Theory and Applications of Categories,
Vol. 30, No. 28, 2015, pp. 9851000.
SKEW-MONOIDAL REFLECTION AND LIFTING THEOREMS
In memory of Brian Day
STEPHEN LACK AND ROSS STREET
This paper extends the Day Reection Theorem to skew monoidal categories. We also provide conditions under which a skew monoidal structure can be lifted
to the category of Eilenberg-Moore algebras for a comonad.
Abstract.
1. Introduction
In the rst part of the paper, we squeeze some more results out of Brian Day's PhD thesis
[2]. The question with which the thesis began was how to extend monoidal structures along
dense functors, all at the level of enriched categories. Brian separated the general problem
into two special cases.
The rst case concerned extending along a Yoneda embedding,
which led to promonoidal categories and Day convolution [3]. The second case involved
extending along a reection into a full subcategory: the Day Reection Theorem [4].
While the thesis was about monoidal categories, we can, without even modifying the
biggest diagrams, adapt the results to skew monoidal categories.
Elsewhere [5, 8] we
have discussed convolution. Here we will provide the skew version of the Day Reection
Theorem [4].
The beauty of this variant is further evidence that the direction choices
involved in the skew notion are important for organizing, and adding depth to, certain
mathematical phenomena.
In the second part of the present paper, the skew warpings of [5] are slightly generalized
to involve a skew action; they can in turn be seen as a special case of the skew warpings
of [6].
Under certain natural conditions these warpings can be lifted to the category
of Eilenberg-Moore coalgebras for a comonad.
monoidal structures.
In particular, this applies to lift skew
For idempotent comonads, we compare the result with our skew
reection theorem.
Both authors gratefully acknowledge the support of the Australian Research Council Discovery Grant
DP130101969; Lack acknowledges with equal gratitude the support of an Australian Research Council
Future Fellowship
Received by the editors 2015-03-10 and, in revised form, 2015-07-09.
Transmitted by R.J. Wood. Published on 2015-07-12.
2010 Mathematics Subject Classication: 18D10; 18A40, 18D15, 18D20.
Key words and phrases: skew monoidal category, reective subcategory, warping, comonad.
c Stephen Lack and Ross Street, 2015. Permission to copy for private use granted.
985
STEPHEN LACK AND ROSS STREET
986
2. Skew monoidal reection
Recall from [9, 5, 8] the notion of
involves a functor
(left) skew monoidal structure
⊗ : X × X −→ X ,
an object
I ∈ X,
on a category
X.
It
and natural families of (not
necessarily invertible) morphisms
αA,B,C : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C),
satisfying ve coherence conditions.
ρA : I → A ⊗ I,
It was shown in [1] that these ve conditions are
independent.
Recall, also from these references, that an
A
λA : I ⊗ A → A,
opmonoidal structure on a functor L : X →
consists of a natural family of morphisms
¯
ψX,Y : L(X ⊗ Y ) → LX ⊗LY
and a morphism
normal when ψ0
ψX,Y
ψ0 : LI → I¯ satisfying
three axioms. We say the opmonoidal functor is
is invertible. We say the opmonoidal functor is
strong when ψ0
and all
are invertible. However, in this paper, a limited amount of such strength, in which
ψ are invertible, will be important.
¯ ᾱ, λ̄, ρ̄) are skew monoidal
¯ I,
(X , ⊗, I, α, λ, ρ) and (A , ⊗,
only certain components of
Suppose
categories.
Suppose L a N : A → X is an adjunction with unit η : 1X ⇒ N L
and invertible counit ε : LN ⇒ 1A . Suppose X is skew monoidal. There exists a skew
monoidal structure on A for which L : X → A is normal opmonoidal with each ψX,N B
invertible if and only if, for all X ∈ X and B ∈ A , the morphism
2.1. Theorem.
L(ηX ⊗ 1N B ) : L(X ⊗ N B) → L(N LX ⊗ N B)
(2.1)
is invertible. In that case, the skew monoidal structure on A is unique up to isomorphism.
Proof. Suppose
A
opmonoidal with the
has a skew monoidal structure
ψX,N B
¯ ᾱ, λ̄, ρ̄)
¯ I,
(⊗,
for which
L
is normal
invertible. We have the commutative square
¯
LX ⊗LN
B
ψ −1
L(X ⊗ N B)
¯
LηX ⊗1
L(ηX ⊗1)
/
¯
LN LX ⊗LN
B
/ L(N LX
ψ −1
⊗ N B)
in which the vertical arrows are invertible. The top arrow is invertible with inverse
¯ .
