A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS
OF GRAY-CATEGORIES
JOHN BOURKE AND NICK GURSKI
Abstract. It was argued by Crans that it is too much to ask that the category
of Gray-categories admit a well behaved monoidal biclosed structure. We make
this precise by establishing undesirable properties that any such monoidal biclosed
structure must have. In particular we show that there does not exist any
tensor product making the model category of Gray-categories into a monoidal
model category.
1. Introduction
The category 2-Cat of small 2-categories admits several monoidal biclosed structures.
One of these, usually called the Gray-tensor product ⊗p, has a number of
appealing features.
(1) Given 2-categories A and B the morphisms of the corresponding 2-category
[A, B] are pseudonatural transformations: the most important transformations
in 2-category theory.
(2) Each tricategory is equivalent to a Gray-category [11]: a category enriched in
(2-Cat, ⊗p).
(3) The Gray tensor product equips the model category 2-Cat with the structure
of a monoidal model category [15]; in particular, it equips the homotopy
category of 2-Cat with the structure of a monoidal closed category.
Gray-categories obtain importance by virtue of (2). Instead of working in a general
tricategory it suffices to work in a Gray-category – for an example of this see [7].
They are more manageable than general tricategories, and differ primarily from
strict 3-categories only in that the middle four interchange does not hold on the
nose, but rather up to coherent isomorphism.
Bearing the above in mind, it is natural to ask whether there exists a tensor product
of Gray-categories satisfying some good properties analogous to the above ones.
This topic was investigated by Crans, who in the introduction to [4] claimed that
it is too much to ask for a monoidal biclosed structure on Gray-Cat that captures
weak transformations, and asserts that this is due to the failure of the middle four
interchange.
The folklore argument against the existence of such a monoidal biclosed structure
Date: March 29, 2015.
2000 Mathematics Subject Classification. Primary: 18D05, 18D15, 18D35.
The first author acknowledges the support of the Grant agency of the Czech Republic, grant
number P201/12/G028. The second author acknowledges the support of the EPSRC in the form
of an Early Career Fellowship. Both authors thank the University of Sheffield for support via the
Mathematical Sciences Research Centre.
1
2 JOHN BOURKE AND NICK GURSKI
– discussed in [23] and likely originating with James Dolan – is fairly compelling
and goes as follows. (For simplicity we here confine ourselves to the symmetric
monoidal case.) Given Gray-categories A and B such a structure would involve
a Gray-category [A, B] whose objects should be strict Gray-functors1
and whose
1-cells η : F → G should be weak transformations. Such a transformation should
involve at least 1-cell components ηa : Fa → Ga and 2-cells
Fa Fb
Ga Gb
Fα //
ηa

ηb

Gα
//
ηα
xÐ zzzzzz
satisfying coherence equations likely involving 3-cells. Which equations should be
imposed?
If we view η : F → G ∈ [A, B] as a Gray-functor 2 → [A, B] it should correspond,
by closedness, to a Gray-functor η : A → [2, B] where η(a) = ηa : Fa → Ga whilst
η(α) should encode the above 2-cell ηα; the requirement that the Gray-functor
η : A → [2, B] preserve composition should then correspond to asking that the
condition
Fa Fb Fc
Ga Gb Gc
Fα // Fβ
//
ηa

ηb

ηc

Gα
//
Gβ
//
ηα
xÐ zzzzzz
ηβ
xÐ zzzzzz =
Fa Fc
Ga Gc
F(βα)
//
ηa

ηc

G(βα)
//
ηβα
xÐ zzzzzz
(1.1)
holds for η. Now given η : F → G and µ : G → H satisfying (1.1) we expect to
define the composite µ ◦ η : F → H ∈ [A, B] to have component (µ ◦ η)α as left
below
Fa Fb
Ga Gb
Ha Hb
Fα //
ηa

ηb

Gα
//
ηα
xÐ zzzzzz
µa

µb

Hα
//
µα
xÐ zzzzzz
Fa Fb Fc
Ga Gb Gc
Ha Hb Hc
Fα //
ηa

ηb

Gα
//
ηα
xÐ zzzzzz
µa

µb

Hα
//
µα
xÐ zzzzzz
Fβ
//
ηc

Gβ
//
µc

Hβ
//
µβ
{Ó 
ηβ
xÐ zzzzzz
.
Having done so, one asks whether µ ◦ β satisfies (1.1) and this amounts to the
assertion that the two ways of composing the four 2-cells above right – vertical
followed by horizontal and horizontal followed by vertical – coincide. But in a
general Gray-category they do not. The conclusion is that that we cannot define
a category [A, B] of Gray-functors and weak transformations.
The above argument, though convincing, is not quite precise. Our goal here is to
describe heavy and undesirable restrictions that any monoidal biclosed structure
on Gray-Cat must satisfy. This is the content of our main result, Theorem 4.2.
1In fact biclosedness forces the objects to be the strict Gray-functors – see Proposition 2.1 and
Remark 2.2.
A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES 3
This result immediately rules out the possibility of the sort of tensor product that
one initially hopes for: in which 2 ⊗ 2 is the non-commuting square
(0, 0) (1, 0)
(0, 1) (1, 1)
//
 
//
xÐ zzzzzz
8dzzzzzz
with some kind of equivalence connecting the two paths (0, 0) (1, 1). In particular
we use Theorem 4.2 to rule out the possibility of a monoidal biclosed structure
on Gray-Cat capturing weak transformations or yielding a monoidal model struc-
ture.
In establishing these results our line of argument is rather different to that outlined
above. In fact our techniques are based upon [10], in which it was shown
that Cat admits exactly two monoidal biclosed structures. The approach there
was to observe that biclosed structures on an algebraic category amount to “double
coalgebras” in the same category, and so to enumerate the former it suffices
to calculate the latter, of which there are often but few. On Cat this amounts
to the enumeration of certain kinds of double cocategories in Cat and we give a
careful account of this background material from [10] in Section 2. Section 3 recalls
the various known tensor products on 2-Cat. Our main result is Theorem 4.2 of
Section 4. Corollary 4.3 applies this result to describe a precise limitation on the
kinds of transformations such a biclosed structure can capture, and Corollary 4.4
applies it to rule out the existence of a monoidal model structure. We conclude
by discussing a related argument of James Dolan.
The authors would like to thank John Baez, James Dolan, Michael Shulman and
Ross Street for useful correspondence concerning a preliminary version of this ar-
ticle.
2. Two monoidal biclosed structures on Cat
It was shown in [10] that there exist precisely two monoidal biclosed structures
on Cat and an understanding of this result forms the starting point of our analysis
of the Gray-Cat situation. The treatment in [10] leaves certain minor details to the
reader. Since these details will be important in Section 4, we devote the present
section to giving a full account – but emphasise that nothing here is original.
Cat is cartesian closed and this accounts for one of the monoidal biclosed structures.
Given categories A and B the corresponding internal hom [A, B] is the
category of functors and natural transformations. This forms a subcategory of
[A, B]f , the category of functors and mere transformations: families of arrows of
which naturality is not required. [A, B]f is the internal hom of the so-called funny
tensor product 2
, which also has unit the terminal category 1. The funny tensor
2The terminology “funny tensor product” follows [21].
4 JOHN BOURKE AND NICK GURSKI
product A B is the pushout
obA × obB obA × B
A × obB A B
//

