Theory and Applications of Categories, Vol. 20, No. 6, 2008, pp. 74–84.
COHERENCE FOR PSEUDODISTRIBUTIVE LAWS REVISITED
F. MARMOLEJO AND R. J. WOOD
Abstract. In this paper we show that eight coherence conditions suffice for the definition of a pseudodistributive law between pseudomonads.
1. Introduction
The definition of a pseudodistributive law of one pseudomonad over another is given in
[Marmolejo, 1999]. We find there the four familiar diagrams given in the original definiton
of a distributive law of [Beck, 1969], but commutativity is replaced by invertible 2-cells.
These are required to satisfy coherence conditions; precisely the issue that [Marmolejo,
1999] addressed. There we find nine coherence conditions and a justification of why they
should suffice. In the thesis [Tanaka, 2005] it is suggested that the nine coherence axioms
given in that paper are incomplete, in the sense that one of the coherence axioms is
missing, a concern echoed in [Tanaka, Power, 2006], now in the form one axiom may be
missing. This kind of criticism casts doubt on the whole integrity of [Marmolejo, 1999]
without pointing out where the mistake might be. We find this unacceptable. It is our
contention that [Marmolejo, 1999] is fundamentally correct, if a little conservative in its
efforts to provide a complete set of axioms. For here we show that in fact eight of the
nine axioms of [Marmolejo, 1999] suffice.
We show here that the contentious condition (H-2) of [Tanaka, 2005] and, by duality,
(coh 8) of [Marmolejo, 1999] are redundant. Our technique involves showing that given
pseudomonads D on A and U on B, a lifting of a 2-functor F : A → B to the corresponding
2-categories of pseudoalgebras is classified by a strong transformation r : U F → F D
together with two invertible modifications subject to two coherence conditions. Readers
familiar with [Street, 1972] will at once recognize (F, r, · · ·) as a pseudo version of the
notion of morphism of monads. When we apply this to the particular case A = B and
D = U, the corresponding classifying transformation r : U F → F U is what was defined
in [Tanaka, 2005] as a pseudo-distributive law of U over F — except that the condition
(H-2) in that paper turns out to be redundant.
Incidentally, while it is true that [Marmolejo, 1999] does not give the definition of
The first author acknowledges financial support from the DGAPA of UNAM, which made possible
his sabbatical stay at Dalhousie University, and also the hospitality of this latter institution. The second
author thanks NSERC Canada for financial support.
Received by the editors 2008-01-10 and, in revised form, 2008-03-13.
Transmitted by Ross Street. Published on 2008-03-13.
2000 Mathematics Subject Classification: 18C15,18D05.
Key words and phrases: pseudomonads, pseudodistributive laws, coherence.
c F. Marmolejo and R. J. Wood, 2008. Permission to copy for private use granted.
74
COHERENCE FOR PSEUDODISTRIBUTIVE LAWS REVISITED
75
pseudoalgebra for a pseudomonad, such a definition does appear in [Marmolejo, 1997].
Perhaps this reference should have been more prominent. We also point out [Marmolejo,
2004], where we find a study of the algebras for the composite pseudomonad resulting
from a distributive law.
For simplicity, we work in the context of 2-categories, 2-functors, strong transformations and modifications, but the results are true in the general context of bicategories,
homomorphisms of bicategories, etc. Also for simplicity, we refrain in this paper from
producing higher dimensional structures as in [Marmolejo, 1999] or [Tanaka, 2005], but
we do feel that the paper would be incomplete without a proof of the equivalence between
liftings and the transformations that classify them, so we include this in the paper.
2. From transitions to liftings to algebras
Let A and B be 2-categories, U = (U, u, n, βU , ηU , µU ) a pseudomonad on A, and D =
(D, d, m, βD , ηD , µD ) a pseudomonad on B. Thus
/ U2 o
U EE
U
y
EE
yy
EE βU  [c ??????ηU
y
EE 
???? yyyy
EE{ 
E
n
yy 1
1
EE
yy
EE
y
y
" |y
U
Uu
uU
/ U2
Un
U3
nU
n
} / U,
µU
U2
n
with βU , ηU , and µU invertible modifications satisfying the coherence conditions found in,
say, [Marmolejo, 1997] and similary for D. Let F : A → B be a 2-functor.
2.1. Definition. A transition from U to D along F : A → B is a strong transformation
r : DF → F U together with invertible modifications
/ DF
F KK
KK
KK ω1 
KK  r
K
F u KK{ 
dF
K%
/ DF U
Dr
D2 F
 F n

