MONADS AS EXTENSION SYSTEMS
—NO ITERATION IS NECESSARY
F. MARMOLEJO AND R. J. WOOD
Abstract. We introduce a description of the algebras for a monad in terms of
ex-tension systems, similar to the one for monads given in [Manes, 1976]. We rewrite distributive laws for monads and wreaths in terms of this description, avoiding the it-eration of the functors involved. We give a profunctorial explanation of why Manes’ description of monads in terms of extension systems works.
1. Introduction
For adjoint functors S ⊣ H, it has been well known since [Eilenberg & Moore, 1965] that monad structures on S are in bijective correspondence with comonad structures on H. Moreover, it is shown in [Eilenberg & Moore, 1965] that if (S,H) is a corresponding
(monad, comonad) pair then the category of S-algebras is isomorphic to the category of Hcoalgebras via a functor that identifies the forgetful functors. After [Street, 1972] it has
been clear that these results of [Eilenberg & Moore, 1965] are actually part of the formal theory of monads, the definitions making sense in any 2-category and the theorems being provable in any suitably complete 2-category. It was acknowledged in [Lack & Street, 2002] that the formal theory of monads is easily adjusted to the greater generality of bicategories, although it suffices to prove most results in a general 2-category. Where possible we take the latter point of view in this paper.
The bijective correspondence of the nullary data η: 1 S ε:H 1
/
/
/
/
for monads and comonads is accomplished by a single application of taking mates with respect to the adjunction S ⊣ H (in any 2-category). For the binary components, it is useful to consider the correspondence of the data as a three-step mating process:
µ:SS S ξ:S HS λ:SH H
δ:H HH
/
/
/
/
/
/
/
/
The second author gratefully acknowledges continuing financial support from the Canadian NSERC. Received by the editors 2009-10-25 and, in revised form, 2010-04-01.
Transmitted by F. W. Lawvere. Published on 2010-04-03. 2000 Mathematics Subject Classification: 18C15, 18C20, 18D05.
Key words and phrases: extension systems, monads, distributive laws, wreaths, profunctors. c
This leads us to contemplate not only monads S= (S, η, µ), and comonads H= (H, ε, δ)
but also 3-tuples (S ⊣ H, η, ξ) and (S ⊣ H, ε, λ). For each of the latter two, it is a simple matter to determine three equations so that the correspondences of the data above extend to the resulting equational structures. We give such equations for an (S ⊣H, η, ξ), which we then call an extension system, in Section 9. The experienced reader will see immediately how to prescribe equations making an (S ⊣ H, ε, λ) what we would call a lifting system, although we will say little, explicitly, about these. We will speak ofξ as an extension operator, which terminology has already been used, for a special case, in [Manes & Mulry, 2007].
Suppose that (S ⊣ H, η, ξ) is an extension system on an object C in a bicategory K.
Then, for every A, B :T→Cin K, we have the composite functions
K(T,C)(B, SA)→ K(T,C)(B, HSA)→ K(T,C)(SB, SA)
where the first factor is given by composition withξAand the second by taking mates with respect to S ⊣ H. This composite, which we will call (−)S
, satisfies equations, which we will give in Section 2, but no longer requires thatShave a right adjoint inK. Accordingly, we generalize the definition of extension system to include the case where S does not necessarily have a right adjoint and show in Section 2 that, given η : 1C → S : C → C
inK, there is a bijective correspondence between monads (S, η, µ) and extension systems (S, η,(−)S
).
In [Manes, 1976], Exercise 1.3, p. 32, monads in Cat were presented as extension
systems in which the data η: 1C → S:C→ Con a category Crequired only that S be
initially given as an object function|S|:|C| → |C|andηas a function defined on|C|with
no a priori naturality requirement. We are able to analyse this simplification in Cat by
considering the canonical embedding of Cat inPro, the bicategory of profunctors. Here
we use the fact that any categoryCis, inPro, the Kleisli object for a canonical monadC
on|C|. We discuss this in Section 9. We remark that when considering extension systems
in Cat, we can always regard the situation as taking place in Pro, where every functor
in Cat has a right adjoint, and exploit the simpler form that extension systems take in
the presence of a right adjoint.