//
where obA and obB are the discrete categories with the same objects as A and
B respectively. Using this formula one sees that the funny tensor product is
symmetric monoidal and, furthermore, that the induced maps
A B → A × B
from the pushout equip the identity functor 1 : Cat → Cat with the structure of
an opmonoidal functor. 3
2.1. Cocontinuous bifunctors with unit. The terminal category 1 is the unit
for both monoidal biclosed structures. Let us recall why this must be so.
Firstly we call a category C equipped with a bifunctor ⊗ : C × C → C, object
I and natural isomorphisms lA : I ⊗ A → A and rA : A ⊗ I → A such that
lI = rI : I ⊗ I → I a bifunctor with unit. If C is cocomplete and ⊗ : C × C → C
cocontinuous in each variable we will often refer to it as a cocontinuous bifunctor
and, when additionally equipped with a unit in the above sense, as a cocontinuous
bifunctor with unit.
Proposition 2.1. Any cocontinuous bifunctor ⊗ with unit on Cat has unit the
terminal category 1.
Proof. For a category C, let End(1C) denote the endomorphism monoid of the identity
functor 1C. In such a setting the morphism of monoids End(1C) → End(I)
given by evaluation at I admits a section. This assigns to f : I → I the endomorphism
of 1C with component
A
r−1
A
// A ⊗ I
A⊗f
// A ⊗ I
rA
// A
at A. The identity functor 1 : Cat → Cat has a single endomorphism: for given
α ∈ End(1Cat) naturality at each functor 1 → A forces αA to be the identity on
objects, whilst naturality at functors 2 → A forces αA to be the identity on arrows.
All but the initial and terminal categories admit a non-trivial endomorphism – a
constant functor – so that the unit I must be either 0 or 1. Cocontinuity of A ⊗ −
forces A ⊗ 0 ∼= 0 and so leaves 1 as the only possible unit.
Remark 2.2. The above argument generalises easily to higher dimensions: in particular,
each monoidal biclosed structure on 2-Cat or Gray-Cat must have unit the
terminal object. The argument in either case extends the above one, additionally
using naturality in maps out of the free living 2-cell/3-cell as appropriate.
This restriction on the unit forces, in each case, the objects of the corresponding
internal hom [A, B] to be the strict 2-functors or Gray-functors respectively. In
3Monoidal categories with unit 1 are sometimes called semicartesian. In fact the funny and
cartesian tensor products are respectively the initial and terminal semicartesian monoidal structures
on Cat.
A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES 5
the case of 2-Cat this is illustrated by the natural bijections 2-Cat(1, [A, B]) ∼=
2-Cat(1 ⊗ A, B) ∼= 2-Cat(A, B). The Gray-Cat case is identical.
2.2. Double coalgebras versus cocontinuous bifunctors. Locally finitely presentable
categories are those of the form Mod(T) = Lex(T, Set) for T a small category
with finite limits. A standard reference is [1]. Examples include Cat, 2-Cat
and Gray-Cat.
Our interest is in monoidal biclosed structures on such categories. We note that a
monoidal structure ⊗ on Mod(T) is biclosed just when the tensor product − ⊗ −
is cocontinuous in each variable: this follows from the well known fact that each
cocontinuous functor Mod(T) → C to a cocomplete category C has a right adjoint.
In practice we will use cocontinuity of bifunctors rather than biclosedness.
Going beyond Mod(T) = Lex(T, Set) one can consider the category Mod(T, C) =
Lex(T, C) for any category C with finite limits, or if C has finite colimits the category
Comod(T, C) = Rex(Top, C) of T-comodels – if T is the finite limit theory for
categories one obtains the categories Cat(C) and Cocat(C) of internal categories
and cocategories in this way. The restricted Yoneda embedding y : Top → Mod(T)
preserves finite colimits and is in fact the universal T-comodel in a cocomplete category:
for cocomplete C restriction along y yields an equivalence of categories
Ccts(Mod(T), C) Comod(T, C)
where on the left hand side Ccts denotes the 2-category of cocomplete categories
and cocontinous functors. At a cocomplete trio let Ccts(A, B; C) denote the category
of bifunctors
A, B → C
cocontinuous in each variable. Evidently we have an isomorphism Ccts(A, B; C)
Ccts(A, Ccts(B, C)) and applying this together with the above equivalence (twice)
gives:
Proposition 2.3. For C cocomplete the canonical functor
Ccts(Mod(T), Mod(T); C) → Comod(T, Comod(T, C))
is an equivalence of categories.
On the right hand side are T-comodels internal to T-comodels, or double T-
comodels.
2.3. Double cocategories. We are interested in the special case of the above
where Mod(T) Cat. Then double T-comodels are double cocategories and in
this setting we will give a more detailed account. A cocategory A in C is a category
internal to Cop. Such consists of a diagram in C:
A1 A2 A3
d //
ioo
c
//
p
//
m //
q
// (2.1)
satisfying, to begin with, the reflexive co-graph identities id = 1 = ic. A3 is
the pushout A2 +A1 A2 with pc = qd, and m the composition map: this satisfies
md = pd and mc = qc. The map m is required to be co-associative, a fact we
make no use of, and co-unital: this last fact asserts that m(i, 1) = 1 = m(1, i)
where (i, 1), (1, i) : A3 A2 are the induced maps from the pushout characterised
6 JOHN BOURKE AND NICK GURSKI
by the equations (i, 1)p = di, (i, 1)q = 1, (1, i)p = 1 and (1, i)q = ci.
The universal cocategory S in Cat, corresponding to the restricted Yoneda embedding,
is given by:
1 2 3
d //
ioo
c
//
p
//
m //
q
// (2.2)
and we sometimes refer to it as the arrow cocategory. It is the restriction of the
standard cosimplicial object ∆ → Cat; in particular 2 and 3 are the free walking
arrow and composable pair:
2 =(0 1)
f
// 3 =(0 1 2)
g
// h // .
The various maps involved are order-preserving and characterised by the equations
d < c and p < m < q under the pointwise ordering.
A double cocategory is, of course, a cocategory internal to cocategories. Since
colimits of cocategories are pointwise, this amounts to a diagram
A1,1 A2,1 A3,1
A1,2 A2,2 A3,2
A1,3 A2,3 A3,3
d1
h
//
i1
h
oo
c1
h
//
p1
h
//
m1
h
//
q1
h
//
d2
h
//
i2
h
oo
c2
h
//
p2
h
//
m2
h
//
q2
h
//
d3
h
//
i3
h
oo
c3
h
//
p3
h
//
m3
h
//
q3
h
//
c1
v

i1
v
OO
d1
v

q1
v

m1
v

p1
v

c2
v

i2
v
OO
d2
v

q2
v

m2
v

p2
v

c3
v

i3
v
OO
d3
v

q3
v

m3
v

p3
v

(2.3)
in which all rows and columns A−,n and Am,− are cocategories, and each trio of
the form f−
h or f−
v is a morphism of cocategories. We sometimes refer to the commutativity
of the dotted square as the middle four interchange axiom in a double
cocategory.
Observe that if A and B are cocategories in C and ⊗ : C × C → D is cocontinuous
in each variable then the pointwise tensor product of A and B yields a double
cocategory A ⊗ B. This follows from the fact that each of Ai ⊗ − and − ⊗ Bj
preserves cocategories. In particular given a bifunctor ⊗ : Cat×Cat → C cocontinuous
in each variable the corresponding double cocategory in C of Proposition 2.3
A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES 7
is S ⊗ S as below:
1 ⊗ 1 2 ⊗ 1 3 ⊗ 1
1 ⊗ 2 2 ⊗ 2 3 ⊗ 2
1 ⊗ 3 2 ⊗ 3 3 ⊗ 3
d⊗1
//
i⊗1oo
c⊗1
//
p⊗1
//
m⊗1 //
q⊗1
//
d⊗2
//
i⊗2oo
c⊗2
//
p⊗2
//
m⊗2 //
q⊗2
//
d⊗3
//
i⊗3oo
c⊗3
//
p⊗3
//
m⊗3 //
q⊗3
//
1⊗c