{ 
/ FU
ω2
mF
F U,
/ F U2
rU
DF
r
(1)
that satisfy the following coherence conditions:
/ D2 F
KK Dω1zzzzz
KK zzz
KK x
Dr
DF u KKK
K% DF KK
/ D2 F
@@
@@ ηD F −1 @@
@@ } mF
@
1 @@
@@
@@
DF@
DdF
DF
Dr
/ DF U
r
rU
2
{{{ F U
{{{{{
{
{
Fn
{
y {{{
/ FU
ω2
r
=
DdF
DF U
F U MM
r −1
MMM uttttttt
MMMv~ t
rU
F U u MMM&
−1
2
F ηU
F U
1
~ '
Fn
FU
(2)
76
F. MARMOLEJO AND R. J. WOOD
D3 F:
D2 r
/ D2 F U
::
::DrU
:
::
::
::
::
2
mDF
:: Dω2 DF U::
:
::
::DF n
DmF :
:
/
2
DF U =
F
Dr
D2 F< µD F zD
z
z
<<
z
rU
<< y zzzzzz
mF
<<
F U2
<
mF <<
z
ω
2
z
<<
z
zzzz F n
< y zzz
/ FU
DF
D3 F
D2 r /
D2 F ULL
mF U
LLDrU
LLL
L%
DF U 2LL
mDF
LLDF n

rU 2 LLL

L%


−1

ω2 U 
{ 
3 rn yyyyy DF U



F
U
y
/
LLLx yy
D2 F MMDr DF U OO { 
LLL
OOO F nU
MMM
rU
O
MMM
F U n LL% rU O'
MMM
F U 2 LFLµyUyyyyyyy F U 2
M
mF MMMM
Lx
MMM ω2yyyyy LLLL
y
y
L Fn
M& x y
Fn % / FU
DF
mr
r
−1
(3)
r
In the case A = B a transition from U to U is the same thing as a distributive law of
U over F in the sense of Definition 5.1 in [Tanaka, 2005] without the coherence condition
(H-2). We will show shortly that this condition follows from the other two.
A transition induces a lifting of F to algebras, Fb, whose definition is given by the
following proposition. (See [Marmolejo, 1997] for the definition of algebras.)
2.2. Proposition. Let (r, ω1 , ω2 ) be a transition from U to D along F : A → B. We
define Fb : AU → BD as follows. For an object


uA /
Ua /
A EE
U A U 2A
UA


EE a1 EE 
a 
a
E
a
2
, nA


1 EEE

E" } 
/A
A
UA a
in AU , we define Fb(A, a, a1 , a2 ) as

dF A /
A
 F AEEE ω DF
EE 1 
EE 
rA
E

F uA EE
E

" 
F UA ,

 1 F a1zzzzzz

x zz
Fa


& FA
2
DrA/

DF A 

rU A ra−1  rA 

 
{ 
/
mF A
F U 2A F U a F U A 


Fn

ω2 A F
a



F a2 




{ 
/
/
DF A r F U A F a F A.
D FA
DF U A
DF a /
COHERENCE FOR PSEUDODISTRIBUTIVE LAWS REVISITED
77
For (f, f1 ) : (A, a, a1 , a2 ) → (B, b, b1 , b2 ) in AU , we define Fb(f, f1 ) as


DF f
/
DF A
DF B 




rf−1  rB 
rA




 