εLX ⊗1
So the bottom arrow is invertible.
Conversely, suppose each
L(ηX ⊗ 1N B ) is invertible.
Wishing
L to become opmonoidal
with the limited strength, we are forced (up to isomorphism) to put
¯ = L(N A ⊗ N B)
A⊗B
and
I¯ = LI ,
SKEW-MONOIDAL REFLECTION AND LIFTING THEOREMS
and to dene the constraints
ᾱ, λ̄, ρ̄
by commutativity in the following diagrams.
L((N A ⊗ N B) ⊗ N C)
Lα
Lλ
L(ηI ⊗1)
/
LN A
L(η⊗1)
/
εA
L(1⊗η)
L(N LI ⊗ N A)
/ L(N L(N A ⊗ N B) ⊗ N C)
L(N A ⊗ (N A ⊗ N C))
L(I ⊗ N A)
987
ᾱ
/ L(N A ⊗ N L(N B
εA
LN A
Lρ
λ̄
L(N A ⊗ I)
A
⊗ N C))
L(1⊗ηI )
/
/A
ρ̄
L(N A ⊗ N LI)
The denitions make sense because the top arrows of the squares are invertible (while
the bottom arrows may not be).
Now we need to verify the ve axioms.
The proofs
all proceed by preceding the desired diagram of barred morphisms by suitable invertible
morphisms involving only
εA , LηX , ηN A ,
or
L(ηX ⊗ 1N B ),
then manipulating until one
can make use of the corresponding unbarred diagram.
The biggest diagram for this is the proof of the pentagon for
ᾱ.
Fortunately, the proof
in Brian Day's thesis [2] of the corresponding result for closed monoidal categories has the
necessary Diagram 4.1.3 on page 94 written without any inverse isomorphisms, so saves
us rewriting it here. (The notation is a little dierent with
ψ
in place of
N
and with some
of the simplications we also use below.)
It remains to verify the other four axioms. The simplest of these is
λ̄LI ρ̄LI =
=
=
=
=
=
=
=
λ̄LI ρ̄LI εLI LηI
λ̄LI L(1 ⊗ ηI )LρN LI LηI
λ̄LI L(1 ⊗ ηI )L(ηI ⊗ I)LρI
λ̄LI L(ηI ⊗ I)L(1 ⊗ ηI )LρI
εLI LλN LI L(1 ⊗ ηI )LρI
εLI LηI LλI LρI
1LI L(λI ρI )
1LI .
For the other three, to simplify the notation (but to perhaps complicate the reading),
we write as if
N
were an inclusion of a full subcategory, choose
L
so that the counit is an
STEPHEN LACK AND ROSS STREET
988
identity, and write
XY
for
X ⊗Y.
Then we have
λ̄B ⊗C
¯ ᾱLI,B,C L(η(LI)⊗B
¯ 1C )L((ηI 1B )1C ) =
=
=
=
=
=
λ̄B ⊗C
¯ L(1ηBC )LαLI,B,C L((ηI 1B )1C )
λ̄B ⊗C
¯ L(1LI ηBC )L(ηI 1BC )LαI,B,C
λ̄B ⊗C
¯ L(ηI 1BC )L(1I ηBC )LαI,B,C
LλBC LαI,B,C
L(λB 1C )
¯ C )L(η(LI)B 1C )L((ηI 1B )1C )
(λ̄B ⊗1
¯ C on right cancellation.
λ̄B ⊗C
¯ ᾱLI,B,C = λ̄B ⊗1
¯ λ̄C )ᾱA,LI,C (ρ̄A ⊗1
¯ C ) = 1A⊗C
of the axiom (1A ⊗
¯ ,
yielding the axiom
For the proof
we can look at Diagram
4.1.2 on page 93 of [2]. The required commutativities are all there once we reverse the
direction of the right unit constraint which Day calls
r
instead of
ρ.
For the nal axiom, we have
ᾱA,B,LI ρ̄A⊗B
=
¯
=
=
=
=
ᾱA,B,LI L(ηAB 1LI )L(1AB ηI )LρAB
L(1A ηBLI )LαA,B,LI L(1AB ηI )LρAB
L(1A ηBLI )L(1A (1B ηI ))LαA,B,I LρAB
L(1A ηBLI )L(1A (1B ηI ))L(1A ρB )
¯ B .
