b 00
b0u

b 01

au0 + bu0

b 10
bu u
bu 1

b1u

b 11

a00
a0u

a01

b 10
cuu
au1 + bu1

b1u

b 11

The subset
(3) cuu = auu +1 buu + a00 ! a01 ! b10 ! b11,
is formed of those 4-tuples (x00, x01, z10, z11) for which there is some pair (y, y') ' a10 ! a11 = b00 !
b01 such that (x00, x01, y, y') ' auu and (y, y', z10, z11) ' buu. In other words, auu +1 buu is an
ordinary composition of relations, provided we view auu and buu as binary relations, as follows:
(4) auu: a00 ! a01

a10 ! a11,

buu: b00 ! b01

b10 ! b11.

This proves that 1-concatenation is strictly categorical, i.e. strictly associative, with strict units
a1
provided by the following degeneracy e1(a) of an ordinary relation a: a0

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9

a0
au

(5)

e1(a0)

a0
au

(e1a)uu

a1

e1(a1)

(e1a)u0 = e1(a0),
(e1a)uu = ,"au),

a1

(e1a)u1 = e1(a1).

,"au) = {(x0, x1, x0, x1) ' a0 ! a1 ! a0 ! a1 | (x0, x1) ' au},
The same holds for 2-concatenation, which can be defined in the symmetric way, or by
transposition and the previous operation:
(6) a +2 a' = s1(s1(a) +1 s1(a')),

e2a = s1e1(a).

We proceed analogously in higher dimension. An n-cube, or n-cubical relation, is a family a =
(at), indexed by the n-tuples t = (t1,..., tn) ' {0, 1, u}n and satisfying the following conditions.
(a) If t ' 2n, then at is a set. Otherwise, let w = |t| be its weight, i.e. the number of u's in the ntuple; then at is a 2w-ary relation
(7) at + At = - ai1...in,
the cartesian product being indexed by those n-tuples i = (i1,..., in) ' 2n where ij #coincides with tj
when the latter is 0 or 1 (there are 2w such n-tuples). For instance, a0u + a00 ! a01, as in diagram
(1).
(b) If t' is a multi-index obtained by replacing one occurrence of u in t with 0 or 1 (with weight w'
= w – 1), then the corresponding projection ptt' must send the 2w-ary relation at into the 2w'-ary
relation at'
(8) ptt': At

At',

p tt'(at) + at'.

We define now a transversal map f: a

b, as a natural transformation on the discrete category

2n which is 'coherent' with the 'multiple' relations inside a and b:
(9) f = (fi): a

b: 2n

Set,

f i : ai

bi

(i = (i1,..., in) ' 2n),

(coherence condition) for every multi-index t ' {0, 1, u}n, the mapping ft: At
Bt defined by the
cartesian product of the components fi singled out in (7), carries the subset at into bt.
Faces are easily defined, using the maps
(10)

%#i : {0, 1, u}n–1

{0, 1, u}n,

%#i (t1,..., tn–1) = (t1,..., #,..., tn–1)

(# = 0, 1),

Transpositions come from permuting the factors of {0, 1, u}n.
Degeneracies are defined inductively, extending (5) (here at is written as a(t) and # ' 2)
(11)

(e1a)(#, t2,..., tn) = a(t2,..., tn),
(e1a)(u, #,..., tn) = (e1%1#a)(#,..., tn),
(e1a)(u, u,..., tn) = ,"a(u,..., tn)).

Here, a special transversal map (see 1.5.5) amounts to an inclusion of subsets of cartesian products
of the vertices.

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10

The importance of transversal maps (in the present strict case, where comparisons are not needed)
is the same as for double categories: the category (or 2-category) Rel(Ab) of relations of abelian
groups even lacks finite products and coproducts, while the double category Rel(Ab) of abelian
groups, their homomorphisms and relations has all double limits and colimits (cf. [GP1]). Similarly,
the cubical categories of relations (on sets or abelian groups) have all cubical limits and colimits (cf.
[G5]).

2. From symmetric cubical categories to globular categories (the strict case)
A = ((An), (%#i ), (ei), (si)) is always a symmetric cubical set. In such a structure, we single out the
globular cubes (in direction 1), which form a globular set Glb(A). Note that, even though
transpositions do not appear in the definition of a globular cube, without them we would have nonequivalent notions of globularity in the various directions.
Similarly, a basic symmetric cubical category A gives a globular !-category Glb(A), while a
symmetric cubical category A has a sort of 'cylindrical' !-category Cyl(A).
2.1. Globular cubes. An n-cube x of the symmetric cubical set A is said to be globular, or an nglobe (in direction 1), if it satisfies the following condition:
(a) for each i = 1,..., n and # = ±, the i-directed face %#i (x) belongs to the image of the iterated
An–1.
degeneracy (e1)i–1: An–i
Here (e1)i–1 is an abuse of notation, for the composite
e1 ... e1 = ei–1 ... e1: An–i

...

An–2

An–1.

Thus, all 0-cubes and 1-cubes are globular; the following diagrams show the cases n = 2, 3

(1)

x2–

x2–

x2+

x+1
x+2

x1–

x

x3–

1

%2#(x) = e1(x2#),

2

x3–
e1e1(x3–)

(2)

x2–

x1–
x+3

x3–

1

x3–
x+2

x+3

e1(x+2)

3

2

x+2

%2#(x) = e1(x2#),
%3#(x) = e1e1(x3#).

x+3

In the second, the 1-directed faces x1# are arbitrary 2-cubes; the 2-indexed faces e1(x2#) are
degenerate in direction 1; the 3-indexed faces e1e1(x3#) are totally degenerate.
One can notice the following points, made precise in the lemma below.