We define algebras for an extension system and, interpolating the aforementioned theorem of [Eilenberg & Moore, 1965], show that if (S, η, µ) and (S, η,(−)S
) correspond then the categories of algebras for each are isomorphic via an arrow that identifies the forgetful arrows. Thus we are able to think of extension systems and their algebras as no more than an alternate presentation for monads. However, there is an important over-arching reason to consider monads in this way. Extension systems allow us to completely dispense with the iteratesSSand SSS of the underlying arrow. No iteration is necessary. A moment’s reflection on the various terms of terms and terms of terms of terms that occur in practical applications suggest that this alone justifies the alternate approach. We give examples in Section 8.
wreaths, Sections 5 and 7. Here we are able to make use of alternate formulations of distributive laws first given in [Marmolejo, Rosebrugh, Wood, 2002]. In anticipation of further work, we note that extension systems in higher dimensional category theory provide an even more important simplification of monads. For even in dimension 2, some of the tamest examples are built on pseudofunctors that are difficult to iterate.
2. Extension systems in a 2-category
In the Introduction we motivated the idea of an extension systemin terms of monad-like data with underlying arrow S:C → C in a 2-category, in the case that S is part of
an adjunction S ⊣ H. However, it is the non-elementary definition, that we can state without the assumption of a right adjoint for S, that is most useful for our work. After a preliminary Definition and Lemma we take this as our starting point.
We work in a 2-category K. Let
C
D
E
S zz== z z z z z
U // T
"
"
E E E E E E E
ε
(1)
be a 2-cell (in K). For any span of arrows (C, D) :T → C;D, pasting ε at S defines a
family of functions
(−)#D,C:K(T,D)(D, SC)→ K(T,E)(T D, U C)
whose effect on din K(T,D)(D, SC) is the pasting composite
T
C
D
E
C EEE"" E E E E
D
/
/
S
<
<
y y y y y y y
U // T
"
"
E E E E E E E
ε
d
This 2-cell, whose full name is d#D,C, will often be written simply as d#. The family of functions (−)#D,C respects whiskering at T, meaning that for any X:S→T,
d#X = (dX)# (2)
and respects blistering at D, meaning that for any b:B →D:T→D,
(db)#=d#·T b (3)
2.1. Definition. A pasting operator
(−)#:K(T,D)(1, S)→ K(T,E)(T, U)
is a family of functions
(−)#D,C:K(T,D)(D, SC)→ K(T,E)(T D, U C)
2.2. Lemma. For arrows S, T, U configured as is in (1), pasting operators
K(T,D)(1, S)→ K(T,E)(T, U)
are in bijective correspondence with 2-cells T S→U.
Proof. Given a pasting operator (−)#:K(T,D)(1, S)→ K(T,E)(T, U), we have
(1S)#S,1C:T S→U.
Moreover, it is easy to see that any d:D → SC arises by whiskering 1S at C by C and
blistering the result at SC by d. Thus any (−)#:K(T,D)(1, S) → K(T,E)(T, U) is completely determined by (1S)#S,1C and the latter can be any 2-cell T S → U. It follows that the assignment (−)# 7→(1
S)#S,1C is a bijection.
2.3. Definition. Let C be an object in a 2-category K. An extension system on C
consists of an arrow S:C→C, a 2-cell η: 1C →S, and a pasting operator
(−)S
:K(T,C)(1, S)→ K(T,C)(S, S)
that we call the S-extension operator. This data is subject to the following equations, for
every C, B, A:T→C, f:B →SA, and g:C →SB,
ηS= 1S, (4)
B ηB //
f
!
!
C C C C C C C
C SB
fS
SA,
(5)
and
SC g
S
/
/
(fS
g)S
"
"
F F F F F F F
F SB
fS
SA.
(6)
2.4. Theorem. For η: 1C → S:C → C in a 2-category K, there is a bijective
corre-spondence between extension systems (S, η,(−)S
) and monads (S, η, µ). Proof. By Lemma 2.2 we have a bijection between pasting operators (−)S
and 2-cells µ:SS → S. Let (S, η, µ) be a monad. The correspondence of Lemma 2.2 provides fS
=µA·Sf :SB →SA. Now (4) is one of the unit monad axioms, while (5) is fS·ηB =µA·Sf ·ηB =µA·ηSA·f =f
using the other unit monad axiom, and (6) is
fS·gS =µA·Sf ·µB·Sg =µA·µSA·SSf ·Sg =µA·SµA·SSf ·Sg =µA·S(µA·Sf ·g) = (fS
using monad associativity; so that (S, η,(−)S
) is an extension system. On the other hand, if (S, η,(−)S
) is an extension system, the correspondence of Lemma 2.2 provides µ = 1SS. The first monad equation is µ ·ηS = 1SS ·ηS = 1S
From now on we do not need to distinguish between monads and extension systems. If (S, η,(−)S
) and (S, η, µ) correspond, we write (S, η,(−)S
) =S= (S, η, µ) and use freely
the equations relating both. Note too that, forb:B →D:T→C, we have
Sb = (ηD·b)S, (7)
which we leave as a simple exercise.