1A ⊗ρ̄
L is dened by ψ0 = 1 : LI → I¯ and ψX,Y =
L(ηX ⊗ ηY ) : L(X ⊗ Y ) → L(N LX ⊗ N LY ). The three axioms for opmonoidality are
easily checked and we have each ψX,N B = L(1N LX ⊗ ηN B )L(ηX ⊗ 1N B ) invertible.
The desired opmonoidal structure on
3. A reective lemma
In this section we state a standard result in a form required for later reference. For the
sake of completeness, we include a proof.
Assume we have an adjunction
L a N: A → X
with unit
η : 1X ⇒ N L and counit
the counit ε is invertible.
ε : LN ⇒ 1A .
Assume
3.1. Lemma.
For Z ∈ X , the following conditions are equivalent:
N
is fully faithful; that is, equivalently,
(i) there exists A ∈ A and Z ∼
= N A;
(ii) for all X ∈ X , the function X (ηX , 1) : X (N LX, Z) → X (X, Z) is surjective;
(iii) the morphism ηZ : Z → N LZ is a coretraction (split monomorphism);
(iv) the morphism ηZ : Z → N LZ is invertible;
(v) for all X ∈ X , the function X (ηX , 1) : X (N LX, Z) → X (X, Z) is invertible.
SKEW-MONOIDAL REFLECTION AND LIFTING THEOREMS
Proof.
989
(i) ⇒ (ii)
∼
=
X (X, Z)
1
/
X (X, N A)
X (ηX ,1)
X (X, Z) o
X (N LX, Z) o
∼
=
∼
=
/
A (LX, A)
N
X (N LX, N A)
(ii) ⇒ (iii) Take X = Z and obtain ν : N LZ → Z with X (ηZ , 1)ν = 1Z .
(iii) ⇒ (iv) If νηZ = 1 then (ηZ ν)ηZ = 1ηZ , so, by the universal property
have ηZ ν = 1.
(iv) ⇒ (v) The non-horizontal arrows in the commutative diagram
X (ηX ,1)
X (N LX, Z)
X (1,ηZ )
/
X (ηX ,1)
X (N LX, N LZ)
i
of
ηZ ,
we
X (X, Z)
/X
6
X (1,ηZ )
(X, N LZ)
∼
=
N
A (LX, LZ)
are all invertible, so the horizontal arrows are invertible too.
(v) ⇒ (i)
(v) ⇒ (ii)
ηZ .
Clearly
and the invertible
and we already have
(ii) ⇒ (iii) ⇒ (iv),
so take
A = LZ
4. Skew closed reection
The Reection Theorem [4] also deals with closed structure.
If, for objects
Y
isomorphism
and
Z
the functor
X (− ⊗ Y, Z)
is representable, say via a natural
X (X ⊗ Y, Z) ∼
= X (X, [Y, Z]),
[Y, Z] a left internal hom. Recall from Section 8 of [8] that
Z , so that − ⊗ Y has a right adjoint, then X becomes left skew closed.
we call the representing object
if this exists for all
Suppose L a N : A → X is an adjunction with unit η : 1X ⇒ N L and
invertible counit ε : LN ⇒ 1A . Suppose X is skew monoidal and left internal homs of
the form [N B, N C] exist for all B, C ∈ A . The morphisms (5.7) are invertible for all
X ∈ X and B ∈ A if and only if the morphisms
4.1. Theorem.
η[N B,N C] : [N B, N C] → N L[N B, N C]
(4.2)
are invertible for all B, C ∈ A . In that case, the skew monoidal structure abiding on A ,
as seen from Theorem 2.1, is left closed. Also, the functor N is strong left closed.
STEPHEN LACK AND ROSS STREET
990
Proof. Consider the following commutative diagram.
A (L(N LX ⊗ N B), C)
∼
=
A (L(η⊗1),1)
/
A (L(X ⊗ N B), C)
X (N LX ⊗ N B, N C)
∼
=
X (X ⊗ N B, N C)
X (N LX, [N B, N C])
∼
=
/
X (ηX ,1)
∼
=
X (X, [N B, N C])
Invertibility of the arrows (5.7) is equivalent to the invertibility of the top horizontal
arrows. This is equivalent to invertibility of the bottom horizontal arrows. By Lemma 3.1,
this is equivalent to invertibility of the arrows (4.2).