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11

- There is no condition on the 1-directed faces x1# = %1#(x) ' An-1 (since (e1)0 = id(An–1)).
- Provided that n ) 2, we have %2#(x) = e1(x2#), for some x2# ' An–2; but both x2# are determined by
x, as the 1-faces of its 1-faces x1$ (independently of $ = ±):
x2# = %1$e1(x2#) = %1$%2#(x) = %1#%1$(x) = %1#(x$1).
- And so on. Finally, the highest-directed faces %n#(x) are totally degenerate, produced by two vertices
# #

%n#(x) = (e1)n–1(p#),

#

#

#

#

2 % (x),
p# = xn# = % 1 % 12 ... % 1n(x) = % 1n ... %n–1
n

(where the indices #$,..., #n are arbitrary).
In the symmetric cubical category A, a globular cube is defined in the same way. A transversal
map f: x
y is said to be cylindrical if it satisfies the same condition (a); then the cubes x, y are
globular, while f itself has a sort of 'cylindrical' geometry, analysed in 2.7. We say that f is special
cylindrical if, moreover, it is special as defined in 1.5, i.e. if all its 2n vertices are identities.
2.2. Lemma and Definition (Globular faces). Let x be an n-globe of the symmetric cubical set A.
Its i-indexed faces %#i (x) are degenerate in the first i–1 directions, and can be obtained as follows
(independently of #$,..., #i = ±):
(1) %#i (x) = (e1)i–1(d#i (x)),
# #

#

#

#

#

d#i (x) = x#i = % 1 % 12 ... % 1 i(x) = % 1 i ... %i–12 % i (x).
The (n–i)-dimensional cube d#i (x) is globular. It will be called a globular face of x of dimension
n–i. In particular, the 1-directed faces
(2) %1#(x) = d1#(x) = x1#,
will be called the main faces of x. They have the same faces
(3) %$j (x1#) = (e1)j–1(x$j ),
independently of #. (We will see in the next lemma that these faces determine the faces of x.)
Proof. By hypothesis, for each i = 1,...n, there exist two cubes x#i such that %#i (x) = (e1)i–1(x#i ).
But then x#i is determined as in formula (1):
x#i = %#12 ... %#1 i e1 ... e1(x#i ) = %#12 ... %#1 i %#i (x) = %#1 %#12 ... %#1 i (x).
Moreover the (n–i)-cube x#i is globular, because its j-directed faces
%$j (x#i ) = %$j %#1 %#12 ... %#1 i (x) = %#1 %#12 ... %#1 i (%$j +i x),
#2
#i
j+i–1) = Im(e )j–1.
belong to %#
1
1 % 1 ... % 1 (Im(e1)

Taking i = 1 in the last computation we get the formula (3)
%$j (%1#(x)) = %1#%$j +1(x) = %1#(e1)j(x$j ) = (e1)j–1(x$j ).
2.3. Lemma (Inductive form). Let x be an n-cube of the symmetric cubical set A, with n ) 1. Then
x is globular if and only if:
(i) the main faces %1#(x) are (n–1)-globes,

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12

(ii) the other faces are degenerate in direction 1, i.e. %$i +1(x) ' Im(e1), for i < n and $ = ±.
Moreover, when x is globular, its main faces %1#(x) have the same boundary.
The second condition above can be equivalently replaced with:
(ii') %$i +1(x) = e1 %$i %1#(x)

(for all i, #, $).

Proof. First, condition (ii') obviously implies (ii). Conversely, if %$i +1(x) = e1(u), then:
u = %1# e1(u) = %1# %$i +1(x) = %$i %1#(x).
Now, let x be an n-globe, so that %#i (x) = (e1)i–1(x#i ). We already know from the previous lemma
that the main faces %1#(x) = d#1 (x) = x#1 are (n–1)-globes and have the same boundary. Property (ii)
also holds: %$i +1(x) ' Im(e1)i + Im(e1).
Conversely, let x satisfy (i) and (ii'). Then %$i +1(x) = e1 %$i %#1 (x). But %#1 (x) is an (n-1)-globe,
whence %$i +1(x) belongs to e1(Im(e1)i–1) = Im(e1)i, so that x is an n-globe.
2.4. Theorem (Closure properties). In a symmetric cubical set, globular cubes are closed under e1
and all faces %#i . In a (possibly basic) symmetric cubical category they are also closed under all iconcatenations. In a symmetric cubical category, globular transversal maps are closed under transversal
composition and all i-concatenations; the same holds for the special ones.
Proof. First, it is obvious that e1 preserves globular cubes (because %#i e1 = e1%#i +1).
We already know from the previous lemma that the (n–i)-dimensional cube d#i (x) is a globular
cube; it follows that %#i (x) = (e1)i–1(d#i (x)) is also.
Now, in a basic symmetric cubical category, let us start from noting that all 1-cubes are globular
and obviously closed under 1-concatenation. Suppose that globular cubes of dimension n–1 are
closed under concatenation in all directions (i.e. 1,..., n–1). Now, consider the i-concatenation x +i y
of two globular n-cubes, with i & n, and let us prove that it is globular, using the previous lemma.
Let us recall the formulas
%#j (x +i y) =

ej(x +i y) =

%#j (x) +i–1 %#j (y),

if j < i & n,

%#j (x) +i %#j (y),

if i < j & n.

ej(x) +i+1 ej(y),

if j & i & n,

ej(x) +i ej(y),

if i < j & n+1.

As to property (i), we must check that the main faces %1#(x +i y) are (n–1)-globes. Indeed, for i =
1, %1#(x +1 y) is x or y, and is globular; for i > 1 , %1#(x +i y) is a concatenation of lower globular
cubes, hence is globular by the inductive hypothesis.
As to property (ii), suppose that %$j (x) = e1(u) and %$j (y) = e1(v), with j > 1. We must check
various cases for %$j (x +i y).
(a) Case j < i (with j > 1 and i > 2):
%$j (x +i y) = %$j (x) +i–1 %$j (y) = e1(u) +i–1 e1(v) = e1(u +i–2 v).
(b) Case 1 < i < j:

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13

%$j (x +i y) = %$j (x) +i %$j (y) = e1(u) +i e1(v) = e1(u +i–1 v).
(c) Case 1 = i < j:
%$j (x +1 y) = %$j (x) +1 %$j (y) = e1(u) +1 e1(v) = e1(u),
where the existence of e1(u) +1 e1(v) means that: u = %1+e1(u) = %1– e1(v) = v.
(d) Case i = j. This is obvious, because %$i (x +i y) is x or y.
The last assertion, about transversal maps, is proved in the same way (for i-concatenations), or is
obvious (for transversal composition).
2.5. The globular set associated to a symmetric cubical one. Starting from the symmetric cubical
set A, we begin by constructing the associated globular set GlbA of its globular cubes.
An n-globe of GlbA is a globular n-cube of A. Faces and degeneracies come from the 1-directed
faces of cubes (because of Theorem 2.4):
(1) d#: GlbnA

d# = %1#, e = e1.