3. Algebras for extension systems
Notwithstanding the last paragraph, in the spirit of [Eilenberg & Moore, 1965], we give a definition of algebras for an extension system.
3.1. Definition. For (S, η,(−)S
) an extension system on C and X an object, both in
K, an (S, η,(−)S
)-algebra with domain X is a pairB= (B,(−)B), where B:X→C and
(−)B
:K(T,C)(1, B)→ K(T,C)(S, B)
is a pasting operator that we call the B-extension operator, subject to the following
equa-tions, for every h:Y →BD and k:Z →SY :T→C,
)-algebras with domain X is a
It is easy to see that (S, η,(−)S
)-algebras with domain X and their homomorphisms
form a categoryK(X,(C,(S, η,(−)S
)) equipped with a forgetful functor to K(X,C). We
recall the (S, η, µ)-algebras with domain X as described in [Street, 1972] or [Marmolejo,
1997] and write K(X,(C,(S, η, µ))) for these.
3.2. Theorem. The categoriesK(X,(C,(S, η,(−)S))) and K(X,(C,(S, η, µ))) are
iso-morphic via a functor that identifies the forgetful functors.
Proof. By Lemma 2.2, pasting operators (−)B
:K(T,C)(1, B) → K(T,C)(S, B) are in
bijective correspondence with 2-cells β:SB →B. It suffices to show that the equations for algebras and their homomorphisms in either sense correspond to those in the other sense. Let (B,(−)B
) be an (S, η,(−)S
)-algebra and consider (B,1BB), where 1BB arises
from (−)B
as in Lemma 2.2. We have 1BB·ηB= 1B by (8), and
1B
B
·S1B
B = 1B
B
·(ηB·1B
B )S
= (1B
B
·ηB·1B
B )B
= (1B
B )B = 1B
B
·(1SB)
S = 1B
B
·1S
S
B = 1B
B
·µB,
by (7), (9), (8), (9) and (2). Ifp: (B,(−)B
)→(A,(−)A
) is a homomorphism of (S, η,(−)S )-algebras then we have
1AA·Sp= 1AA·(ηA·p)S= (1AA·ηA·p)A=pA=p·1BB,
by (7), (9), (8), and (10), showing that we also havep: (B,(1B)B)→(A,(1A)A), a
homo-morphism of (S, η, µ)-algebras.
On the other hand, if (B, β) is an (S, η, µ)-algebra then, for h:Y → BD:Y → C,
definehB =SY Sh // SBD βD // BD. Now (8) ishB·ηY =βD·Sh·ηY =βD·ηBD·h=
h, and (9) is
(hB
·k)B
=βD·SβD·S2h·Sk=βD·µBD·S2h·Sk =βD·Sh·µY ·Sk =hB·kS.
Ifp: (B, β)→(A, α) is an (S, η, µ)-homomorphism, then (p·h)A
=α·Sp·Sh=p·β·Sh= p·hB
establishes (10) showing that we also have an (S, η,(−)S
)-homomorphism. Thus we do not need to distinguish between (S, η,(−)S
)-algebras and (S, η, µ)-algebras and speak simply of S-algebras, freely using all the equations now at hand.
4. The 2-category Mnd(K)
LetKbe a 2-category. Recall from [Street, 1972] that an object of the 2-category Mnd(K) consists of a pair (C,S) whereC is an object of K and S is a monad on C, that a 1-cell
that the diagrams
F
ηSF
}
}
{{{{ {{{{
F ηT
!
!
C C C C C C C C
SF λ //F T
S2F Sλ //
µSF
SF T λT //F T2
F µT
SF λ //F T
commute, and that a 2-cell (F, λ)→(F′, λ′) : (D,T)→(C,S) consists of a 2-cellϕ:F →
F′ inK such that the diagram
SF λ //
Sϕ
F T
ϕT
SF′ λ′ //F
′T
commutes.
In the spirit of Proposition 3.4 in [Marmolejo, Rosebrugh, Wood, 2002] (where it is done for distributive laws), we have the following lemma.
4.1. Lemma. GivenF :D→C, there is a bijection between 2-cellsλ:SF →F T making
(F, λ) : (D,T) → (C,S) a 1-cell of Mnd(K) and 2-cells α:SF T → F T making (F T, α)
an S-algebra satisfying the equation
SF T2 αT //
SF µT
F T2
F µT
SF T α //F T.