{ 
/ FUB  ,
F f, F U A


F Uf




F
a
F
b

F
f

1








{ 
/
FA
FB
Ff
For ξ : (f, f1 ) → (g, g1 ) in AU , define Fbξ = F ξ. Then Fb : AU → B D is a 2-functor such
that the diagram
Fb
AU
A
/ BD
(4)
/B
F
commutes, where the vertical arrows are the usual forgetful functors.
Proof. We prove that Fb is well defined on objects and leave the rest to the reader. Thus,
we must show that Fb(A, a, a1 , a2 ) is a D-algebra. Paste ηD F A−1 on the left of the second
coordinate of the above pasting. Use (2). Since (A, a, a1 , a2 ) is a U-algebra, we can replace
the pasting of F ηU A−1 and F a2 by F a ◦ F U a1 . Then replace the pasting of ruA −1 , ra−1
and F U a1 by rA ◦ DF a1 . This gives us one of the equations.
For the other, paste D of the second coordinate of the above pasting with the second
coordinate of the above pasting, and with µD F A. Use (3). Using that (A, a, a1 , a2 ) is a
−1
U-algebra, replace the pasting of DF a2 , ra−1 , rnA
, F a2 and F µU A by the pasting of ra−1 ,
−1
−1
rU−1a , F a2 , F n−1
a and F a2 . Conclude the calculation replacing the pasting of Dra , rU a ,
−1
−1
F na and ω2 U A by the pasting of ω2 A, mF a and ra−1 .
We show now the missing condition.
2.3.
Theorem.
dDF
DF
DF3
dDF
/ D2 F
33
33
33 βD F 33 33 ~ 1 33
33
33
33
Dr
/ DF U
r
F U KK
rU
mF
DF
FU
2
ω2 Fn
} / FU
r
=
/ D2 F
dr zzz Dr
zzzz
z
y zz / DF U
KKdF U
KK ω1 S 
KK  rU
KK{ 
F uU KK%
2
F
U
F βU 1
Fn
$
FU
(5)
78
F. MARMOLEJO AND R. J. WOOD
Proof. Lemma 9.1 of [Marmolejo, 1997], applied to Fb(U A, nA, βU A, µU A) for every possible A gives us
zzz DrU
zzzz
y zzz / DF U 2
drU
rU
DF U
:
dDF U
/ D2 F U
::
::
:: βD F U ::
:: ~ :
1 ::
::
::
::
: DrU
/ DF U 2
rU 2
DF U
F U3
F nU
} / F U2
ω2 U
rU
F U2
 rU




{ 
/ F U2 =
−1
rn
mF U
/ DF U
DF n
F Un
dF U 2
dF n zzz DF n
Fn
zzzz
y zzz / DF U
F U KK
KKdF U
KK ω1 U 
KK  rU
KK{ 
F uU KK%
2
F
U
F βU 1
 F n
 { 
/ FU
F µU
Fn
/ D2 F U
dDF U
DF U
Fn
$
F U.
Precompose both sides with DF u. On top of both sides paste first with dDF u , then with
Dru , and then with DF ηU . On the left and down of both sides paste first with ru−1 and
then with F ηU−1 .
On the left replace the pasting of Dru , dDF u , ω2 U , βD F U and ru−1 by the pasting of
rU−1u , F n−1
u , ω2 and βD F . Observe that the pasting of all the 2-cells left with the exception
of βU F and ω2 is an identity.
On the right, replace the pasting of Dru , dDF u , drU and ru−1 by the pasting of dr and
dF U u . Now the pasting of DF ηU , dF U u , dF n and F ηU−1 cancels out.
3. Transitions and liftings are essentially the same
3.1. Definition. Transitions (r, ω1 , ω2 ) and (s, π1 , π2 ) from U to D along F are said
to be coherently isomorphic if there is an invertible α : r → s such that
Dr
/
F KK
 DF
KK


KK { 
KK s ks α
ω1 =
K
F u KKK
% dF
π1
2
D F
ω2 =
r
F U,
Dα
rU
+
3 DF U
Ds
r
α−1
s
*
sU
mF
DF
αU
4 FU
2
 F n

{ 
+ 3 F U.
π2
3.2. Proposition. If α : (r, ω1 , ω2 ) → (s, π1 , π2 ) is a coherent isomorphism between
transitions from U to D along F , and G, H : AU → B D are the corresponding induced
liftings, then there is a 2-isomorphism ψ : G → H such that
G
A
U
ψ
H
*
4 BD
/B
COHERENCE FOR PSEUDODISTRIBUTIVE LAWS REVISITED
79
is the identity, where the rightmost arrow is the usual forgetful functor.
Proof. For any (a1 , a2 ) = (A, a, a1 , a2 ) ∈ AU , define ψ(a1 , a2 ) = (IdF A , F a ◦ αA).
We now produce a transition from a lifting. Assume G : AU → B D such that
G
AU
A
F
/ BD
/B
commutes. For every A ∈ A we have (βU A, µU A) = (U A, nA, βU A, µU A) ∈ AU . Thus we
have the D-algebra G(βU A, µU A) = (F U A, G(βU A, µU A)0 , G(βU A, µU A)1 , G(βU A, µU A)2 ):


DG(βU A,µU A)0
dDU A /
/
2
DF U A
DF U A
D F UA

 F UA


G(βU A,µU A)1 
G(βU A,µU A)0 , mF U A
G(βU A,µU A)2  G(β A,µ A)0  .