Glbn–1A :e,

We have thus a globular set ((GlbnA), (d#), (e))
d#

(2)

... GlbnA

e

d#

Glbn–1A

d#

...
e

e

d#

Glb1A

e

Glb0A = A0.

This means that faces and degeneracies satisfy the globular identities:
(3) d#d– = d#d+,

d#e = id,

and indeed we already know that %1#%1$(x) = d2#(x) does not depend on $ (cf. 2.2.1).
In any globular set G = ((Gn), (d#), (e)) one can define the higher faces (or iterated faces) d#i ,
for i = 1,..., n (and independently of #$,..., #n)
(4) d#i : Gn

d#i = d#1 d#12 ... d#1 i.

Gn–i,

For G = GlbA, this definition plainly agrees with the globular faces defined above (in 2.2). An nglobe x ' Gn can be written in the following form, showing all its globular faces:
(5) x: d1–x

+
–
1 d1x: d2x

+
2 d2x:

... dn–x

+
n dnx.

We write Glb (resp. TrnGlb) the category of globular sets (resp. n-truncated globular sets) and
their (obvious) morphisms.
2.6. Compositions. If A is a basic symmetric cubical category, we make the globular set GlbA (on
the underlying symmetric cubical set A) into a globular category GlbA, by defining its
compositions.
Let x, y be two n-globes of A, consecutive in direction i = 1,..., n. This is equivalently expressed
by the following conditions, globular or cubical respectively
(1) d+i(x) = d–i(y),

%+i(x) = %–i(y),

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14

because %#i = (e1)i–1d #i (by 2.5.2). Therefore, we define their i-composition x +i y as in A: the
result is globular, because of Theorem 2.4.
Since our compositions are strictly associative and unitary, GlbA is indeed a globular category.
If A is an (extended) symmetric cubical category, the same arguments apply to cylindrical
transversal maps (between globular cubes), and we obtain a category object CylA in the category of
globular categories. This will be called a cylindrical category.
By a further restriction to special cylindrical transversal maps (between globular cubes) we obtain a
'special' category object SCylA in the category of globular categories, where the 'transversal'
morphisms between vertices are identities. This will be called a special cylindrical category.
Cylindrical transversal maps have indeed a cylindrical shape: for instance, a cylindrical 2-map f: x
x' (where the faces %2#(f) are 1-degenerate)
p'
e1(f––)

f––

p'
0

b'

p

p

a

x

f

+

q'

b

q

1
2

f++
q

can be pictured as a solid cylinder on a 2-globe
p
a

f––

p'
f+

x

2

q

f++

1 b: p

f–: a
f+: b

a', f––: p
b', f++: q

1 b': p'

2 q,
2 q',

0

b'

b

x: a
x': a'
1

q'

p',
q'.

After its transversal 0-faces x, x', the map f has the following globular faces
(3) f# = d1#(f): %1#(x)

%1#(x'),

f## = d2#(f): d2#(x)

d2#(x')

(# = ±).

It is special cylindrical if its vertices f## = d2#(f) are identities.
2.7. Globular relations. The basic cubical category !Rel(0) of cubical relations yields a globular
category Glb(!Rel(0)), where a 2-globe a consists of a 2-cubical relation (as in 1.8.1) whose 2-faces
are 1-degenerate. Writing a: a' 1 a": x 2 y, we have
- two sets x, y,
- two (binary) relations a' + x ! y, a" + x ! y,
- and one quaternary relation a satisfying the following conditions (where ,(x) is the diagonal of x
! x)

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15

x
a
(1)

a'

a"

1
2

a + x ! y ! x ! y,
(p1, p3)(a) + ,(x), (p2, p4)(a) + ,(y),
(p1, p2)(a) + a',

(p2, p4)(a) + a".

y
On the other hand, the cubical category !Rel (including transversal maps) yields a cylindrical
category Cyl(!Rel). Recalling the description of a special transversal map of !Rel as an inclusion
of subsets of cartesian products of the vertices (at the end of 1.8), the special cylindrical category
SCyl(!Rel) can be called an ordered !-category.
As to low-dimensional truncated cases, the double category of relations 2Rel = tr1(!Rel) stays
unchanged under Cyl (since all 1-cubes are globular), while SCyl(2Rel) is the ordinary 2-category
of relations (an ordered category). One level up, the triple category of relations 3Rel = tr2(!Rel)
yields:
- a structure Cyl(3Rel), which is a category object in the category of 2-categories,
- an ordered 2-category SCyl(3Rel) (having consistent orderings of 1-cells and 2-cells with the same
0-dimensional faces).

3. Weak symmetric cubical categories
We now recall the definition of a weak symmetric cubical category, introduced in [G1, G4].
3.1. Symmetric precubical categories. First, a basic symmetric precubical category
(1) A = ((An), (%#i ), (ei), (si), (+i)),
is a symmetric cubical set with compositions, satisfying the consistency axioms (cub.1-2) of 1.4, where
compositions are consistent with transpositions (in the sense of 1.7.2). Notice that we are not
assuming that the i-compositions behave in a categorical way or satisfy interchange, in any sense, even
weak. The morphisms of this structure are obvious.
Second, a symmetric precubical category is a category object A in the category of basic
symmetric precubical categories (as defined above)
%0#
(2)

A(0)

e0

A(1)

c0

A(2)

(# = ±).

Explicitly, this means the following data and axioms.
(wcub.1) A basic symmetric precubical category A(0) = ((An), (%#i ), (ei), (si), (+i)), whose entries are
called n-cubes, or n-dimensional objects of A.
(wcub.2) A basic symmetric precubical category A(1) = ((Mn), (%#i ), (ei), (si), (+i)), whose entries are
called n-maps of A, or also (n+1)-cells.