Moreover, if under the given bijectionλandαcorrespond, givenF :D→C, andλ′ andα′
correspond, given F′:D →C, then a 2-cell ϕ:F →F′ gives a 2-cell ϕ: (F, λ)→(F′, λ′)
of Mnd(K) if and only if ϕT:F T →F′T is anS-homomorphism.
Proof. If we start with α, then λ=α·SF η
T. In the opposite direction, given λ, define
α=F µT ·λT.
Denote the 2-category implicitly defined by the above lemma with 1-cells (F, α) : (D,T) → (C,S) by Mnd′(K). Observe that the composition of 1-cells (G, β) : (E,U) →
(D,T) and (F, α) : (D,T)→(C,S) is given by F Gtogether with
SF GU SF ηTGU//SF T GU αGU //F T GU F β //F GU.
4.3. Theorem. Let T = (T, η
T,(−)
T
) be a monad on D, and let S = (S, η
S,(−)
S ) be a monad on C. A 1-cell in Mnd(K) from (D,T) to (C,S) can equivalently be defined as
follows: (F,(−)λ) : (D,T) → (C,S) where F :D → C, and (F T,(−)λ) is an S-algebra,
such that for every u:U →T V :X→D and h:X →F T U:X→C, the diagram
SX hλ //
(F uT ·h)λ
#
#
H H H H H H H H
H F T U F uT
F T V
(11)
commutes.
Furthermore, given (F,(−)λ),(F′,(−)λ′
) : (D,T)→(C,S), then ϕ:F →F′ is a 2-cell
in Mnd(K) if and only if ϕT: (F T,(−)λ)→(F′T,(−)λ′
) is a morphism of S-algebras.
Proof. According to Theorem 3.2 we have that pasting operators (−)λ:K(X,C)(1, F T)
→ K(X,C)(S, F T) that make (F T,(−)λ) an S-algebra are in bijective correspondence
with 2-cells α : SF T → F T that make (F T, α) anS-algebra. So we must show that the
extra equation given in the statement of the theorem is satisfied if and only if the extra equation given in Lemma 4.1 is satisfied.
Given (F,(−)λ) as in the statement of the theorem, we have that
α := (1F T)λ:SF T →F T.
The extra equation isF µT·αT =F(1T
T
)·(1F T2)λ =F(1TT)λ = (1F Tλ·η
SF T·F(1T
T ))λ =
1F Tλ·(ηSF T·F(1T
T ))S
=α·SF µT. Thus (F, α) : (D,T)→(C,S) is a 1-cell in Mnd
′(K).
In the opposite direction, assume that (F, α) : (D,T)→(C,S) is a 1-cell in Mnd′(K).
Then for h:X→F T U :X →Cwe have that
hλ := SX Sh //SF T U αU //F T U. (12)
Foru:U →T V, the commutative diagram
SX Sh //
S(F(uT
)·h)
%
%
SF T U αU //
SF T u
F T U
F T u
SF T2V αT V //
SF µTV
F T2V
F µTV
SF T V αV //F T V
gives us (11).
4.4. Remark. Observe that in the previous theorem we can obtainλ:ST →T Sdirectly from the pasting operator (−)λ as (F η
T)λ to obtain a 1-cell (F, λ) : (D,T) → (C,S) as
5. The 2-category EM(K)
Recall from [Lack & Street, 2002] that the 2-category EM(K) has the same objects and 1-cells as Mnd(K), but the 2-cells (F, λ)→(F′, λ′) : (D,T)→(C,S) are 2-cellsρ:F →F′T
inK such that the diagram
SF λ //
Sρ
F T ρT //F′T2
F′µ
T
SF′T λ′T //F′T2 F′µ
T
/
/F′T
commutes.
5.1. Lemma. If under the bijection given in Lemma 4.1, λ and α correspond, given F :D → C, and λ′ and α′ correspond, given F′:D →C, then a 2-cell ρ:F → F′T is a
2-cell ρ: (F, λ)→(F′, λ′) of EM(K) if and only if the diagram
SF T α //
SρT
F T ρT //F′T2
F′µ
T
SF′T2 α′T //F
′T2
F′µ
T
/
/F′T
commutes.
We present the following description of the 2-cells in EM(K).