U
U





{ 


1
,
F UA
DF U A G(β A,µ A) / F U A
U
U
0
U
For f : A → B in A, we have (U f, n−1
f ) : (βU A, µU A) → (βU B, µU B) in A . Applying
G we obtain


DF U f
/
DF U A
DF U B




−1

G(U
f,n
)

F U f, G(βU A,µU A)0
 G(βU B,µU B)0  : G(βU A, µU A) → G(βU B, µU B)
f






{ 


/
F UA
F
U
B
F Uf
in B D . Given A ∈ A, we define
rA := ( DF A
/ DF U AG(βU A,µU A)0/ F U A, )
DF uA
and, for f : A → B a 1-cell in A, we define
DF A
rf :=
/ DF U A
DF uA
G(βU A,µU A)0
/ F UA
−1

G(U f,n−1

f )

 F U f


DF U f






{ 
{ 
/ DF U B
/ F UB
DF uf
DF f
DF B
DF uB
G(βU B,µU B)0
For any ϕ : f → g : A → B in A,
−1
Dϕ : (Df, Dn−1
f ) → (Dg, Dng ) : (βU A, µU A) → (βU B, µU B)
is a 2-cell in AU . Applying G to Dϕ it follows that F U ϕ is a 2-cell in B D . It is now easy
to see that with the given definitions:
3.3.
Lemma. r : DF → F U is a strong transformation.
80
F. MARMOLEJO AND R. J. WOOD
For A ∈ A, define
/ DF A
dF A
FA
ω1 A :=
DF uA
~ / DF U A
dF uA
F uA
F UA
dF U A
G(βU A,µU A)0
~
- F UA
G(βU A,µU A)1
1
Observe that (nA, µU A) : (βU U A, µU U A) → (βU A, µU A) in AU . Apply G to (nA, µU A)
and define ω2 A as the pasting
D2 F A
D2 F uA
/ D2 F U A
DG(βU A,µU A)0
mF U A
mF A





{ 
/ DF U A
m−1
F uA
DF A
DF uA
/ DF U A DF uU A / DF U 2 A G(βU U A,µU U A)0 / F U 2 A
NNN
NNN DF βU A 

NNN
 DF nA
1 NNNN{ 
N& 
−1
DF U ARR G(nA,µU A)  F nA
RRR

vvv
RRR
{ 
G(βU A,µU A)2vvvvvv
R
R
RR
vv
G(βU A,µU A)0 RRRR
vvvvv
R)
v~ vvvv
/ F U A.
G(βU A,µU A)0
3.4. Proposition. If G : AU → BD is a lifting of F : A → B to algebras and r, ω1 , and
ω2 are defined as above, then (r, ω1 , ω2 ) is a transition from U to D along F .
3.5. Theorem. Let (r, ω1 , ω2 ) be a transition from U to D along F and write G : AU →
B D for the corresponding lifting. If (s, π1 , π2 ) is the transition induced by G, then (r, ω1 , ω2 )
and (s, π1 , π2 ) are coherently isomorphic. Let G : AU → B D be a lifting to algebras of
F : A → B and write (r, ω1 , ω2 ) for the transition from U to D along F induced by G. If
H is the lifting induced by (r, ω1 , ω2 ), then there is a 2-natural isomorphism ψ : G → H
such that U D ◦ ψ is the identity.
Proof. In one direction, define α : r → s as the pasting
DF
r
ωu DF u
DF U
/ FU
rU
F Uu
F ηU
/ F U2
1


{ 
/ F U.
Fn
In the other, start with (A, a, a1 , a2 ) ∈ AU . Now (a, a2 ) : (βU A, µU A) → (a1 , a2 ) is a 1-cell
in AU . We define ψ(a1 , a2 ) as