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16

(wcub.3) Symmetric cubical functors %0# and e0, called 0-faces and 0-degeneracy, with %0#.e0 = id.
Also here, an n-map is written as f: x
x', where %0– f = x, %+0f = x' are n-cubes. Every nx. Note that %0# and e0 preserve cubical faces (%#i ,
dimensional object x has an identity e0(x): x
with i > 0), cubical degeneracies (ei), transpositions (si) and cubical concatenations (+i). In
particular, given two i-consecutive n-maps f, g, their 0-faces are also i-consecutive and we have:
(3) f +i g: x +i y

x' +i y'

(for f: x

x', g: y

y'; %+if = %–ig).

(wcub.4) A composition law c0 which assigns to 0-consecutive n-maps f: x
x', h: x'
x" (of
the same dimension), an n-map hf: x
x" (also written h.f). This composition law is (strictly)
categorical, and forms a category A n = (An, Mn, %0#, e0 , c0 ). It is also consistent with the basic
symmetric precubical structure, in the following sense
(4) %#i (hf) = (%#i h).(%#i f),

ei(hf) = (eih)(eif),

si(hf) = (sih)(sif),
%–i f

%–i h

–f

–h

(h +i k).(f +i g) = hf +i kg,
x

x"
0

y

–g

–k

%+ig

%+ik

y"

i

The last condition is the (strict) middle-four interchange between the strict composition c0 and
any weak one.
x' is said to be special if its 2n vertices %#f are identities, where:

An n-map f: x
(5) %#f: %#x

%#x'

%# = %1#1 %2#2 ... %n#n

(# i = ±).

In degree 0, this just means an identity.
3.2. Comparisons. We now define a weak symmetric cubical category A as a symmetric precubical
category (3.1), which is further equipped with invertible special transversal maps, playing the role of
comparisons for units, associativity and cubical interchange, as follows. (We only assign the
comparisons in direction 1; all the others can be obtained with transpositions.)
(wcub.5) For every n-cube x, we have a special invertible n-map .1x, which is natural on n-maps
and has the following faces (for n > 0)
(1) .1x: (e1%1–x) +1 x
%1#.1x = e0%1#x,

x

(left-unit 1-comparison),
%#j .1x = .1%#j x

(1 < j & n)

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17
%1–x

%1–x
0

e0%1–x
%+jx
e1%1–x
%–jx

x

.1%+jx

. 1 % –j x
%–jx

x

j
1

e0%+1x

%+
1x

(1 < j & n).

%+
1x

The naturality condition means that, for every n-map f: x
commutes
.1x

(e1%1–x) +1 x
(2)

%+jx

x', the following square of n-maps

x

(e1%1–f) +1 f

f

(e1%1–x') +1 x'

.1x'

x'

(wcub.6) For every n-cube x, we have an invertible special n-map /1x, which is natural on n-maps
and has the following faces (the naturality diagram, similar to diagram (2), is not written down)
(3) /1x: x +1 (e1%+1x)

x,

(right-unit 1-comparison),

%1#/1x = e0%1#x,

%#j /1x = /1%#j x
%1–x

(1 < j & n),

%1–x
0

e0%1–x
%+jx
x
%–jx

x

/1%+jx

%+jx

/1%–jx

e1%1+x

%–jx

%+
1x

j
1

e0%+1x

(1 < j & n).

%+
1x

(wcub.7) For three 1-consecutive n-cubes x, y, z, we have an invertible special n-map 01(x, y, z),
which is natural on n-maps and has the following faces
(4) 01(x, y, z): x +1 (y +1 z)

(x +1 y) +1 z

%1–01(x, y, z) = e0%1–x,
%#j 01(x, y, z) = 01(%#j x, %#j y, %#j z)

(associativity 1-comparison),
%+101(x, y, z) = e0%1+z,

(1 < j & n),

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18
%1–x

%1–x

e0%1–x

%+jx

%+jx
x +1 y

%+jy
%–j x

0

%+jy
%–j x

x
01%+j

j

01%–j

1

%+jz

%+jz

z

%–j y

%–j y
y +1 z

%–j z

% –j z

e0%+1z

%+
1z

(1 < j & n).

%+
1z

(wcub.8) Given four n-cubes x, y, z, u which satisfy the boundary conditions making the following
concatenations possible, we have an invertible n-map 11(x, y, z, u), which is natural on n-maps and has
the following faces (partially displayed below)
(5) 11(x, y, z, t): (x +1 y) +2 (z +1 t)

(x +2 z) +1 (y +2 t)

%1–11(x, y, z, t) = e0(%1–x +2 %1–z),
%2–11(x, y, z, t) = e0(%2–x +1 %2–y),
%#j 11(x, y, z, t) = 11(%#j x, %#j y, %#j z, %#j t)
%2–x

%1–x

x +1 y

%1–z

z +1 t
%2+z

%2+t

(2 < j & n),

%2–y
%+
1y

e0
e0

%+
1t

(interchange 1-comparison),

%+111(x, y, z, t) = e0(%1+y +2 %+1t),
%+212(x, y, z, t) = e0(%2+z +1 %+2t),

%1–x

e0

%1–z

%2–x

%2–y

x

y

+2

+2

z

t

0

%+
1y

1

%+
1t

2

e0
%2+z

%2+t

(wcub.9) Finally, these comparisons must satisfy some conditions of coherence, listed in 3.3 below.
There is a more general 'u-lax' case, dealt with in [G6], which will not be used here. For a u-lax
symmetric cubical category we do not assume the comparisons for left and right unitarity to be
invertible (writing them as directed towards simpler expressions, as above); but we still require that the
comparisons for associativity and interchange be invertible. The main example is the cubical structure
Sng(X) of singular cubes of a topological space, with transversal maps defined by reparametrisations.
In that version, the axioms above are denoted as (ucub.5-9).
3.3. Coherence. The coherence axiom (wcub.9) means that the following diagrams of transversal
maps commute (assuming that all the cubical compositions make sense).

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19

(i) Coherence pentagon for 0 = 01:
(x +1 y) +1 (z +1 u)
(1)

0

0

x +1 (y +1 (z +1 u))

((x +1 y) +1 z) +1 u

1+0

0+1

0

x +1 ((y +1 z) +1 u)

(x +1 (y +1 z)) +1 u

(ii) Coherence conditions for the unit comparisons:
e1%1–x +1 (x +1 y)
(2)

0

(e1%1–x +1 x) +1 y

.