For the penultimate sentence of the Theorem, we now have the natural isomorphisms:
¯ C) ∼
A (A⊗B,
= X (N A ⊗ N B, N C)
∼
= X (N A, [N B, N C])
∼
= X (N A, N L[N B, N C])
∼
= A (A, L[N B, N C])
[B, C] = L[N B, N C]
∼
N [B, C] = N L[N B, N C] = [N B, N C].
yielding the left internal hom
Our notation for a right adjoint to
X ⊗−
for
A.
For the last sentence, we have
is
X (X ⊗ Y, Z) ∼
= X (Y, hX, Zi) .
The
right internal hom hX, Zi
may exist for only certain objects
Z.
In general, the
existence of right homs in a left skew monoidal category does not give a left or right
skew closed structure.
When they do exist, we can reinterpret a stronger form of the
invertibility condition (5.7) of Theorem 2.1.
Suppose L a N : A → X is an adjunction with unit η : 1X ⇒ N L and
invertible counit ε : LN ⇒ 1A . Suppose X is skew monoidal, and left internal homs of
the form [Y, N C] and right internal homs of the form hX, N Ci exist. The invertibility of
one of the following three natural transformations implies invertibility of the other two:
4.2. Theorem.
L(ηX ⊗ 1Y ) : L(X ⊗ Y ) → L(N LX ⊗ Y ) ;
(4.3)
η[Y,N C] : [Y, N C] → N L[Y, N C] ;
(4.4)
hηX , N Ci : hN LX, N Ci → hX, N Ci .
(4.5)
SKEW-MONOIDAL REFLECTION AND LIFTING THEOREMS
991
Proof. Consider the commutative diagram (4.6). Invertibility of any one of the horizontal
families in the diagram implies that of the other two. Invertibility of the arrows (4.3) is
equivalent to the invertibility of the top horizontal family. By Lemma 3.1, invertibility
of the middle horizontal family is equivalent to invertibility of the arrows (4.2). By the
Yoneda Lemma, invertibility of the bottom horizontal family is equivalent to invertibility
of the arrows (4.5).
A (L(N LX ⊗ Y ), C)
∼
=
A (L(η⊗1),1)
/
A (L(X ⊗ Y ), C)
X (N LX ⊗ Y, N C)
∼
=
∼
=
X (X ⊗ Y, N C)
X (ηX ,1)
X (N LX, [Y, N C])
/
X (Y, hN LX, N Ci)
X (1,hηX ,1i)
/
∼
=
(4.6)
X (X, [Y, N C])
∼
=
∼
=
X (Y, hX, N Ci)
5. An example
This is an example of the opposite (dual) of Theorem 2.1 which we enunciate explicitly
as Proposition 5.1 below. Instead of a reection we have a coreection. To keep using
left skew monoidal categories we also reverse the tensor product. For a monoidal functor
R: X → A ,
we denote the structural morphisms by
ϕ0 : I → RI
and
ϕX,Y : RX ⊗ RY → R(X ⊗ Y ) .
Suppose R ` N : A → X is an adjunction with counit ε : N R ⇒ 1X
and invertible unit η : 1A ⇒ RN . Suppose X is left skew monoidal. There exists a left
skew monoidal structure on A for which R : X → A is normal monoidal each ϕN A,Y
invertible if and only if, for all A ∈ A and Y ∈ X , the morphism
5.1. Proposition.
R(N A ⊗ εY ) : R(N A ⊗ N RY ) → R(N A ⊗ Y )
(5.7)
is invertible.
µ : U → O. For an object A of the slice category Set/U ,
u ∈ U . We have an adjunction
Consider an injective function
we write
Au
for the bre over
R ` N : Set/U → Set/O
(N A)i =
P
Au and (RX)u = Xµ(u) with invertible unit. The ith comP
ponent of the counit εX : N RX → X is the function
µ(u)=i Xµ(u) → Xi which is the
identity of Xi when i is in the image of µ.
dened by
µ(u)=i
STEPHEN LACK AND ROSS STREET
992
Let
C
be a category with
dening the tensor
X ⊗Y
obC = O.
Then
Set/O
becomes left skew monoidal on
by
(X ⊗ Y )j =
X
Xi × C (i, j) × Yj
i
and the (skew) unit
I by Ij = 1.