G(βU A,µU A)0
/
DF@ U=A
FUA
==


==
==


DF a
=
DF
uA
F
a
=
=
IdF A ,
.
== G(a,a2 )−1 ==


==
==
DF a1


} / DF A
/ FA
DF A
1
G(a ,a )
1
2 0
COHERENCE FOR PSEUDODISTRIBUTIVE LAWS REVISITED
81
4. Op-transitions
Dually, we have op-transitions:
4.1. Definition. An op-transition from U to D along F is a strong transformation
r : F U → DF together with invertible modifications
/ FU
F KK
KK
KK ω1 
KK  r
K
dF KK{ 
K% DF,
Fu
/ DF U
rU
F U2
/ D2 F
Dr
 mF

{ 
/ DF
ω2
Fn
FU
r
(6)
that satisfy the following coherence conditions:
/ F U2
KK ω1 Uzzzzz
KK zzz
KK x
rU
dF U KKK
K% F U KK
/ F U2
@@
@@ F βU @@
@@ } Fn
@
1 @@
@@
@@
F U@
F uU
FU
F U :3
rU 2
::
::
::
::
::
:
F nU ::
rU
r
Dr
2
{{{D F
{{{{{
{
{
mF
{
y {{{
/ DF
ω2
DF U
DF MM
d
MMM tttrttttt
MMMv~
Dr
MM
dDF M&
βD F D 2 F
=
r
/ DF U 2
::
::DrU
:
2
D F :U
:: mF U
::
:
/
2
DF U =
FU
rU
F U 2< F µ−1
z
U
z
zz
<<
Dr
<< y zzzzzz
Fn
<<
2
D F
<
F n <<
ω2 zzz
<<
zzzz mF
< y zzz
/ DF
FU
F Un
/ DF U
ω2 U
r
We obtain the dual of Theorem 2.3
F U3
F uU
~ 1
rU 2 /
mF
DF
DF U 2LL
DF n
F Un
'
LLDrU
LLL
L%
D2 F ULL
LLmF U

D2 r LLL


L%

Dω2  3
mr yy
{ 

y

y
y DF U

F U 2 MMrU / DF U OO {  D F LLLLx yyy
OOODmF
LLL
MMM
Dr
OO'
MMM
mDF L% MMM Dr
2 µD F −1 yy 2
MM
D F LL yyyyyy D F
F n MMM
MMM ω2 yyyy LLLx L
LL mF
MM& x yyyy
mF % / DF
FU
rn
(7)
r
(8)
82
4.2.
F. MARMOLEJO AND R. J. WOOD
Proposition.
F Uu /
F U2
33
33 −1
33F ηU 33 F n
33 ~ 1 33
33
33
33
F U3
rU
FU
/ DF U
−1
ru
zzz rU
r
zzzz
y zzz / DF U
DF KK
Dr
KKDF u
KK Dω1 
KK  Dr
KK{ 
DdF KK%
ηD F −1 D 2 F
1
=
D2 F
/ F U2
F Uu
FU
ω2 mF
} / DF
(9)
mF
r
$
DF
5. Distributive laws
We are now in a position to explain why there are eight coherence conditions for a pseudodistributive law.
5.1. Proposition. A distributive law of U over D consists of a transition (r : U D →
DU, ω1 , ω3 ) from U to U along D, together with an op-transition (r, ω2 , ω4 ) from D to D
along U that satisfy the following coherence conditions:
=U
{{
u {{{
{
{{
{{
1 CC
CC
CC
C
d CCC
!
D
Ud /
DD ω2 yU D
DD yyyyy
DD x yy
r
dU DDD
D" d−1
DU
u
<
zzzz
zz
zz Du
zz
U< 2 DEE
n
DU 2
dn } U
=
dU
(10)
Du
y
EEU r
yy
EE
yy U ω2
y
E"
y
/
2
U DU
U NNN U dU
NNN ω2 U 
NNN 
rU
N 
dU 2 NN{ NN
& U 2d
= U DD
DD
{{
DDU d
DD
{
{
DD
{
{
"
{
u
d
1 CC
U
D
z<
CC
z
z
CC
uD zzz r
C
d CCC
zz ω1 !
zz ~ / DU,
D
u {{{
Dn
/ DU
U2
U 2d
/ U 2D
OOOU r
OOO
O'
U DU
n
nD
=
} / UD
rU
nd
U
Ud
DU
OOO Dn
O
ω2 r OOO'
~ 2 DU,
dU
ω3
2
(11)
COHERENCE FOR PSEUDODISTRIBUTIVE LAWS REVISITED
y
EErD
yy
EE
yy ω1 D
y
E"
y
/
2
DU D
D NNN DuD
NNN Dω1 
NNN 
Dr
N { 
D2 u NNNN
& uD2
D2
U< D2EE
m
} D
Du
/ U D2
OOOrD
OOO
O'
DU D
m
Um
} / UD
=
Dr
um
D2 U
m−1
u uD2
83
D
mU
D U
OOO mU
O
ω1 r OOO'
~ 2 DU,
uD
/ DU
(12)
ω4
2
Du
and
U 2 D3 2
U 2 D2
U rD
/ U DU D
rU D
nD2
DU 2 DOO
U rD/
U DU3 D
rU D/
2
DU 3D
33
33
33
33
33DU r
33
2
nD
rr 33
33
U Dr 33
3
3
33 2
DrU /
U m
3
/
2
2
33
U D3
U D 3U rDU DU DU
D2 U3 2
33
3
33
33
3
33U mU
33mU 2
33 −1 33
33
33
ω4 U 33 nm 33 U ω4
3
3
33 / U DU
/ DU 2
33 U 2 D
Um 3
Ur
rU
33
33 nD
ω3 zzzz Dn
33
zzz
=
y zzz
/ DU
UD
(13)
r
ODU
OOOr
O'
DnD
DU DUOO
ODrU
OOO
/
2
O'
DU DOO
U D OO rD
OOO
OOO
D2 U 2 OO 2
OODr
OOO
O
OmU
OOO Dω3 
OOO
OOO