.+1

x +1 y

x +1 (e1%1–y +1 y)
(3)

0

(x +1 e1%+1x) +1 y

1+.

/+1

x +1 y

x +1 (y +1 e1%+1y)
(4)

0

(x +1 y) +1 e1%+1y

1+/

/

x +1 y
(iii) Coherence hexagon for 1 = 11 and 0 = 01
0+0

(x +1 (y +1 z)) +2 (x' +1 (y' +1 z'))

((x +1 y) +1 z) +2 ((x' +1 y') +1 z')

1

(5)

1

(x +2 x') +1 ((y +1 z) +2 (y' +1 z'))

((x +1 y) +2 (x' +1 y')) +1 (z +2 z')

1+1

1+1

(x +2 x') +1 ((y +2 y') +1 (z +2 z'))

0

((x +2 x') +1 (y +2 y')) +1 (z +2 z')

(iv) Coherence conditions for 1 = 11, . = .1 and / = /1
(e1%1–x +1 x) +2 (e1%1–y +1 y)

.+.

/+/

x +2 y

1

(6)

1

+
(x +2 y) +1 (e1% +
1 x +2 e 1 % 1 y)

(e1%1–x +2 e1%1–y) +1 (x +2 y)
e1%1–(x +2 y) +1 (x +2 y)

+
(x +1 e1%+
1 x) +2 (y +1 e1 % 1 y)

x +2 y
.

/

(x +2 y) +1 e1%+1(x +2 y)

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20

(The equality in the left and right column of this diagram follows from the 'geometric interactions'
of axiom (cub.2), in 1.4.)
Truncation works as described in1.8. Again, a 1-truncated symmetric weak cubical category has no
transpositions and is the same as a pseudo double category.
3.4. Remarks. As in 1.6(b), given two positive integers i & n, the weak symmetric cubical category A
has an associated weak double category Ani, whose double cells are the transversal n-maps of A;
again, the two composition laws are the transversal composition and i-concatenation of A.
The axioms (wcub.5-7) and (wcub.9)(i)(ii) just express this fact, for i = 1 (which is sufficient,
because of symmetry).
3.5. Unitarity. The weak symmetric cubical category A is said to be unitary if, for every cube x, its
comparisons .1x and /1x are transversal identities, namely e0(x). (By symmetry, the same holds
for every cubical i-composition.)
Then, for a transversal map f: x

y, the naturality of .1 and /1 proves that:

(i) e1%1–f +1 f = f = f + e1%+1f.
e1%1–x +1 x

x +1 e1%+1x

x

e1%1–f +1 f

f

e1%1–y +1 y

y

f +1 e1%+1f
y +1 e1%+1y

But there is a more general notion, that is sufficient to ensure that globular cubes be closed under
concatenation, and can often be 'forced' on weak symmetric cubical categories without heavy
modifications. We say that A is preunitary if, for every 1-degenerate cube e1(x), its comparisons
.1(e1(x)) and /1(e1(x)) are transversal identities, actually the same: e0e1(x): e1(x) +1 e1(x)
e1(x).

4. Examples: cubical cospans and cubical profunctors
We recall, from [G1], the construction of the weak symmetric cubical category !C osp(X) of
cubical cospans on a category with pushouts X. Then, starting from the structure !Cosp(Emb) of
cubical cospans of full embeddings of categories, we give a construction of the weak symmetric cubical
category !C at of cubical profunctors. (This modifies a similar construction given in [G4]: besides
using more general transversal maps, we also correct an error of that paper with respect to cubes.)
Here, an embedding of categories always means a faithful functor, injective on objects.
4.1. Cubical cospans. We follow a construction given in [G1] for cubical cospans. Modifications of
this structure for higher dimensional cobordism can be found in [G2, G3].
Let X be a category with pushouts, and equipped with a choice of them, called distinguished
pushouts. A cospan in X is a diagram of shape
(1)

u = (u–: X–

X0

X+ :u+),

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21

X+; they are composed with distinguished pushouts, forming a
viewed as a morphism u: X–
bicategory; or, also, the weak arrows of a larger structure, the pseudo double category Cosp(X), as in
[GP1].
The model of the construction is the formal cospan ", together with its cartesian powers "n
(-1,-1)

(0,-1)

(1,-1)

1
2

(2)

–1

0

1

",

(-1, 0)

(0, 0)

(1, 0)

(-1, 1)

(0, 1)

(1, 1)

" 2.

(In these diagrams, identities and composed arrows are understood.)
An n-cubical cospan in X is thus defined as a functor x: "n

X.

Faces and degeneracies are obvious, and these diagrams form a cubical set. Moreover, for i = 1,...,
n, there are compositions x +i y of i-consecutive n-cubes, computed by distinguished pushouts.
These operations behave 'categorically' and satisfy interchange in a weak sense, up to suitable
comparisons.
To make room for the latter, the n-th component of !Cosp(X)
(3) C ospn(X) = Cat("n, X),
is not reduced to the set of functors x: " n
X (the n-cubes, or n-dimensional objects, of the
X.
structure), but is the category of such functors and their natural transformations f: x
x': " n
The latter are the n-maps of the structure, or (n+1)-cells.
The comparisons are invertible special n-maps (between different realisations of the same colimit);
but general n-maps are also important, e.g. to define limits and colimits (cf. [G1], Section 4.6, or [G5]
for a general theory of cubical limits).
4.2. Remarks. Let us make precise that our choice of pushouts in X assigns, to each span (f, g) one
distinguished pushout (f', g') (in a symmetric way, i.e. consistently with permutation of pairs)
f

1

x
(1)

g

f'
g'

x

1

1

x

1

x

We assume a first unitarity constraint:
(i) each square of identities is a distinguished pushout,
so that !C osp(X) is preunitary (3.5).
If one assumes a stronger unitarity constraint (as in [G1]):
(ii) the distinguished pushout of the span (f, 1) is (1, f), and symmetrically,