5.2. Theorem. Given 1-cells(F,(−)λ),(F′,(−)λ′) : (D,T)→(C,S)in EM(K), a 2-cell
(F,(−)λ)→(F,(−)λ′
)in EM(K)can be defined as anS-algebra morphismβ: (F T,(−)λ)→
(F′T,(−)λ′
) such that for every u:U →T V :X→D, the diagram
F T U βU //
F uT
F′T U
F′uT
F T V βV //F′T V
commutes.
Proof. Assume we have β:F T →F′T as in the statement of the theorem. Define
The commutative diagram
gives us the equation in Lemma 5.1.
In the opposite direction, assume that ρ: (F, λ) → (F′, λ) : (D,T) → (C,S) satisfies
the equation in Lemma 5.1. Define
β := (F T ρT //F′T2 F F′T).
′µ
T
/
/ (14)
Then the commutative diagram
SF T α //
6.1. Proposition. Given monads T and S on C in a 2-category K, there is a
bijec-tive correspondence between distribubijec-tive laws λ:ST → T S of S over T and S-algebras
α:ST S →T S that satisfy the commutativity of the diagrams
ST S2ST µS//
and the commutativity of the diagram:
SX hλ //
using (7), (15) and (9). Now a direct use of (16) produces (11).
The corresponding 2-cell α = 1T Sλ is, according to Theorem 4.3, an S-algebra and it
satisfies the first of the equations in Proposition 6.1. The second one is given by α·SηTS = 1T S
In the opposite direction, assume we have an S-algebra α:ST S → T S that satisfies
the conditions of Proposition 6.1. Its corresponding pasting operator in h:X →T SU is given by
The proofs of the next two propositions are left to the reader.
6.3. Proposition. Given a distributive law λ of S over T, the composite monad is
given by T◦λ S= (T S, T η
S ·ηT,((−) λ)T
).
There is also a result closer to “compatible structures”:
6.4. Proposition. LetSand Tbe monads on C. There is a bijection between
distribu-tive laws ofSoverTand monad structures(T S, T η
, and the diagram
7. Wreaths
Recall from [Lack & Street, 2002] that given a monad S = (S, η, µ) on an object Cof K
and a 1-cell T:C→C, a wreath consists of 2-cells
σ: 1C→T S, λ:ST →T S, ν:T2 →T S,
that satisfy the commutativity of the following diagrams:
T such that (T S, α) is an S-algebra and the following diagrams commute:
7.2. Theorem. Given a monad S = (S, η,(−)S
) on C in K and a 1-cell T :C→C, a
wreath can be equivalently defined as follows:
1. A 2-cell σ: 1C→T S in K.
Assume that we have the conditions of the statement of the theorem. Then α= 1T Ss
andγ = 1T St. We check that the conditions of Proposition 7.1 are satisfied. The fact that
(T,(−)s) : (C,S) → (C,S) is a 1-cell in Mnd(K) means, according to Theorem 4.3, that
(T S, α) is an S-algebra and that the first of the diagrams in Proposition 7.1 commutes.
the fact that (−)t respects blistering. The fifth is γ·T α·σT S = 1
T St·T(1T Ss)·σT S =
(1T Ss)t·σT S = 1T S, using (17). The sixth isγ·T α·αT S = (1T Ss)t·1T ST Ss= ((1T Ss)t)s =
(αt)s = (γ·T α)s = (γs·ηT2S·T α)s =γs·(ηT2S·T α)S
=α·Sγ·ST α. And the last is γ·T α·γT S =αt·1
T ST St = (1T Ss)t·1T ST St= ((1T Ss)t)t = (αt)t= (γ·T α)t=γt·T2α =
γ·T γ·T2α.
In the opposite direction assume we have σ: 1C→T S, α:ST S →T S and γ:T2S→
T S that satisfy the conditions of Proposition 7.1. Then for f:B → T SA we have that fs = αA ·Sf, and ft = γA ·T f, that is, (−)s is the pasting operator corresponding
to α and (−)t is the pasting operator corresponding to γ. The fact that (T S, α) is an
S-algebra together with the first commutative diagram of Proposition 7.1 means that
(T,(−)s) : (C,S) →(C,S) is a 1-cell in Mnd(K). Now, (σA)t =γA·T σA=T ηA, using
the triangle from Proposition 7.1. Furthermore
T(hS)·σB =T µA·T Sh·σB =T µA·σSA·h=αA·SσA·h= (σA)s·h,
using the second commutative diagram from Proposition 7.1, and
(fs)t·σB =γA·T αA·T Sf ·σB =γA·T αA·σT SA·f =f,
using the fifth commutative diagram of Proposition 7.1 give us (17). (18) is given by
T(hS)·gt=T µA·T Sh·γB·T g =T µA·γSA·T2Sh·T g =γA·T2µA·T2Sh·T g = (T(hS)·g)t
using the third commutative diagram from Proposition 7.1. Finally, the commutative diagrams
SC
S((fs)t·g)
Sg
/
/ST SB αB //
ST Sf
T SB
T Sf
ST ST SAαT SA//
ST αA
T ST SA
T αA
ST2SA
SγA
T2SA
γA
ST SA αA //T SA,
T C
T((fs)t·g)
T g
/
/T2SB γB //
T2Sf
T SB
T Sf
T2ST SAγT SA//
T2αA
T ST SA
T αA
T3SA
T γA
T2SA
γA
T2SA γA //T SA,
give us (19).