1⊗i

1⊗d

1⊗q

1⊗m

1⊗p

2⊗c

2⊗i
OO
2⊗d

2⊗q

2⊗m

2⊗p

3⊗c

3⊗i
OO
3⊗d

3⊗q

3⊗m

3⊗p

.
(2.4)
The cartesian product of categories 2 × 2 is of course the free commutative square
(0, 0) (1, 0)
(0, 1) (1, 1)
(0,f)

(f,0)
//
(1,f)

(f,1)
//
=
(0, 0) (1, 0)
(0, 1) (1, 1)
(0,f)

(f,0)
//
(1,f)

(f,1)
//
=
whilst the funny tensor product 2 2 is the non-commutative square. These are
depicted above. The double cocategory structures S × S and S S are clear in
either case.
2.4. Only two biclosed structures. In the case of both the funny and cartesian
products, the associativity constraint (A ⊗ B) ⊗ C ∼= A ⊗ (B ⊗ C) is the only
possible natural isomorphism. (This is clear at the triples (1, 1, 1), (2, 1, 1), (1, 2, 1)
and (1, 1, 2), thus at all triples involving only 1’s and 2’s by naturality, and these
force the general case.) Similar remarks apply to the unit isomorphisms. To show
that these are the only monoidal biclosed structures, it therefore suffices to prove:
Proposition 2.4. The funny tensor product and cartesian product are the only
two cocontinuous bifunctors with unit on Cat.
Any such bifunctor has unit 1 by Proposition 2.1. Such bifunctors correspond to
double cocategories as in (2.4) in which the top and left cocategories S ⊗ 1 and
1 ⊗ S are isomorphic to S. Accordingly, the above result will follow upon proving:
Theorem 2.5. Up to isomorphism there exist just two double cocategories in Cat
whose top and left cocategories coincide as the arrow cocategory S in Cat. They
are S S and S × S.
The following result from [17] is a helpful starting point.
8 JOHN BOURKE AND NICK GURSKI
Lemma 2.6. Each cocategory in Set is a co-equivalence relation and, consequently,
the co-kernel pair of its equaliser.
We sketch the proof. To begin with, given a cocategory A in Set as in (2.1), one
should show that the maps d, c : A1 A2 are jointly epi. The pushout projections
p, q : A2 A3 certainly are, so given x ∈ A2 there exists y ∈ A2 such that py = mx
or qy = mx. So either x = m(i, 1)x = dix or x = m(1, i)x = cix; therefore d and
c are jointly epi and A a co-preorder. We omit details of the symmetry. Because
Setop
is Barr-exact, being monadic over Set by [19], each co-equivalence relation
in Set is the co-kernel pair of its equaliser.
Let us call O the cocategory in Set:
1 2 3
d //
ioo
c
//
p
//
m //
q
//
which is the co-kernel pair of ∅ → 1, so that d and c are the coproduct inclusions.
Because colimits of cocategories are pointwise each set X gives rise to a cocategory
X.O by taking componentwise copowers (so X.O2 = X.2 etc). By exactness a
morphism of cocategories f : A → O corresponds to a function Eq(dA, cA) → ∅
between equalisers, and a unique such exists just when Eq(dA, cA) = ∅. Since A1
is the co-kernel pair of its equaliser it follows that A2 = A1.2 and that f =!.O for
! : A1 → 1. Since each cocategory present in a double cocategory comes equipped
with a map to either its top or left cocategory it follows that:
Corollary 2.7. There exists only one double cocategory in Set whose left and top
cocategories coincide as the cocategory O; namely the cartesian product O × O.
Proof of Theorem 2.5. By Proposition 2.3 we are justified in denoting such a double
cocategory S ⊗ S as in (2.4).
(1) Since the functor (−)0 : Cat → Set preserves colimits it preserves (double)
cocategories, and takes the arrow cocategory S to O. By Corollary 2.7 we have
an isomorphism of double cocategories (S ⊗ S)0
∼= O × O and it is harmless
to assume that they are equal. In particular we then have (2 ⊗ 2)0 = 2 × 2.
Let us abbreviate d ⊗ 2 by d2
h and so on, as in (2.3). We then have a partial
diagram of 2 ⊗ 2:
(0, 0)
d2
vf
//
d2
hf

(1, 0)
c2
hf

(0, 1)
c2
vf
// (1, 1)
(2.5)
which gives a complete picture on objects. There may exist more arrows than
depicted, but all morphisms are of the form (a, b) → (c, d) where a ≤ c and
b ≤ d. If, for instance, there were an arrow α : d2
h1 = (0, 1) → (0, 0) = d2
h0 we
would obtain i2
hα : 1 → 0 ∈ 2 but no such arrow exists, and one rules out the
other possibilities in a similar fashion.
(2) The next problem is to show that each of d2
h, c2
h, d2
v and c2
v is fully faithful,
which amounts to the assertion that there exists a unique arrow on each of
the four sides of 2 ⊗ 2 (as in (2.5)) and no non-identity endomorphisms. The
A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES 9
four cases are entirely similar and we will consider only d2
h. Certainly d2
h is
faithful. Suppose that p2
h were full when restricted to the full image of d2
h.
Then α : d2
hx → d2
hy gives m2
hα : m2
hd2
hx → m2
hd2
hy. Since m2
hd2
h = p2
hd2
h we
obtain β : d2
hx → d2
hy such that p2
hβ = m2
hα. But then α = (i2
h, 1)m2
hα =
(i2
h, 1)p2
hβ = d2
hi2
hβ so that d2
h is full as well as faithful.
So it will suffice to show that p2
h is full on the objects (0, 0) and (0, 1) in the
image of d2
h. This involves computing the pushout 3 ⊗ 2 of categories and
we can reduce this to a simpler computation involving graphs. To this end
observe that 2 = 1 ⊗ 2 is free on the graph 2G = {0 → 1}. By adjointness d2
h
and c2
h correspond to maps d , c : 2G U(2 ⊗ 2) and we obtain a diagram:
F(2G) FU(2 ⊗ 2) F(P)
1 ⊗ 2 2 ⊗ 2 3 ⊗ 2
d2
h
//
c2
h
//
p2
h
//
q2
h
//
Fd //
Fc
//
Fp
//
Fq
//
1
 