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22
the weak symmetrical cubical category !Cosp(X) becomes unitary, which would simplify even more
our procedures of globularisation, in the next section. But we prefer not to oversimplify this example.
4.3. Profunctors as cospans. Now, to deal with cubical profunctors, we start from cubical cospans of
full embeddings of (small) categories.
x1 has a collage x0, which
The crucial point is the fact that an ordinary profunctor x: x–1
consists of the sum of the categories x–1 and x1, supplemented with new homs x0(a, b) = x(a, b),
for a in the domain and b in the codomain. (Formally, the collage of a profunctor is a double colimit,
the cotabulator, in the weak double category of categories, functors and profunctors, see [GP1].)
Thus, the profunctor x can be described as a cospan
(1)

x–1

x–

x0

x+

x1

satisfying the following conditions (which imply that x–, x+ have disjoint images)
(i) x–, x+ are full embeddings,
(ii) the embeddings x–, x+ cover all the objects of x0,
(iii) there are no arrows in x0 going from an object coming from x1 to one coming from x–1.
We have already seen how a profunctor yields such a cospan. Conversely, given the cospan (1), the
profunctor is reconstructed as:
(2) x: (x–1)op ! x1

Set,

(a, b)

x0(x–(a), x+(b)).

Note that condition (ii) is not closed under concatenation; we shall modify this operation (in 4.5).
In particular a functor f: x
f: xop ! y

y is identified with the profunctor

Set,

f(a, b) = y(f(a), b),

and can be described as a cospan x
z
y where z is the coproduct x + y, supplemented with
new arrows z(a, b) = y(f(a), b), for each object a of x and b of y.
4.4. Cospans of categories. We begin from considering condition 4.3(i). Let C = ! C osp(Cat) be
the weak symmetric cubical category of cubical cospans of functors between small categories. Let
Emb be the category of small categories and their full embeddings, and E the transversally full
substructure of C whose cubes belong to !Cosp(Emb).
Thus, an n-cube of E is a functor x: " n
Cat.
y: " n

Emb, but an n-map is a natural transformation f: x

To show that this is legitimate, we prove that Emb has pushouts, which are also pushouts in Cat.
Given a span of full embeddings A
X
B, let us rename the items of A and B so that
these functors are full inclusions and X = A 2 B. (The letters a, a', #, #' will denote objects and
arrows of A, while b, b', $, $' belong to B and x, x' are objects of X.)
Now, the pushout W contains the obvious set-theoretical union A 3 B, supplemented with:

141
23

x
b, for #: a
x in A and $: x
b in B, modulo the equivalence
(a) new arrows [$#]: a
x' in X such that #' = 4# in
relation generated by identifying $# = $'#' if there exists some 4: x
A and $ = $'4 in B,
(b) and, symmetrically, new arrows [#$]: b

x

a.

The composition in W is easily defined, as in the following examples:
(1) [$#].#' = [$(##')],
[#'$'].[$#] = #'.($'$).#%

for #': a'

a, #: a

x, $: x

b,

for #: a

x, $: x

b, $': b

x', #': x'

where the last composition is in A (since $'$: x

a',

x' belongs to the full subcategory X = A 2 B).

4.5. Essential cubical cospans. We now take condition 4.3(ii) into account, and construct a weak
symmetric cubical category E' of essential cubical cospans of full embeddings; it is contained in E,
but has different cubical compositions. (It might be viewed as a quotient of E, but using such a
construction would be longer.)
Emb is essential if, in each category x(t)
We say that an n-cospan (of full embeddings) x: "n
n
(t ' Ob" ), every object is the image of some object in a category occupying a vertex of the cube.
(Plainly, there is nothing to check about such categories.)
Emb has an associated essential cospan x̂, obtained as follows: the
Every n-cospan x: " n
vertices of the cube (marked with bigger bullets, in the 2-dimensional case below) are left unchanged,
but we replace each other category x(t) (t ' Ob"n) with the full subcategory x̂(t) determined by the
objects which are the image of some object of a vertex

(1)

Essential cubical cospans form a structure E' that inherits from E faces, degeneracies,
transpositions and transversal maps. But we redefine the i-composition x +̂ i y of i-consecutive
essential n-cubical cospans as the essential n-cubical cospan associated to their i-composition in E
(2) x +̂i y = (x +i y)ˆ.
It is easy to see that the associativity comparisons restrict to (invertible) comparisons
(3) 0̂i(x, y, z): x +̂ i (y +̂i z)

(x +̂ i y) +̂i z,

and similarly for unitarity and interchange.
Finally, E' is again a weak symmetric cubical category.
4.6. Cubical profunctors. Finally, we take care of condition 4.3(iii), and construct a weak symmetric
cubical category !Cat of cubical profunctors; it is contained in E', but has different degeneracies.

142
24
We say that an n-cube x of E' is an n-profunctor if, for each ordinary cospan which appears in x
in a given direction i
(1)

x(t')

x(t)

x(t")

(t'i = –1, ti = 0, t"i = 1; t'j = tj = t"j for j * i ),

condition (iii) of 4.3 is satisfied: there are no arrows in x(t) going from an object coming from x(t")
to one coming from x(t').
Cubical profunctors inherit from E' faces, transpositions, transversal maps, compositions and
comparisons for associativity and interchange. Degeneracies make some problems (as it is also the
case for cospans in the domain of cobordism, see [G3]). Indeed, already in degree 1, the degenerate
cospan of a (non-empty) category
e1(x) = (x = x = x),
is essential but does not satisfy condition (iii). However (as in [G3]) we can replace degeneracies with
cylindrical degeneracies: the 1-cube E1(x) on the category x is the following essential cospan of
disjoint embeddings (which is also reduced):
x–

(2)

x

x+

x0

x

where the category x0 = x ! 2 is the collage of the identity profunctor of x (and x–, x+ are the
obvious embeddings). It is easy to see that E1(x) is a weak identity for concatenation with 1profunctors (but not with general 1-cubes of E'). Similarly one defines the cylindrical degeneracy
E1(x) of any n-cube.
We restrain the construction of composition so that E1(x) +1 E1(x) = E1(x).
Finally, we have obtained a preunitary weak symmetric cubical category ! C at of cubical
profunctors, contained in E'; the embedding preserves all the structure, except degeneracies (and their
comparisons), and is transversally full.
Notice that the bicategory of ordinary profunctors can be realised as a strict 2-category, as recalled
in [GP1]: a profunctor u: A
B can be defined as a colimit-preserving functor
(3) u^: SetA

SetB,

u^(F)(b) =

a

u(a, b) ! F(a).