The associativity constraint
α : (X⊗Y )⊗Z → X⊗(Y ⊗Z)
is dened by the component functions
X
Xi × C (i, j) × Yj × C (j, k) × Zk →
i,j
X
Xi × C (i, k) × Yj × C (j, k) × Zk
i,j
induced by the functions
C (i, j) × C (j, k) → C (i, k) × C (j, k)
(a : i →Pj, b : j → k) to (b ◦ a : i → k, b : j → k). Dene λY : I ⊗ Y → Y to have
j -component i C (i, j) × Yj → Yj whose restriction to the ith injection
Pis the second
projection onto Yj . Dene ρX : X → X ⊗ I to have j -component Xj →
i Xi × C (i, j)
equal to the composite of Xj → Xj × C (j, j), x 7→ (x, 1j ), with the j th injection.
taking
This provides an example of Proposition 5.1. In fact, it satises the stronger condition
of the dual to Theorem 4.2. To see that
R(X ⊗ εY ) : R(X ⊗ N RY ) → R(X ⊗ Y )
is invertible, since
N
is fully faithful, we need to prove
G(X ⊗ εY ) : G(X ⊗ GY ) → G(X ⊗ Y )
G = N R is the idempotent comonad generated by the reection.
(GX) = Uj × Xj where Uj is the bre of µ over j ∈ O. Since µ is injective,
Uj ∼
= Uj ⊗ Uj , so
is invertible where
Notice that
G(X ⊗ GY )j = Uj × (X ⊗ GY )j
X
= Uj ×
Xi × C (i, j) × (GY )j
i
=
X
∼
=
X
Uj × Xi × C (i, j) × Uj × Yj
i
Uj × Xi × C (i, j) × Yj
i
= Uj × (X ⊗ Y )j
= G(X ⊗ Y )j .
SKEW-MONOIDAL REFLECTION AND LIFTING THEOREMS
The resultant left skew structure on
Set/U
993
has tensor product
¯ v = R(N A ⊗ N B)v
(A⊗B)
= (N A ⊗ N B)µ(v)
X
=
(N A)i × C (i, µ(v)) × (N B)µ(v)
i
∼
=
X
Au × C (µ(u), µ(v)) × Bv .
u
Of course we can see that this is merely the left skew structure on
Set/U
arising from
u ∈ U and whose morphisms u → v are
µ(u) → µ(v) in C ; that is, the category arising as the full image of the functor
the category whose objects are the elements
morphisms
µ: U → C .
As an easy exercise the reader might like to calculate the monoidal structure
¯
RX ⊗RY
→ R(X ⊗ Y )
on
R
and check that these components are not invertible in general while, of course, they
are for
X = N A.
6. Skew warpings riding a skew action
We slightly generalize the notion of skew warping dened in [5] to involve an action. This
is actually a special case of skew warping on a two-object skew bicategory in the sense of
[6].
Let
C
denote a left skew monoidal category. A
left skew action of C
on a category
A
is an opmonoidal functor
C −→ [A , A ] , X 7→ X ? −
(6.8)
where the skew monoidal (in fact strict monoidal) tensor product on the endofunctor
category
[A , A ]
is composition. The opmonoidal structure on (6.8) consists of natural
families
αX,Y,A : (X ⊗ Y ) ? A −→ X ? (Y ? A)
and
λA : I ? A −→ A
(6.9)
subject to the three axioms (6.10), (6.11), (6.12).
α?1
(X ⊗ (Y ⊗ Z)) ? A
/ (X
α
((X ⊗ Y ) ⊗ Z) ? A
α
/X
? ((Y ⊗ Z) ? A)
1?α
/
⊗ Y ) ? (Z ? A)
α
X ? (Y ? (Z ? A))
(6.10)
STEPHEN LACK AND ROSS STREET
994
/
α
(I ⊗ Y ) ? A
&
λ?1
Y ?A
x
(6.11)
λ
/X
α
(X ⊗ I) ? A
I ? (Y ? A)
O
? (I ? A)
ρ?1
X ?A
A category
A
A
/
1
equipped with a skew action of
C
T : A −→ C ;
(b) an object
K
of
X ?A
skew C -actegory.
on
A
consists of the following data:
A;
(c) a natural family of morphisms
v0 : T K −→ I ;
(d) a morphism
(6.12)
1?λ
is called a
skew left warping riding the skew action of C
(a) a functor
vA,B : T (T A ? B) −→ T A ⊗ T B
in
C;
and,
(e) a natural family of morphisms
kA : A −→ T A ? K ;
such that the following ve diagrams commute.