k

where is an identity on objects and full functor, given by the counit of the
adjunction F U. P is the pushout of d and c in the category of graphs so
that the top row is a pushout of categories. Since bijective-on-objects and full
functors form the left class of a factorisation system on Cat they are closed
under pushout in Cat2
; thus the induced map k between pushouts is bijectiveon-objects
and full too, and it is harmless to suppose that it is the identity
on objects. By a two from three argument it follows that p2
h will be full on
{(0, 0), (0, 1)} so long as Fp is full on these same objects. In the category
of graphs the pushout P of the monos d and c is easily calculated, and it is
clear that p is a full embedding of graphs where restricted to (0, 0) and (0, 1).
P has the further property that the only morphisms in P having codomain
(0, 0) or (0, 1) also have domain amongst these two objects. Using the explicit
construction of morphisms in FP as paths in P, these two facts are enough
to ensure that Fp is full where restricted to (0, 0) and (0, 1), as required.
(3) Because each of d2
h, c2
h, d2
v and c2
v is fully faithful our knowledge of 2 ⊗ 2
is complete with the exception that we have not determined the hom-set
2 ⊗ 2((0, 0), (1, 1)). The two cases 2 ⊗ 2((0, 0), (1, 1)) = 1, 2 correspond to
the free commutative square and non-commutative square and these do give
double cocategories.
To rule out any other cases suppose that the complement
X = 2 ⊗ 2((0, 0), (1, 1)) \ {c2
hf ◦ d2
vf, c2
vf ◦ d2
hf}
is non-empty. Then the pushout 2 ⊗ 3 is generated by the graph below right
(0, 0)
d2
vf
//
X
##GGGGGGGG
d2
hf

(1, 0)
c2
hf

(0, 0)
p2
hd2
vf
//
Xl
##GGGGGGGG
p2
hd2
hf

(1, 0)
Xr
##GGGGGGGG
q2
hd2
vf
//
p2
hc2
hf

(2, 0)
q2
hc2
hf

(0, 1)
c2
vf
// (1, 1) (0, 1)
p2
hc2
vf
// (1, 1)
q2
hc2
hf
// (2, 1)
10 JOHN BOURKE AND NICK GURSKI
subject to the equations asserting the commutativity of the left and right
squares whenever the square does in fact commute in 2⊗2. We have two copies
Xl and Xr of X as depicted. At α ∈ X we must have m2
hα : (0, 0) → (2, 1)
and also (i2
h, 1)m2
hα = (1, i2
h)m2
hα = α. This implies that the path m2
hα must
involve both a morphism of Xl and a morphism of Xr, but there exists no
such path.
3. Monoidal biclosed structures on 2-Cat
There are at least five monoidal biclosed structures on 2-Cat. By Remark 2.2
any monoidal biclosed structure on 2-Cat has unit 1 and, correspondingly, the
objects of any possible hom 2-category must be the 2-functors. The 1-cells between
these are, in the five cases, 2-natural, lax natural, pseudonatural, oplax natural
and not-necessarily-natural transformations; in each case, modifications are the
2-cells.
Lax(A, B)
&&NNNNNNNNNNN
A ⊗l B
$$JJJJJJJJJJ
[A, B]
88qqqqqqqqqq
//
&&MMMMMMMMMM
Ps(A, B) // [A, B]f A B
::uuuuuuuuuu
//
$$IIIIIIIIII A ⊗p B // A × B
Oplax(A, B)
88pppppppppp
A ⊗o B .
::tttttttttt
The five homs are connected by evident inclusions and forgetful functors as in the
diagram above left. The arrows therein are the comparison maps of closed functor
structures on the identity 1 : 2-Cat → 2-Cat. By adjointness (of the A⊗− [A, −]
variety) there is a corresponding diagram of functors on the right, each of which
is now the component of an opmonoidal structure on 1 : 2-Cat → 2-Cat. Reading
left to right and top to bottom we have the funny tensor products, lax/ordinary
and oplax Gray tensor products [12], and the cartesian product.
The tensor product ⊗l is monoidal biclosed; indeed
2-Cat(A, Lax(B, C)) ∼= 2-Cat(A ⊗l B, C) ∼= 2-Cat(B, Oplax(A, C)) .
Likewise ⊗o is biclosed, but neither tensor product is symmetric; rather, they are
opposite: A ⊗l B ∼= B ⊗o A. The other cases are symmetric monoidal.
Viewing 2 as a 2-category in which each 2-cell is an identity, let us calculate 2⊗l 2
and 2 ⊗p 2. Both have underlying category the non-commuting square 2 2 and
are depicted, in turn, below.
(0, 0) (1, 0)
(0, 1) (1, 1)
(f,0)
//
(0,f)

(1,f)

(f,1)
//
xÐ zzzzzz
(0, 0) (1, 0)
(0, 1) (1, 1)
(f,0)
//
(0,f)

(1,f)

(f,1)
//
∼= (3.1)
We do not know whether the above five encompass all monoidal biclosed structures
on 2-Cat, but can say that the techniques of Section 2 are insufficient to determine
A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES 11
whether this is true. Those techniques worked because each cocontinuous bifunctor
with unit on Cat underlies a unique monoidal structure, whereas on 2-Cat there
exist such bifunctors not underlying any monoidal structure. For an example let
CB1(2-Cat) denote the category of cocontinuous bifunctors on 2-Cat with unit 1.
It is easily seen that the forgetful functor CB1(2-Cat) → [2-Cat2
, 2-Cat] creates
connected colimits. In particular the co-kernel pair of → ⊗l yields an object ⊗2
of CB1(2-Cat), with 2 ⊗2 2 given by
(0, 0) (1, 0)
(0, 1) (1, 1)
(f,0)
//
(0,f)

(1,f)

(f,1)
//
v~ tttttt
v~ tttttt
.
One can show that this tensor product is not associative up to natural isomorphism
as follows. Any natural isomorphism would have component at (2 ⊗2 2) ⊗2 2 →
2 ⊗2 2(⊗22) the associativity isomorphism (2 2) 2 → 2 (2 2) on underlying
categories, but direct calculation shows that this last isomorphism cannot be
extended to 2-cells.
4. The case of Gray-categories
The inclusion j : Cat → 2-Cat that views each category as a locally discrete
2-category allows us to view the cocategory S of (2.2) as a cocategory in 2-Cat,
so that we obtain the tensor product double cocategories S ⊗l S and S ⊗p S corresponding
to the lax and ordinary Gray tensor product. The 2-categories 2 ⊗l 2
and 2 ⊗p 2, depicted in (3.1), each consist of a non-commuting square and 2-cell –
invertible in the second case – within. Homming from S ⊗l S into a 2-category C
yields the double category 2-Cat(S ⊗l S, C) of quintets in C [9], whose squares are
lax squares in C as below left. The fact that the middle four interchange axiom
holds in the double cocategory S⊗l S corresponds, by Yoneda, to the fact that the
two ways of composing a diagram of 2-cells as below right
. .
. .
//
 //
xÐ zzzzzz
. . .
. . .
. . .
// //
// //
// //