8. Monads, algebras, distributive laws and wreaths in
Cat
giving a monadSon a categoryCas an extension system, that a mere function|S|:|C| →
|C| and a mere C-arrow valued function η, defined on |C| with no a priori naturality
requirement, sufficed in the presence of an extension operator. Thus fewer axioms are required for extension systems on categories but we do have to show the naturality of the transformations that we introduce. However, formally it is very similar to what we have done so far in a general 2-category, and most of the proofs are similar to the proofs already given. Thus, in this section we present the precise statements for this important particular case and give the extra arguments necessary for the naturality of the various transformations. In the next section, we analyze the extra structure onCat that enables
the description of extension systems as functions, in terms of profunctors. First we recall Exercise 1.3.12, page 32, of [Manes, 1976]:
8.1. Theorem. A monad S on a category C can be defined as follows: A function
|S|:|C| → |C|, for every A ∈ C, an arrow ηA:A → SA, and for every morphism
:SX →B subject to the commutativity of the following diagrams (with h:X →B and y:Y →SX):
C subject to the commutativity of the diagram (where h:X →B):
Proof. Given (B,(−)B
) be monads on the category
C. A distributive law of S over T can be defined as follows. For every A in C an S -and α is natural.
8.4. Theorem. Given a monad S = (S, η,(−)S
) on a category C and an endofunctor
T :C→C, a wreath can be equivalently given as follows:
is a morphism of S-algebras for every h:B →SA. That is, the diagrams
5. For every g:C →T SB, h:B →SA, k:C →B and f:B →T SA, the diagrams
using (21) and (20). For the naturality of α we have:
T Sℓ·αB=T((ηSA·ℓ)
using the equations of (20). And for the naturality of γ we have
T Sℓ·γB =T((ηSA·ℓ)
S
)·1T SBt= (T Sℓ)t=γA·T2Sℓ.
8.5. Example. Beck’s original example [Beck, 1969] has Sthe free monoid monad and T the free abelian group monad. ThusSA is the underlying set of the free monoid in A,
SA={[a1, . . . , an]|n∈N, aj ∈A, j = 1, . . . , n},
ηS:A→SA is such that
ηS(a) = [a],
and for h:A→SB, we have that
where ∼ denotes concatenation ([b1, . . . , bk] ∼ [c1, . . . , cℓ] = [b1, . . . , bk, c1, . . . , cℓ]). On
the other hand, T A is the underlying set of the free group with A generators. Thus the elements of T A are formal sums X
a∈A
where this last sum is taken in the abelian group T B. Now, T SB has a monoid structure given by
X
On the other hand
(fλ)T
It takes just a moment to realize that to see that these are the same, it suffices to show that the operation ∗ distributes over the addition inT SA, but this is easy.
shows that we have a distributive law.
8.6. Example. Another example from [Beck, 1969] hasT arbitrary onC with
coprod-ucts, and S the constants monad. (It is done for the category of sets in Beck’s paper but
it works in the given context.) That is, take a fixed object C in C, define for every A,
SA=A+C,ηSA=iA:A →A+C the canonical injection of the first summand, and for
h:A→SB define hS
:SA→SB as the unique arrow such that the diagram
A ηSA//
commute. The commutative diagram
the commutative diagram
. And the commutative diagram
A ηSA //
· fλ. Therefore we have a distributive law, where λA:ST A→T SAis the unique arrow that makes the diagram
By the naturality of ηT, we have
(ηT(M ×A)·ηMA)
λ(m, a) =T(m∗(−)×A)((η
T(M ×A)·ηMA)(a))
= (T(m∗(−)×A)·ηT(M ×A))(e, a)
=ηT(M ×A)(m∗(−)×A)(e, a)
=ηT(M ×A)(m, a).