vA,B ⊗1
T (T A ? B) ⊗ T C
/
O
(T A ⊗ T B) ⊗ T C
vT A?B,C
T A ⊗ (T B ⊗ T C)
T (T (T A ? B) ? C)
T (vA,B ?1)
αT A,T B,T C
O
1⊗vB,C
T ((T A ⊗ T B) ? C)
(6.13)
T A ⊗ T (T B ? C)
4
*
T αT A,T B,C
vA,T B?C
T (T A ? (T B ? C))
vK,B
TK
7 ⊗ TB
&
I ⊗ TB
T (T K ? B)
T (v0 ?1B )
T (I ⊗ B)
v0 ⊗1T B
T λB
/
λT B
TB
(6.14)
SKEW-MONOIDAL REFLECTION AND LIFTING THEOREMS
vA,K
T (T A ? K)
/
O
TA ⊗ TK
T kA
TA
/
ρT A
vA,B ?1K
T (T A ?O B) ? K
/
1T A ?kB
v0 ?1K
TK O? K
TA ⊗ I
(T A ⊗ T B) ? K
/
1K
αT A,T B,K
(6.16)
T A ? (T B ? K)
/
I ?K
kK
K
(6.15)
1⊗v0
kT A?B
TA ? B
995
/
λK
(6.17)
K
6.1. Example. A skew warping on a skew monoidal category (in the sense of [5]) is just
the case where
A =C
with tensor as action.
Just as in Proposition 3.6 of [5], we obtain a skew monoidal structure from a skew
warping.
A skew left warping riding a left skew action of a left skew monoidal
category C on a category A determines left skew monoidal structure on A as follows:
6.2. Proposition.
¯ = T A ? B;
(a) tensor product functor A⊗B
(b) unit K ;
(c) associativity constraint
vA,B ?1C
T (T A ? B) ? C −→ (T A ⊗ T B) ? C
αT A,T B,C
−→
T A ? (T B ? C) ;
(d) left unit constraint
v ?1
λ
0
B
B
T K ? B −→
I ? B −→
B ;
(e) right unit constraint
k
A
A −→
TA ? K .
¯ K) −→ (C , ⊗, I).
There is an opmonoidal functor (T, v0 , vA,B ) : (A , ⊗,
6.3. Example. Skew warpings are more basic than skew monoidal structures in the
following sense. Just pretend, for the moment, that we do not know what a skew monoidal
(or even monoidal) category is, except that we would like endofunctor categories to be
examples.
For any category
A,
the endofunctor category
C = [A , A ]
acts on
A
by
evaluation; as a functor (6.8), the action is the identity. A left skew warping riding this
action could be taken as the denition of a left skew monoidal structure on
A.
STEPHEN LACK AND ROSS STREET
996
7. Comonads on skew actegories
C,
For a left skew monoidal category
left skew
C -actegories
let
CatC
denote the 2-category whose objects are
as dened in Section 6.
A morphism is a functor
F: A → B
equipped with a natural family of morphisms
γX,A : X ? F A −→ F (X ? A)
(7.18)
such that (7.19) and (7.20) commute.
γ
(X ⊗ Y ) ? F A
α
X ? (Y ? F A)
1?γ
/
/
X ? F (Y ? A)
I ? FA
γ
/
F ((X ⊗ Y ) ? A)
/
γ
(7.19)
Fα
F (X ? (Y ? A))
F (I ? A)
λ
&
(7.20)
Fλ
FA
strong
Such a morphism is called
when each γX,A is invertible. A 2-cell ξ : (F, γ)
C
in Cat is a natural transformation ξ : F ⇒ G such that (7.21) commutes.
X ? FA
1?ξA
γX,A
X ? GA
γX,A
/
/
⇒ (G, γ)
F (X ? A)
(7.21)
ξX?A
G(X ? A)
CatC .
As usual with actions, there is another way to view the 2-category
as the homcategory of a 1-object skew bicategory
ΣC
Regard
C
in the sense of Section 3 of [6]. A
left skew C -actegory is an oplax functor A : ΣC → Cat. A morphism (F, γ) : A → B
C
in Cat can be identied with a lax natural transformation between the oplax functors.
The 2-cells are the modications.
C
We are interested in comonads (A , G, γ, δ, ε) in the 2-category Cat . These are objects
C
of the 2-category Mnd∗ (Cat ) as dened in [7]. Alternatively, they are oplax functors
ΣC → Mnd∗ (Cat). For later reference, apart from the conditions for being a comonad on
A and the conditions (7.19) and (7.20), we require commutativity of (7.22).