xÐ zzzzzz
xÐ zzzzzz
xÐ zzzzzz
xÐ zzzzzz
(4.1)
in a 2-category always coincide.
In a Gray-category it is not necessarily the case that the two composite 2-cells
coincide and, correspondingly, neither S⊗lS nor S⊗pS remains a double cocategory
upon taking its image under the inclusion j : 2-Cat → Gray-Cat.4
This lack of
double cocategories in Gray-Cat ultimately limits the biclosed structures that can
exist on Gray-Cat – see Theorem 4.2 below.
4Specifically, the inclusion fails to preserve the pushouts 3 ⊗l 3 and 3 ⊗p 3.
12 JOHN BOURKE AND NICK GURSKI
4.1. Gray-categories and sesquicategories. Since the above problem concerns
the failure of the middle four interchange for 2-cells, it manifests itself at the simpler
two-dimensional level of sesquicategories [21]. Let us recall Gray-categories,
sesquicategories and their inter-relationship.
A Gray-category is a category enriched in (2-Cat, ⊗p) and we write Gray-Cat
for the category of Gray-categories. A sesquicategory C is a category enriched
in (Cat, ). In elementary terms such a C consists of a set of objects; for each
pair B, C of objects a hom-category C(B, C) of 1-cells and 2-cells, together with
whiskering functors C(f, C) : C(B, C) → C(A, C) and C(B, g) : C(B, C) → C(B, D)
for f : A → B and g : C → D such that (gα)f = g(αf) where defined. We write
Sesquicat for the category of sesquicategories.
The forgetful functor (−)1 : 2-Cat → Cat has both left and right adjoints. Furthermore
(−)1 : (2-Cat, ⊗p) → (Cat, ) is strong monoidal and is consequently
the left adjoint of a monoidal adjunction [14]. It follows that the lifted functor
(−)2 : Gray-Cat → Sesquicat, which forgets 3-cells, is itself a left adjoint. This was
observed in [16]. In particular (−)2 preserves colimits and so double cocategories.
We have a commutative triangle of truncation functors:
Gray-Cat Sesquicat
Cat
(−)2
//
(−)1
##GGGGGGGGGG
(−)1
{{wwwwwwwwww
j
ccGGGGGGGGGG j
;;wwwwwwwwww
in which both functors to Cat have fully faithful left adjoints. Denoted by j in
either case, these adjoints view a category as a Gray-category or sesquicategory
whose higher dimensional cells are identities.
In what follows the term pre-double cocategory will refer to a diagram as below
left:
A1,1 A2,1 A3,1
A1,2 A2,2 A3,2
A1,3 A2,3
d1
h
//
i1
h
oo
c1
h
//
p1
h
//
m1
h
//
q1
h
//
d2
h
//
i2
h
oo
c2
h
//
p2
h
//
m2
h
//
q2
h
//
d3
h
//
i3
h
oo
c3
h
//
c1
v

i1
v
OO
d1
v

q1
v

m1
v

p1
v

c2
v

i2
v
OO
d2
v

q2
v

m2
v

p2
v

c3
v

i3
v
OO
d3
v

A2,2 A3,2
A2,3 A3,3
q3
v

m3
v

p3
v

q2
v

m2
v

p2
v

p3
h
//
m3
h
//
q3
h
//
p2
h
//
m2
h
//
q2
h
//
in which all columns Am,− and rows A−,n of length three are cocategories, and
each trio of the form f−
h or f−
v is a morphism of cocategories.
Being a double cocategory is merely a property of a pre-double cocategory: A3,3
is uniquely determined as the pushout A2,3 +A1,3 A2,3 = A3,2 +A3,1 A3,2 and
A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES 13
f3
h = f2
h +f1
h
f2
h and f3
v = f2
v +f1
v
f2
v for f ∈ {p, m, q}. In fact, the only distinction
is that in a pre-double cocategory the dotted square expressing middle
four interchange need not commute.
The important example for us, I, is a pre-double cocategory in Sesquicat. Its image
under (−)1 : Sesquicat → Cat is S S whilst each of its component sesquicategories
In,m is locally indiscrete, i.e., there is a unique invertible 2-cell between
each parallel pair of 1-cells. This suffices for a full description, but for the reader’s
convenience we point out that I2,2 is the pseudo-commutative square 2 ⊗p 2 and
indeed that I is the image of the pre-double cocategory underlying S ⊗p S under
the inclusion 2-Cat → Sesquicat.
In order to see that I is not a double cocategory in Sesquicat it suffices, by Yoneda,
to exhibit a configuration of shape (4.1) of invertible 2-cells in a sesquicategory for
which the two possible composites do not coincide. This is straightforward.
Proposition 4.1. There exist only two double cocategories in Sesquicat whose
top and left cocategories coincide as the arrow cocategory S. They are the double
cocategories S S and S × S in Cat viewed as double cocategories in Sesquicat.
Proof. By Proposition 2.3 each double cocategory in Sesquicat is the tensor double
cocategory S⊗S associated to a cocontinuous bifunctor ⊗ : Cat×Cat → Sesquicat,
and accordingly we will use the tensor notation of (2.4).
The forgetful functor (−)1 : Sesquicat → Cat is cocontinuous and so preserves double
cocategories. It also preserves S and so, by Theorem 2.5, either (S⊗S)1 = S S
or (S ⊗ S)1 = S × S. In particular (2 ⊗ 2)1 = 2 2 or (2 ⊗ 2)1 = 2 × 2. Our main
task is to show that all 2-cells in 2 ⊗ 2 are identities and we begin by treating the
two cases together.
Let us firstly show that each of d2
h, c2
h, d2
v and c2
v is a full embedding of sesquicategories:
thus all 2-cells on the four sides of 2 ⊗ 2 are identities. The argument
here is a straightforward adaptation of the second part of the proof of Theorem 2.5
and we only outline it. As there, it suffices to show that d2
h is fully faithful (now
on 2-cells as well as 1-cells) and for this it is enough to show that p2
h is full when
restricted to the full image of d2
h. In the Cat-case of Theorem 2.5 this amounted
to studying a pushout of categories. This was simplified to computing a pushout
of graphs by using the adjunction F U between categories and graphs, that the
counit of this adjunction is bijective on objects and full, and that such functors
are closed under pushouts in Cat2
. In the sesquicategory case, we argue in a similar
fashion, but replace the category of graphs by the category Der of derivation
schemes [21]: such are simply categories A equipped with, for each parallel pair
f, g : A → B ∈ A, a set A(A, B)(f, g) of 2-cells. There is an evident forgetful
functor V : Sesquicat → Der which has a left adjoint G = G2G1. The first component
G1 adds formally whiskered 2-cells fαg for triples appropriately aligned
– as in f : A → B, α ∈ A(B, C)(h, k) and g : C → D – whilst G2 applies the
free category construction to each hom graph G1A(A, B). Each component of the
counit GV → 1 is an isomorphism on underlying categories and locally full. Such
sesquifunctors form the left class of a factorisation system on Sesquicat and are
therefore closed under pushout. By employing these facts as in Theorem 2.5 the
problem reduces to calculating a simple pushout in Der.
Now suppose (2 ⊗ 2)1 = 2 2. Consider the pre-double cocategory I discussed
14 JOHN BOURKE AND NICK GURSKI
above in which I2,2 = 2 ⊗p 2 is the free pseudo-commutative square and in which
both of I3,2 and I2,3 are locally indiscrete. Indiscreteness induces a unique morphism
F : S ⊗ S → I of pre-double cocategories whose image under (−)1 is the
identity. The fact that F is a morphism of pre-double cocategories implies that of
the six faces of the cube:
2 ⊗ 3 I2,3
2 ⊗ 2 3 ⊗ 3 I3,3
3 ⊗ 2 I3,2
F2,3
//
F3,2
//
2⊗m ??
m⊗2 ????
m⊗3
????
3⊗m
??
m3
h
????
m3
v
??
F3,3
//
2 ⊗ 3 I2,3
2 ⊗ 2 I2,2 I3,3
3 ⊗ 2 I3,2
F2,3
//
F3,2
//
2⊗m ??
m⊗2 ????
F2,2
//
m3
h
????
m3
v
??
m2
v ??
m2
h
???? (4.2)
the five left-most are commutative. In particular, the outer paths of the right
diagram commute as do its two leftmost squares. If in 2 ⊗ 2 there existed a 2-cell
as below (or in the opposite direction)
(0, 0) (1, 0)
(0, 1) (1, 1)
(f,0)
//
(0,f)