This realisation might perhaps be used for another construction of !Cat as a strict symmetric
cubical category.

5. Globularisation in the weak case
For a weak symmetric cubical category A, a globularisation procedure has to make use of the
transversal maps, that contain the comparisons. This can be done associating to any transversal n-map
its companion, an (n+1)-cube.
Companions in double categories have been introduced and studied in [GP2-4]. They are related
with connections in cubical sets, in the sense of Brown-Higgins [BH1, BH2, ABS].

143
25
In this section we suppose, for the sake of simplicity, that A is preunitary, as defined in 3.5. This
ensures that the globes of A are closed under cubical compositions.
x+ is an (n+1)-cube x such that
5.1. Companions. A companion of a transversal n-map f: x–
+
%n#+1(x) = x#, equipped with (n+1)-maps 5: e1(x–)
0 x and 6: x
0 e1(x )
x–
(1)

en+1

(x–)

e0(x-)

x–
x

5

x–

f

x–
x

x+

x+

0

en+1(x+)

6

x+

n+1

x+
e0(x+)

f

such that 6.5 = en+1(f) and 5 +n+1 6 coincides with e0(x) up to unitarity comparisons, as in the
following diagram:
x–

e0(x-)

x–

en+1(x–)
(2)

e0(x-)
5

x–

x
–1
. n+1
(x)

x+

x

x

e0(x-)

x–

x–

x

=

x

e0(x)

x

en+1(x+)

6

x+
e0(x+)

x–

/n+1(x)

x+

x+
e0(x+)

x–

x+

x+

x+

e0(x+)

The companion is determined up to invertible (n+1)-maps.
We say that A has structural companions if every transversal n-map f: x–
x+ is equipped
with a companion f*, and the global assignment is consistent with faces and degeneracies, in the sense
that:
(3) (%#i (f))* = %#i (f*),

(ei(f))* = ei(f*)

(i = 1,..., n).

Below, we only work with low-dimensional cases, up to 2-truncation (and dimension 3). As a
consequence, we only need companions for transversal 1-maps.
5.2 Low-dimensional cases. Let A be a weak symmetric cubical category.
(i) Truncating A in degree 1 we get a weak 2-cubical category 2A = tr1(A), with one strict direction
and one weak direction; in other words, a weak double category. Its 1-cubes and special transversal
maps form a bicategory.
(ii) Truncating A in degree 2 we get a weak symmetric 3-cubical category 3A = tr2(A), with one strict
direction and two weak directions.
We want now to show that, if A has structural companions, we can use the 2-globes and certain
cylindrical 2-maps to construct a tricategory T, as defined in Gordon, Power and Street [GPS].
(TD1) the objects of T are the 0-globes (or 0-cubes) p, q,... of A.
(TD2) For objects p–, p+ we have a bicategory T(p–, p+) where:

144
26

(a) a 0-cell u is a 1-globe u: p–

+
#
#
1 p (i.e. a 1-cube with %1 u = p ), called a 1-cell of T,

+
–
(b) a 1-cell x: u–
1 u is a 2-globe x: u
e1(p#)), called a 2-cell of T,

+ –
1u :p

+ (i.e. a 2-cube with %# x = u# ,
1

2p

%#2 x =

+ –
+
–
+ –
+ –
+
(c) a 2-cell f: x–
0x :u
1 u is a globular 2-map f: x
0x :u
1u :p
2 p (i.e. a
2-map f: x– 0 x+ whose cubical faces %#i f are transversal identities, for i = 1, 2), called a 3-cell of
T. (Such globular 2-maps are necessarily cylindrical, because

%2#(f) = e0(%$0 %2#(f)) = e0(%2# %0$(f)) = e0(%2# x$) = e0(e1(p#)) = e1(e0(p#)). )
(d) 2-cells of T have a composition law, defined by the 1-composition of 2-cubes (taking into account
the preunitarity property, which ensures that 2-globes are closed under 1-composition):
(for %+1x = %1–y).

(1) x +1 y

(e) 3-cells of T have two composition laws, defined by transversal composition and 1-composition in
A:
(2) f +0 g = gf

(for %+0f = %0–g),

(3) f +1 g

(for %+1f = %1–g).

(f) The 0-composition of 3-cells is categorical, i.e. strictly associative and unitary.
(g) The 1-composition of 2-cells is associative up to its 0-invertible comparisons in A:
(4) 01(x, y, z): x +1 (y +1 z)
p–

p–
x

p+

p–

p–

y
p+

(x +1 y) +1 z,

0

1

z

2

p+

p+

We verify now that 01(x, y, z) is a globular 2-map (i.e. 2-cells of T(p–, p+)):
%1–01(x, y, z) = e0%1–x,

%+101(x, y, z) = e0%1+z,

%2#01(x, y, z) = 01(%2#x, %2#y, %2#z) = 01(e1(p#), e1(p#), e1(p#)) = e0(e1(p#)).
where the penultimate equality comes from a partial unitarity property (cf. 3.3(v)).
Their naturality with respect to transversal maps f: x–

+
–
0 x , g: y

+
–
0 y , h: z

+
0 z holds in

A.
(h) The composition of 2-cells is unitary up to its comparisons in A, which again are globular (also
because of preunitarity).
(TD3) For objects p, q, r, the last composition is defined as a weak functor of bicategories:
(5) C: T(p, q) ! T(q, r)

T(p, r),

C(u: p

1 q, v: q

1 r) = (u +1 v: p

C(x: u–

+
1u :p

–
2 q, y: v

+
1u :q

C(f: x–

+ –
0x :u

+
1u :p

–
2 q, g: y

= (f +2 g: x– +2 y–

0x

++

1 r),
2 r) = (x +2 y: u
+ –
0y :v

+ –
–
2 y : u +1 v

1u

–+

+
1u :q
++

+
1v :p

1v

–

2 r)
2 r).