γ
X ? GA
1?δ
X ? G2 A
γ
/ G(X
? GA)
/
Gγ
/
G(X ? A)
δ
γ
/
G(X ? A)
1?ε
G2 (X ? A)
The Eilenberg-Moore coalgebra construction
adjoint to the 2-functor
X ? GA
(A , G, δ, ε) 7→ A G
&
ε
(7.22)
X ?A
is the 2-functor right
Cat → Mnd∗ (Cat) taking each category to that category equipped
with its identity comonad.
SKEW-MONOIDAL REFLECTION AND LIFTING THEOREMS
997
For each comonad (A , G, γ, δ, ε) in the 2-category CatC , the EilenbergMoore coalgebra category A G becomes a left skew C -actegory with skew action
7.1. Proposition.
a
γX,A
X?a
X ? (A −→ GA) = (X ? A, X ? A −→ X ? GA −→ G(X ? A)) .
This provides the Eilenberg-Moore construction in the 2-category CatC (in the sense of
[7]).
ΣC → Mnd∗ (Cat)
Mnd∗ (Cat) → Cat.
Proof. Compose the oplax functor
with the Eilenberg-Moore 2-functor
Let
U: A G → A
denote the underlying functor
corresponding to
(A , G, γ, δ, ε)
(A, a) 7→ A.
Suppose (T, K, v, v0 , k) is a skew left warping riding the C -actegory
A . Suppose (A , G, γ, δ, ε) is a comonad in the 2-category CatC for which all morphisms
of the form γT A,K and γT A,T B?K are invertible. Then (T U, (GK, δK ), v, v00 , k0 ) is a skew
left warping riding the C -actegory A G of Proposition 7.1, where v00 = v0 ◦ T εK and
0
= γT−1A,K ◦ GkA ◦ a.
k(A,a)
7.2. Proposition.
0
: (A, a) → (T A ? GK, γT A,GK ◦ (1 ? δK )) is a Gk(A,a)
coalgebra morphism. This uses the rst diagram of (7.22), naturality of δ with respect to
kA , and the coassociativity of the coaction a : A → GA.
It remains to verify the ve axioms (6.13), (6.14), (6.15), (6.16), (6.17). Since only v is
Proof. First we need to see that
involved in (6.13), it follows from axiom (6.13) for the original skew warping. For (6.14),
we have the diagram
T (T εK ?1)
T (T GK ? B)
vGK,B
T GK ⊗ T B
which uses naturality of
T εK ⊗1
v
/
T (v0 ?1)
T (T K ? B)
/
/
T (I ⊗ B)
vK,B
TK ⊗ TB
v0 ⊗1
/I
⊗ TB
λT B
/
T λB
TB
and axiom (6.14) for the original skew warping.
The next
diagram proves (6.15).
TA
Ta
/
T GA
T GkA
/
T G(T A ? K)
T εT A?K
1
v
T
(T A ? K)
4
T εA
T kA
TA
ρT A
Precomposing the next diagram with
/
TA ⊗ I o
T γ −1
/
T (T A ? GK)
vA,GK
T (1?εK )
T A ⊗ T GK
vA,K
(
1⊗v0
1⊗T εK
TA ⊗ TK
1 ? b : T A ? B → T A ? GB
proves (6.16). Take note
STEPHEN LACK AND ROSS STREET
998
here of which components of
γ
are required to be invertible.
γ
T A ? GB
1?GkB
/
G(T A ? B)
GkT A?B
/
G(T (T A ? B) ? K)
1
t
γ
G(T (T A ? B) ? K) o
T A ? G(T B ? K)
G(1?kB )
γ
t
G(T A ? (T B ? K))
/
γ −1
T A ? G(T B ? K)
vA,B ?1
(T A ⊗ T B) ? GK
Gα
γ −1
T (T A ? B) ? GK
G(vT A,B ?1)
G((T A ⊗ T B) ? K) o
γ
1?γ −1
/
α
T A ? (T B ? GK)
Then
δK
GK
1
/
'
G2 K
GkGK
/ G(T GK
GεK
GK
GkK
/
? K)
G(T K ? K)
T GK ? GK
T εK ?1
γ −1
/
T K ? GK
v0 ?1
G(v0 ?1)
G(I ? K) o
#
/
G(T εK ?1)
1
γ −1
γ
I ? GK
GλK
GK
1
/
λGK
GK
yields (6.17), which completes the proof.
Under the hypotheses of Proposition 7.2, the functor U : A G → A
preserves the tensor products obtained from the skew warpings via Proposition 6.2 and
becomes opmonoidal when equipped with the unit constraint εI : GI → I .