(1,f)

(f,1)
//
xÐ zzzzzz
then F2,2 : 2⊗2 → I2,2 would be epi, and it would follow that the rightmost square
of (4.2) commutes. Since I is not a double cocategory it does not commute and
accordingly there exists no such 2-cell.
Consequently there may only exist endo 2-cells of the two paths (0, 0) (1, 1) of
2 ⊗ 2. Now at the level of underlying categories the pushout is freely generated
by the graph below right
(0, 0)
(f,0)
//
(0,f)

(1, 0)
(1,f)

(0, 0)
(g,0)
//
(0,f)

(1, 0)
(h,0)
//
(1,f)

(2, 0)
(2,f)

(0, 1)
(f,1)
// (1, 1) (0, 1)
(g,1)
// (1, 1)
(h,1)
// (2, 1)
with the maps p2
h and q2
h including the left and right squares respectively. It follows
that the only 2-cells in 3 ⊗ 2 are of the form (h, 1)(p2
hθ) and (q2
hφ)(g, 0). Suppose
that an endo 2-cell θ of (1, f)(f, 0) exists in 2 ⊗ 2. It follows that m2
hθ is an
endomorphism of (2, f)(h, 0)(g, 0) and so of the form (q2
hφ)(g, 0). But then θ =
(1, i2
h)m2
hθ = (1, i2
h)(q2
hθ) ◦ (1, i2
h)(g, 0) which is an identity 2-cell since (1, i2
h)q2
hθ =
c2
hi2
hθ is one. Similarly each endomorphism of (f, 1)(0, f) is an identity and we
finally conclude that all 2-cells in 2 ⊗ 2 are identities.
Now suppose that (2⊗2)1 = 2×2. As there exists only a single arrow (0, 0) → (1, 1)
the hom-category A = 2 ⊗ 2((0, 0), (1, 1)) is a monoid, and we must prove that it
is the trivial monoid. By comparing the universal property of the pushout 3 ⊗ 2
with that of the coproduct of monoids, we easily obtain the following complete
description of 3⊗2. It has underlying category the product 3×2 and the pushout
A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES 15
inclusions p2
h, q2
h : 2 ⊗ 2 3 ⊗ 2 are full embeddings; thus 3 ⊗ 2((0, 0), (1, 1)) =
A = 3 ⊗ 2((1, 0), (2, 1)) as depicted left below.
(0, 0) (1, 0) (2, 0)
(0, 1) (1, 1) (2, 1)
(g,0)
//
(0,f)

(1,f)

(h,0)
//
(2,f)

(g,1)
//
(h,1)
//
A A
(0, 0) (1, 0)
(0, 1) (1, 1)
(0, 2) (1, 2)
(f,0)
//
(f,1)
//
(0,g)

(1,g)

(0,h)

(1,h)

(f,2)
//
A
A
(0, 0) (1, 0) (2, 0)
(0, 1) (1, 1) (2, 1)
(0, 2) (1, 2) (2, 2)
(g,0)
//
(0,g)

(1,g)

(h,0)
//
(2,g)

(0,h)

(1,h)

(2,h)

(g,2)
//
(h,2)
//
(g,1)
//
(h,1)
//
A A
A A
Moreover 3 ⊗ 2((0, 0), (2, 1)) is the coproduct of monoids
3 ⊗ 2((0, 0), (1, 1)) + 3 ⊗ 2((1, 0), (2, 1)) = A + A
with inclusions given by whiskering. The isomorphic 2 ⊗ 3 is depicted centre
above, also with hom monoid 2 ⊗ 3((0, 0), (1, 2)) the coproduct A + A. Likewise
we calculate that 3 ⊗ 2 and 2 ⊗ 3 embed fully into the pushout 3 ⊗ 3 which is
depicted above right, and that the monoid 3 ⊗ 3((0, 0)(2, 2)) is the coproduct 4.A
with the coproduct inclusions given by whiskering as before.
As, for instance, discussed in Section 9.6 of [3], elements of a coproduct of monoids
Σi∈IBi admit a unique normal form: as words b1 . . . bn where each bi belongs to
some Bj − {e} and no adjacent pair (bi, bi+1) belong to the same Bj. In order for
this formulation to make sense the monoids Bj must have disjoint underlying sets,
or else must be replaced by disjoint isomorphs. In our setting we have natural
choices for such isomorphs given by whiskering: for instance, in the case of 3 ⊗
2((0, 0), (2, 1)) = A + A these are the sets {a(g, 0) : a ∈ A} and {(h, 1)a : a ∈ A}.
We will use uniqueness of normal forms to show that A is the trivial monoid. Let
a ∈ A be a non-trivial element and consider the normal form decomposition m2
ha =
b1 . . . bn which alternates between elements of the sets (h, 1){A} and {A}(g, 0). The
maps (i2
h, 1) and (1, i2
h) evaluate b1 . . . bn to the product Πj=2i(aj) and Πj=2i+1(aj)
respectively. (Here aj is the A-component of bj, and the empty product is identified
with the unit of A.) Since (i2
h, 1)m2
h = 1 = (1, i2
h)m2
h it follows that the normal
form of m2
ha must have length at least 2, as must m2
va by an identical argument.
Consider m3
vm2
ha. Now m3
v = m2
v +m1
v
m2
v maps words (h, 1)a and a(g, 0) of length
1 to (h, 2)(m2
va) and (m2
va)(g, 0) respectively, whose normal forms are alternating
words in (h, 2)A(0, g) and (h, h)A, and in (2, h)A(g, 0) and A(g, g) respectively. In
particular each m3
vbi is an alternating word of length ≥ 2 in one pair, and m3
vbi+1
in the other pair. Therefore the normal form of m3
vm2
ha is simply obtained by
juxtaposing the normal forms of the components (m3
vb1, . . . , m3
vbn). In particular
the first two elements of this normal form must be either an alternating word in
(h, 2)A(0, g) and (h, h)A, or in (2, h)A(g, 0) and A(g, g): a vertically alternating
word. By contrast, the first two elements of the normal form of m3
hm2
va alternate
horizontally. By uniqueness of normal forms we conclude that A must be the
trivial monoid.
Having shown that 2⊗2 is merely a category in the two possible cases, we observe
that since the full inclusion j : Cat → Sesquicat is closed under colimits and ⊗
16 JOHN BOURKE AND NICK GURSKI
cocontinuous in each variable, it follows that the pushouts 3 ⊗ 2, 2 ⊗ 3 and 3 ⊗ 3
must be categories too.
Theorem 4.2. Let ⊗ : Gray-Cat2
→ Gray-Cat be a cocontinuous bifunctor with
unit. Then the underlying bifunctor
Cat2 j2
// Gray-Cat2 ⊗
// Gray-Cat
(−)1
// Cat
coincides with either the cartesian or funny tensor product. Moreover, at categories
A and B, the tensor product A ⊗ B has only identity 2-cells.
Proof. Since the forgetful functor (−)2 : Gray-Cat → Sesquicat is cocontinuous
the composite bifunctor ⊗ :
Cat2 j2
// Gray-Cat2 ⊗
// Gray-Cat
(−)2
// Sesquicat
is a cocontinuous bifunctor. The full theorem amounts to the assertion that ⊗ is
isomorphic to one of j◦×, j◦ : Cat2
Cat → Sesquicat which, by Proposition 2.3,
is equally to show that the double cocategory S ⊗ S is isomorphic to S × S or
S S. By Remark 2.2 the unit for ⊗ is 1 and it follows that S ⊗ S has left and
top cocategories S ⊗ 1 and 1 ⊗ S isomorphic to S. By Proposition 4.1 S ⊗ S must
be isomorphic to one of S S and S × S.
4.2. No weak transformations. The informal argument of the introduction was
against the existence of a monoidal biclosed structure on Gray-Cat capturing weak
transformations. Here we make a precise statement. In Proposition 2.1 we saw
that any monoidal biclosed structure on Gray-Cat has unit 1. Therefore the (left or
right) internal hom [A, B] must have Gray-functors as objects (as per Remark 2.2).
We take it as given that a weak transformation η : F → G of Gray-functors ought
to involve components ηa : Fa → Ga and 2-cells
Fa Fb
Ga Gb
Fα //
ηa