OOO
OOO
'


Dn
OOO{ 
OOO
DU 2
O
O
O'
U m OOOO


m
n 
OOO
ω4 D 2 U


OOO

O
Dn
OOO{
OOO ~ OO
O'
mU O'
/ DU
UD
ω3 D r
Proof. The references (coh1), . . . , (coh9) are to the paper [Marmolejo, 1999]. The
coherence conditions for the transition of U to U along D correspond to the coherence
conditions (coh2) and (coh4). The coherence conditions for the op-transition from D to
D along U correspond to the coherence conditions (coh7) and (coh9). The coherence
condition (coh8) follows from these last two according to Proposition 4.2. (10), (11), (12)
and (13) are (coh1), (coh3), (coh5) and (coh6) rewritten.
References
[Beck, 1969] J. Beck, Distributive Laws, in Seminar on Triples and Categorical Homology Theory, Lecture Notes
in Mathematics 80, 119-140.
84
F. MARMOLEJO AND R. J. WOOD
[Marmolejo, 1999] F. Marmolejo, Distributive Laws for Pseudomonads, Theory and Applications of Categories
5 (1999), No. 5, 91-147
[Marmolejo, 2004] F. Marmolejo, Distributive Laws for Pseudomonads II, Journal of Pure and Applied Algebra
194, No. 1-2, 2004, 169-182.
[Marmolejo, 1997] F. Marmolejo, Doctrines Whose Structure Forms a Fully Faithful Adjoint String, Theory
and Applications of Categories 3 (1997), No. 2, 24-44.
[Street, 1972] R. Street, The Formal Theory of Monads, Journal of Pure and Applied Algebra 2, 1972, 149-168.
[Tanaka, 2005] Miki Tanaka, Pseudo-Distributive Laws and a Unified Framework for Variable Binding, Ph.D.
Thesis, University of Edinburgh, 2005.
[Tanaka, Power, 2006] Miki Tanaka and John Power, Pseudo-Distributive Laws and Axiomatics for Variable
Binding, Higher-Order and Symbolic Computation, Vol. 19, No. 2-3, Springer Netherlands, 2006,
305-337.
Instituto de Matemáticas,
Universidad Nacional Autónoma de México,
Área de la Investigación Cientı́fica,
Ciudad Universitaria, México D.F. 04510,
México.
Department of Mathematics and Statistics,
Dalhousie University,
Chase Building,
Halifax, Nova Scotia,
Canada B3H 3J5.
Email: quico@matem.unam.mx
rjwood@dal.ca
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