Furthermore (fλ·η
MA)(a) =f
λ(e, a) = T(e∗(−)×B)(f(a)) =f(a). Forg =hg
1, g2i:C→ M ×A and (m, c)∈M ×C we have
(fλgM)(m, c) = fλ(gM(m, c)) =fλ(m∗g1(c), g2(c)) = T(m∗g1(c)∗(−)×A)(f(g2(c))) =T(m∗(−)×A)(T(g1(c)∗(−)×A)(f(g2(c))))
=T(m∗(−)×A)((fλg)(c)) = (fλg)λ(m, c).
It is not hard to show that for any m ∈M, fλ ◦(m∗(−)×B) = T(m∗(−)×A)◦fλ. Then for h:C→T(M ×A) and (m, c)∈M ×C we have
((fλ)Th)λ(m, c) = (T(m∗(−)×A)·(fλ)T)(h(c)) = ((ηT(M ×A)·(m∗(−)×A))
T
·(fλ)T
)(h(c)) = (((ηT(M ×A)·(m∗(−)×A))
T
·fλ)T
)(h(c)) = ((T(m∗(−)×A)·fλ)T
)(h(c)) = (fλ·(m∗(−)×B))T (h(c)) = ((fλ)T
·ηT(M ×B)·(m∗(−)×B))
T (h(c))
= ((fλ)T·T(m∗(−)×B))(h(c)) = ((fλ)T·hλ)(m, c). We thus have a distributive law ofM over T.
8.8. Example. An example taken from [Varacca, 2003] has the monad P on Set of
finite non-empty subsets, whose structure is given by, for any set X, P X ={X0 ⊆X|X0 finite non-empty}, ηPX:X → P X is ηPX(x) = {x}, and for any function f:Y → P X,
fP (Y0) =
S
y∈Y0f(y).
On the other hand we have the monad V on Set of indexed valuations, whose
struc-ture is given as follows. For a set X, V(X) has as elements equivalent classes of spans
K X
(0,∞)oo r χ // inSetwithK finite, and the given span is equivalent to the span
K′ X
(0,∞) χ
′
/
/
r′
o
o iff there is a bijection κ:K →K′ such that the diagram
K
X (0,∞)
K′
χ
(
(
P P P P P P r
u
u
kkkkkk
χ′
6
6
n n n n n n r′
i
i
SSSSS κ
commutes. The arrow ηVX:X →V(X) is such that the span associated tox∈X is
1 X.
Take a functionf:Y →V(X). Fory∈Y denote the spanf(y) by (0,∞) Ky X.
where r and χ are the unique functions such that the diagram
(0,∞)
on the same span gives us
a
Then the canonical isomorphism
shows that these spans are the same element of V(X). That is, the diagram
V(Z) g
V
/
/
(fV
g)V
$
$
H H H H H H H H
H V(Y)
fV
V(X)
commutes. We have then shown that we have a monad V.
We now give the distributive law. Take h:Y →P(V(X)), and take a span
J Y.
(0,∞)oo q ψ //
For every choice functionℓ:J →Sj∈Jh(ψ(j)) (that is,ℓ(j)∈h(ψ(j))), ifℓ(j) is the span
Kj X,
(0,∞)oo rj χj //
for j ∈J, we form the span S(ℓ, q, ψ) as
`
j∈JKj X,
(0,∞)oo r χ //
where for k∈Kj we defineχ(k) =χj(k) and r(k) = q(j)·rj(k). We define
hλ((q, ψ)) = {S(ℓ, q, ψ)|ℓ:J → [
j∈J
h(ψ(j)) is a choice function}.
We show that this defines a distributive law of V over P. We must show first that
this definition of hλ produces a structure of V-algebra on P(V(X)). We know that
ηVY(y) = (p1q,pyq). Then a choice function ℓ: 1→h(y) is simply to pick an element in
h(y). Then S(ℓ,p1q,pyq) is this chosen element, thus hλ ◦η
VY =h.
Then a typical element inhλ(gV
((0,∞)oo p I ζ //Z)) is formed as follows. Assume g(z) is given by (24) for everyz ∈Z. Then for every i∈I and every j ∈Jζ(i) we take an element
Kij X
(0,∞) χ
ij
/
/
rij
o
o
inh(ψζ(i)(j)). Then the element of hλ(gV
(p, ζ)) is
(0,∞)oo r `(i,j)∈`i∈IJζ(i)Kij χ //X,
where for k∈Kij, χ(k) = χij(k) and r(k) =qζ(i)(j)·rij(k).