ηb

Gα
//
ηα
xÐ zzzzzz
(or in the opposite direction) to begin with. Even if we restrict our attention to
the simple case that A is merely a category and B a 2-category one expects that
there should exist transformations whose components ηα are not identities. This
is, at least, the case for the pseudonatural transformations that arise in 2-category
theory. The following result shows that this is not possible.
Corollary 4.3. Let ⊗ be a biclosed bifunctor with unit on Gray-Cat, and let [A, B]
denote either internal hom. Suppose that A is a category and B a 2-category.
Then the underlying category of [A, B] is isomorphic to the category [A, B1]f of
functors and unnatural transformations, or to the category of functors and natural
transformations [A, B1].
Proof. To begin, we observe that although the inclusion j : 2-Cat → Gray-Cat
does not have a right adjoint it does have a left adjoint Π2. This sends a Graycategory
A to the 2-category Π2(A) with the same underlying category as A and
A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES 17
whose 2-cells are equivalence classes of those in A: here, a parallel pair of 2-cells
are identified if connected by a zig-zag of 3-cells. We will use the evident fact that
if all 2-cells in A are identities then Π2(A) is isomorphic to the underlying category
A1 of A. We will also use the series of adjunctions
Gray-Cat
Π2
-- 2-Cat
j
nn
(−)1
,, Cat
j
mm
in which the left adjoints point rightwards, and in which the composite left adjoint
is (−)1 : Gray-Cat → Cat.
Now let A be a category and B a 2-category and [A, B] the hom Gray-category
characterised by the adjunction − ⊗ A [A, −]. We have isomorphisms:
Cat(C, ([A, B])1) ∼= Gray-Cat(C, [A, B]) ∼= Gray-Cat(C ⊗ A, B) ∼=
2-Cat(Π2(C ⊗ A), B) ∼= 2-Cat((C ⊗ A)1, B)
in which the action of the inclusions j are omitted. The first three isomorphisms
use the aforementioned adjunctions. By Theorem 4.2 all 2-cells in C ⊗ A are
identities and this gives the fourth isomorphism. Furthermore Theorem 4.2 tells
us that (C ⊗ A)1 is isomorphic to the funny tensor product of categories C A or
to the cartesian product C × A. In the first case we obtain further isomorphisms
2-Cat((C ⊗ A)1, B) ∼= 2-Cat(C A, B) ∼= Cat(C A, B1) ∼= Cat(C, [A, B1]f ) .
Applying the Yoneda lemma to the composite isomorphism Cat(C, ([A, B])1) ∼=
Cat(C, [A, B1]f ) we obtain ([A, B])1
∼= [A, B1]f or, in the second case, an isomorphism
[A, B]1
∼= [A, B1]. Finally, we note that an essentially identical argument
works in the case that [A, B] is the Gray-category corresponding to the adjunction
A ⊗ − [A, −].
4.3. No monoidal model structure. The cartesian product of categories and
Gray tensor product of 2-categories are generally considered to be homotopically
well behaved tensor products. In the former case we have that the cartesian product
of equivalences of categories is again an equivalence; in the latter we have
the analogous fact [15] but also further evidence, such as the fact that every tricategory
is equivalent to a Gray-category [11]. Still further evidence comes from
the theory of Quillen model categories [18]: both tensor products form part of
monoidal model structures [13]. Recall that a monoidal model category consists
of a category C equipped with both the structure of a symmetric monoidal closed
category and a model category; furthermore, these two structures are required to
interact appropriately. There is a unit condition and a condition called the pushout
product axiom, a special case of which asserts that for each cofibrant object A the
adjunction
A ⊗ − [A, −]
is a Quillen adjunction. When the unit is cofibrant, as it often is, this special case
of the pushout product axiom already implies that the homotopy category ho(C)
is itself symmetric monoidal closed, and that the projection C → ho(C) is strong
monoidal.
18 JOHN BOURKE AND NICK GURSKI
In the case of Cat the model structure in question has weak equivalences the equivalences
of categories, whilst the fibrations and trivial fibrations are the isofibrations
and surjective equivalences respectively. In the case of 2-Cat the weak equivalences
are the biequivalences, whilst the fibrations are slightly more complicated to describe
– see [15]. However each object is fibrant, and the trivial fibrations are
those 2-functors which are both surjective on objects and locally trivial fibrations
in Cat.
The model structure on 2-Cat is determined by that on Cat in the following way.
Given a monoidal model category V one can define a V-category A to be fibrant if
each A(a, b) is fibrant in V, and a V-functor F : A → B to be a trivial fibration if it
is surjective on objects and locally a trivial fibration in V. Now a model structure
is determined by its trivial fibrations and fibrant objects. Consequently, when the
above two classes do in fact determine a model structure on V-Cat, the authors of
[2] referred to it as the canonical model structure and described conditions under
which it exists. In particular the model structure on 2-Cat is canonical.
Furthermore it was shown in [16] that Gray-Cat admits the canonical model structure,
as lifted from 2-Cat, and a natural question to ask is whether this forms part
of a monoidal model structure. In 1.8(v) of [2] the authors indicated that it was
unknown whether such a tensor product exists. The following result shows that
this is too much to ask.
Corollary 4.4. There exists no cocontinuous bifunctor ⊗ : Gray-Cat2
→ Gray-Cat
with unit such that for each cofibrant A either of the functors A ⊗ − or − ⊗ A is
left Quillen. In particular there exists no monoidal model structure on Gray-Cat.
Proof. By Corollary 9.4 of [16] a Gray-category is cofibrant just when its underlying
sesquicategory is free on a computad. All we need are two immediate consequences
of this: namely, that 2 is cofibrant and that each cofibrant Gray-category has
underlying category free on a graph.
Now suppose such a bifunctor ⊗ does exist. Since 2 is cofibrant one of 2 ⊗ −
and − ⊗ 2 must be left Quillen. Therefore 2 ⊗ 2 is cofibrant. In particular the
underlying category of 2 ⊗ 2 must be free on a graph. Now (2 ⊗ 2)1 = 2 2 or
2 × 2 by Theorem 4.2 and only the first of these is free on a graph. Consequently
(2 ⊗ 2)1 = 2 2.
Let I denote the free isomorphism 0 ∼= 1 and consider the map j : 2 → I sending
0 → 1 to 0 ∼= 1. By Theorem 4.2 j ⊗ j : 2 ⊗ 2 → I ⊗ I coincides with the funny
tensor product j j : 2 2 → I I on underlying categories, and both 2-categories
have only identity 2-cells. In particular j ⊗ j sends the non-commuting square
(0, 0) (1, 0)
(0, 1) (1, 1)
(0,f)