7.3. Corollary.
Since C is an object of CatC with its own tensor product as skew
action, and since it supports the identity skew warping, for any comonad (C , G, γ, δ, ε) in
the 2-category CatC , Corollary 7.3 applies to give a skew monoidal structure on C G with
U : C G → C opmonoidal.
7.4. Corollary.
7.5. Remark. If the comonad of Corollary 7.4 is idempotent and
then
(G, γ) is strong in CatC
U: CG → C
is a coreection and the dual of Theorem 2.1 applies. The same skew
G
monoidal structure on C
is obtained as in Corollary 7.4. The point is that the diagram
(7.23) commutes by
G
applied to the right-hand diagram of (7.22) and a counit property
of the comonad. So Theorem 2.1 appears to be a stronger result than Corollary 7.4 in the
SKEW-MONOIDAL REFLECTION AND LIFTING THEOREMS
999
idempotent comonad case.
G(1⊗εY )
G(X ⊗ GY )
GγX,GY
/
G(X ⊗ Y )
4
GεX⊗Y
GG(X ⊗ Y )
1
/
(7.23)
δX⊗Y
GG(X ⊗ Y )
8. The example of Section 5 without injectivity
Let
C
be a category with object set
O
and morphism set
E,
and let
ξ: U → O
be a
Composition with ξ induces a comonadic functor
N = ξ! : Set/U → Set/O; write R = ξ ∗ for the right adjoint, given by pullback. The
∗
comonad G = N R = ξ! ξ is given by − ×O U .
The category structure on C induces a skew monoidal structure on Set/O , with tensor
product X ⊗ Y given by:
X
(X ⊗ Y )j =
Xi × C (i, j) × Yj
function (not necessarily injective).
i
and so
X ⊗−
X ×O E ×O −. The unit I is the terminal object 1 : O → O.
G and X ⊗ − involving products in Set/O, it is clear that we
isomorphisms γX,Y : X ⊗ GY ∼
= G(X ⊗ Y ), compatible with the comonad
is given by
From the formulas for
have natural
structure, in the sense that the diagrams (7.22) commute. Almost as easy is compatibility
with the associativity map and left unit constraint in the sense of diagrams (7.19) and
(7.20).
C with object-set O, giving rise to the skew monoidal category
Set/O, and the comonad G = ξ! ξ ∗ on Set/O as required by Corollary 7.4. This gives rise
∗
0
to a skew monoidal structure on Set/U , with unit ξ I ; in other words with unit I equal to
the terminal object 1 : U → U . It is clear from the construction that this tensor product
So we have a category
preserves colimits in each variable.
So from the general theory, it must correspond to
some category A with object-set U .
∗
Since ξ : Set/U → Set/O is opmonoidal, ξ is the object part of a functor
∗
Since ξ preserves the tensor, the functor F is fully faithful.
Thus
A
must in fact be obtained from
ξ: U → C
F : A → C.
via the factorization into a bijective-
on-objects functor followed by a fully faithful functor.
References
[1] Jim Andrianopoulos, Remarks on units of skew monoidal categories,
arXiv:1505.02048v1. 986
[2] Brian J. Day, Construction of Biclosed Categories (PhD Thesis, University of New
South Wales, 1970)
988
http://www.math.mq.edu.au/~street/DayPhD.pdf.
985, 987,
STEPHEN LACK AND ROSS STREET
1000
[3] Brian J. Day, On closed categories of functors,
Lecture Notes in Mathematics 137
(Springer-Verlag, 1970)138. 985
[4] Brian J. Day, A reection theorem for closed categories,
J. Pure Appl. Algebra 2
(1972) 111. 985, 989
[5] Stephen Lack and Ross Street, Skew monoidales, skew warpings and quantum categories,
Theory and Applications of Categories 26
(2012) 385402.
985, 986, 993,
995
[6] Stephen Lack and Ross Street, On monads and warpings,
géométrie diérentielle catégoriques 60 (2014) 244266.
[7] Ross Street, The formal theory of monads,
Cahiers de topologie et
985, 993, 996
J. Pure Appl. Algebra 2 (1972) 149168.
996, 997
[8] Ross Street, Skew-closed categories,
Journal of Pure and Applied Algebra 217 (June
2013) 973988. 985, 986, 989
[9] Kornel Szlachányi, Skew-monoidal categories and bialgebroids,
231 (2012) 16941730.
Advances in Math.
986
Department of Mathematics, Macquarie University NSW 2109, Australia
Email:
steve.lack@mq.edu.au, ross.street@mq.edu.au
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