1u

++

+
1v :p

2 r),

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27
To define the comparison of C with respect to the 1-composition of 2-cells (2-globes of A), take
four 2-cells x, y, z, t disposed as below. Then the interchange comparison of A
11(x, y, z, t): (x +1 y) +2 (z +1 t)

p

p
y
q

1

q

z
r

(x +2 z) +1 (y +2 t),

p

x
q

0

2

t
r

r

yields the comparison we want:
(6) 11(x, y, z, t): C(x +1 y, z +1 t)

0 C(x, z) +1 C(y, t).

This is indeed a globular cell (also because of preunitarity):
%1–11(x, y, z, t) = e0(%1–x +2 %1–z),

%+111(x, y, z, t) = e0(%1+y +2 %+1t),

%2–11(x, y, z, t) = e0(e1(p)) = e1(e0(p)),

%+212(x, y, z, t) = e0(e1(r)) = e1(e0(r)).

(TD4) The unit of an object p is a strict functor of bicategories, defined on the singleton 2-category
1:
(7) U: 1

T(p, p),

U(*) = (e1(p): p

1 p),

U(1: 1

1 1: *

2 *) = (e1e1(p): e1(p)

U(1: 1

0 1: 1

1 1: *

1 e1(p): p

2 *) = (e0e1e1(p): e1e1(p)

2 p),
0 e1e1(p): e1(p)

1 e1(p): p

2 p).

(TD5) For objects p, q, r, s we define a pseudo natural equivalence of pseudo functors:
(8) k: C(1 ! C)

C(C ! 1): T(p, q) ! T(q, r) ! T(r, s)

T(p, s).

First, for a triple of 1-cells (u, v, w) forming an object of T(p, q) ! T(q, r) ! T(r, s), we have a
comparison 2-cell:
(9) k(u, v, w): u +1 (v +1 w)

1 (u +1 v) +1 w: p

2 s,

k(u, v, w) = (0(u, v, w))*,
given by the companion of the 1-map
0(u, v, w): u +1 (v +1 w)

0 (u +1 v) +1 w.

Second, for a triple of 2-cells (x, y, z) forming a 1-cell of T(p, q) ! T(q, r) ! T(r, s)
(10)

x: u–

+
1u :p

y: v–

2 q,

+
1v :q

2 r,

z: w–

+
1w :r

2 s,

we construct a comparison 3-cell:
(11)

k(x, y, z): x +2 (y +2 z) +1 k(u+, v+, w+)

– –
–
0 k(u , v , w ) +1 (x +2 y) +2 z:

u– +1 (v– +1 w–)

2 s.

1 (u

++

+
+
1 v ) +1 w : p

As a first step we consider the cell 7 = 5 +1 (0(x, y, z) +1 6), obtained by 1-composition of the
comparison 3-cell 0(x, y, z) with a unit 5 and counit 6 of companionship for 0(u#, v#, w#)

146
28

u– +1 (v– +1 w–)

e0(-)

e1(-)
u– +

1

(v– +

u– +1 (v– +1 w–)
k–

5
1

w– )

X

– 0(u–,v–,w–)

(u– +

1

v– ) +

0
1

w–

1

0(x, y, z)

Y

u+ +1 (v+ +1 w+)

– 0(u+,v+,w+)

(u+ +1 v+) +1 w+

k# = k(u#, v#, w#),

k+

#6

e1(-)

X = x +2 (y +2 z),

(u+ +1 v+) +1 w+

(u+ +1 v+) +1 w+

e0(-)

Y = (x +2 y) +2 z.

Then we obtain k(x, y, z) by correcting the previous 3-cell 7 with the appropriate comparisons for
units, as follows:
u–+1(v–+1w–)

u–+1(v–+1w –)

. 1–1(X +1 k+)

X + 1 k+

e1(-)

u–+1(v–+1w –)
X

(u++1v+)+1w+

(u++1v+)+1w+

u–+1(v–+1w–)
7

k–

u–+1(v–+1w–)

e 1 k – +1 / 1 Y

(u–+1v–)+1w–

k– +1 Y

Y+1e1(-)
(u++1v+)+1w+

(u++1v+)+1w+

5.3. Lemma (Cospans and companions). The weak symmetric cubical categories !C osp(X) and
! C at have structural companions.
Proof. (a) We first address cubical cospans. The natural transformation f: x–
x+: " n
X can be
n
X. The functors p, 5, 6 defined as follows (on the objects)
viewed as a functor f: 2 ! "
(1) p: "

2,

p(– 1) = 0,

p(0) = p(1) = 1 ,

5: 2 ! "

2,

5(0, i) = 0,

5(1, i) = p(i),

6: 2 ! "

2,

6(0, i) = p(i),

6(1, i) = 1,

yield an (n+1)-cube and (n+1)-maps that satisfy our conditions:
(2) f* = f.(p ! " n): " n+1
X,
5f = f.(5 ! " n): 2 ! " n+1
X,

6f = f.(6 ! " n): 2 ! " n+1

X.

(b) A natural transformation f: x–
x+: "n
Cat between cubical profunctors can be transformed
into an (n+1)-cubical profunctor f*, following the procedure sketched at the end of 4.3 for the 0dimensional case (when f is a functor between small categories.
5.4. Globularisation of cubical cospans. Truncating the cubical structure !Cosp(X) in degree 1,
we get a weak 2-cubical category 2Cosp(X) = tr1(!C osp(X)), with one strict direction and one weak
direction. This coincides with the pseudo (or weak) double category defined and studied in [GP1,
GP2]. Its associated special cylindrical structure SCyl(2C osp(X)) is the ordinary bicategory of
cospans.

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29
Truncating !C osp(X) in degree 2 we get a weak symmetric 3-cubical category 3C osp(X) =
tr2 (! C osp(X)), with one strict direction and two weak directions. Then the procedure expounded in
5.2 gives a tricategory,
The same can be done with !Cat.

References
[ABS] F.A.A. Al-Agl - R. Brown - R. Steiner, Multiple categories: the equivalence of a globular and
a cubical approach, Adv. Math. 170 (2002), 71-118.
[BH1] R. Brown and P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), 233260.
[BH2] R. Brown - P.J. Higgins, Tensor products and homotopies for ! -groupoids and crossed
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