(f,0)
//
(1,f)

(0,f)
//
to a still non-commuting square of isomorphisms.
As described in Section 3 of [16] there is a universal adjoint biequivalence E in
Gray-Cat. It is a cofibrant Gray-category with two objects and both maps 1 E
are trivial cofibrations. In particular E has underlying category free on the graph
A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES 19
(f : 0 1 : u) so that we obtain a factorisation of j:
2
j1
−→ E
j2
−→ I ,
and hence of j ⊗ j:
2 ⊗ 2
l
−→ E ⊗ E
k
−→ I ⊗ I ,
where we have written j1 ⊗ j1 = l, j2 ⊗ j2 = k. The two distinct paths r, s :
(0, 0) (1, 1) of 2 ⊗ 2 give rise to parallel 1-cells l(r), l(s) : l(0, 0) l(1, 1). A
2-cell α : l(r) ⇒ l(s) would yield a 2-cell k(α) : j ⊗ j(r) ⇒ j ⊗ j(s) but no such
2-cell can exist by the above. In particular there exists no 2-cell l(r) ⇒ l(s) in E.
However since 1 → E is a trivial cofibration in Gray-Cat we must have that
1 ∼= 1 ⊗ 1 → 1 ⊗ E → E ⊗ E is a trivial cofibration. By 2 from 3 the map
E ⊗ E → 1 must be a weak equivalence and indeed, since all Gray-categories are
fibrant, a trivial fibration. But this implies that there exists a 2-cell between any
parallel pair of 1-cells in E⊗E and we have just shown that this is not the case.
4.4. Dolan’s objection to Street’s files. In a 1996 email [8] to Ross Street,
James Dolan described an objection to Street’s then conjectural notion of semistrict
n-category – called an n-file [20]. Dolan’s objection, brought to our attention during
the editorial process, is closely related to the arguments of the present paper
and for that reason we discuss it here.
The category n-Cat of strict n-categories is cartesian closed. Taking the union
of the sequence (n+1)-Cat = (n-Cat, ×)-Cat gives the cartesian closed category
V1 = (ω-Cat, ×).5
ω-Cat supports several other monoidal biclosed structures,
including a higher dimensional version of the (lax) Gray tensor product. The following
construction of this tensor product follows [22] and we refer to that paper
for further details and references. The free ω-categories {O(In) : n ∈ N} on the
parity n-cubes In form a dense full subcategory Q of ω-Cat. The subcategory is
monoidal, with tensor product O(In)⊗O(Im) = O(In+m) induced from the tensor
product of cubes. Day’s technique [5, 6] for extending monoidal structures along
a dense functor can, with work, be employed to establish the monoidal biclosed
structure V2 = (ω-Cat, ⊗): here called the Gray-tensor product of ω-categories.
Gray-categories, identifiable as 3-dimensional V2-categories with invertible coherence
constraints, were called 3-files in [20].
Corresponding to the fact that the cartesian product is the terminal semicartesian
tensor product, so V1-categories, just ω-categories, can naturally be viewed
as V2-categories. Accordingly we have a composite functor
Q // ω-Cat V1-Cat // V2-Cat
and the composite is fully faithful. It was conjectured in [20] that this functor
satisfied Day’s conditions whereby the monoidal structure on Q could be extended
to a biclosed one ⊗2 on V2-Cat; then one would take V3 = (V2-Cat, ⊗2), define
4-files as certain 4-dimensional categories enriched in V3, and iterate to higher
dimensions.
5As in [20, 22] an object of ω-Cat is an n-category for some n. We note that ω-Cat is often
used to refer to ω-categories with non-trivial cells in each finite dimension.
20 JOHN BOURKE AND NICK GURSKI
Dolan’s objection6
, expressed in the terms of the present paper, was as follows.
The co-graph O(I0) O(I1) = d, c : 1 2 underlies the standard cocategory S
in Cat, equally a cocategory in ω-Cat or in V2-Cat. Accordingly a biclosed tensor
product ⊗2 would give a double cocategory S ⊗2 S with left and top cocategories
S ⊗2 1 and 1 ⊗2 S isomorphic to S. If ⊗2 were obtained by Day’s technique of
extension then the composite inclusion Q → V2-Cat would be strong monoidal;
hence (S ⊗2 S)1,1 = O(I1) ⊗2 O(I1) ∼= O(I2). But O(I2) is just
(0, 0) (1, 0)
(0, 1) (1, 1) .
//
 
//
xÐ zzzzzz
Dolan observed that there is only one possible double cocategory structure on
O(I2) (with S as its left and top cocategories) and that this does not form a double
cocategory: as in the Gray-Cat setting of Section 4, the middle four interchange
axiom fails to hold. Consequently there cannot exist a biclosed tensor product ⊗2
obtained by extension along Q → V2-Cat.
The key observation above is that the lax square double cocategory in 2-Cat (or
ω-Cat) does not remain one on passing to a world like Gray-Cat (or V2-Cat) in
which the middle 4-interchange doesn’t hold. This is the same observation at the
heart of our key result Proposition 4.1 – whose additional value is that it examines
not just the lax square but all possible double cocategories.
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Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,
Brno 60000, Czech Republic
E-mail address: bourkej@math.muni.cz
School of Mathematics and Statistics, University of Sheffield, Sheffield, United
Kingdom
E-mail address: nick.gurski@sheffield.ac.uk