On the other hand, a typical element of (hλgV
)λ(p, ζ) is formed as follows. For every
i∈I and j ∈Jζ(i) take an element
Kij X
(0,∞) χ
ij
/
/
rij
o
inh(ψζ(i)(j)). Then the element is formed as
(0,∞) `i∈I `
j∈Jζ(i)Kij
r′
o
o
χ′ //X,
where for k ∈Kij (j ∈Jζ(i)) we have thatχ′(k) = χij(k) andr′(k) = p(i)·qζ(i)(j)·rij(k). The canonical isomorphism `i∈I`j∈J
ζ(i)Kij →
`
(i,j)∈`i∈IJζ(i)Kij then tell us that
hλgV
= (hλg)λ.
The condition (ηPV(X)·ηVX)λ = ηPV(X) is easy, since there is only one possible
choice function for a given element in V(X).
Finally, take k:Z →P(V(Y)) and h:Y →P(V(X)). We must check that (hλ)Pkλ =
((hλ)P
k)λ. Take
I Z
(0,∞)oo p ζ // inV(Z). Then a typical element of ((hλ)Pkλ)(p, ζ)
is formed as follows: for every i∈I chose a span (0,∞) Ji Y ψi
/
/
qi
o
o ink(ζ(i)), then
for every j ∈ Ji choose a span (0,∞) Kij X χij
/
/
rij
o
o in h(ψi(j)); then the element in
question is
`
(i,j)∈`i∈IJiKij X
(0,∞)oo r χ //
where for k∈Kij, (j ∈Ji) we have χ(k) = χij(k) and r(k) =p(i)·qi(j)·rij(k).
On the other hand, a typical element in ((hλ)Pk)λ(p, ζ) is formed as follows: for every
i ∈ I we take a span (0,∞) Ji Y ψi
/
/
qi
o
o in k(ζ(i)), then for every j ∈Ji we take a
span (0,∞) Kij X χij
/
/
rij
o
o in h(ψi(j)); then the element in question is
` i∈I
`
j∈JiKij X
(0,∞) χ
′
/
/
r′
o
o
where for k ∈ Kij, (j ∈ Ji) we have χ′(k) = χij(k) and r′(k) = p(i) ·qi(j)· rij(k).
Therefore, (hλ)Pkλ = (hλk)λ. Thus we have a distributive law.
Compare with the proof given in [Varacca, 2003].
8.9. Example. This one is taken from [Manes & Mulry, 2007]. LetSbe the free monoid
monad (described in example 8.5) and let T the submonad of S of nonempty words:
T A={[a1, . . . , an]|n ∈N\{0}, aj ∈A, j = 1, . . . , n},
for any setA. GiveT SA the following binary operation
[U1, . . . , Uk]∗[V1, . . . , Vℓ] = [U1, . . . , Uk−1, Uk∼V1, V2, . . . , Vℓ]
whereU1, . . . , Uk, V1, . . . , Vℓ are elements ofSA, and∼denotes concatenation. It is
imme-diate to see that T SA with this operation is a monoid. Therefore, if for anf:B →T SA we define fλ:SA→T SA such that
we obtain an S-algebra (T SA,( )λ). Now, for [a
1, . . . , ak]∈SA we have
(T ηSA·ηTA)λ([a1, . . . , ak]) = [[a1]]∗ · · · ∗[[ak]] = [[a1, . . . , ak]] =ηTSA([a1, . . . , ak]).
Finally we must show that forf:B →T SA, (fλ)T
:T SB →T SAis a monoid morphism with respect to the operation ∗ on both, T SB and T SA. Let [U1, . . . , Uk],[V1, . . . , Vℓ]∈
T SB. Then (fλ)T
([U1, . . . , Uk]∗[V1, . . . , Vℓ]) = (fλ)T([U1, . . . , Uk−1, Uk∼V1, V2, . . . , Vℓ])
=fλ(U1)∼ · · · ∼fλ(U
k−1)∼fλ(Uk ∼V1)∼fλ(V2)∼fλ(Vℓ)
and
(fλ)T
([U1, . . . , Uk])∗(fλ)T([V1, . . . , Vℓ])
= (fλ(U1)∼ · · · ∼fλ(U
k))∗(fλ(V1)∼ · · · ∼fλ(Vℓ))
=fλ(U1)∼ · · · ∼(fλ(U
k)∗fλ(V1))∼ · · · ∼fλ(Vℓ).
Since it direct to see that fλ(Uk)∗fλ(V1) =fλ(Uk ∼V1), we have a distributive law ofS
over T.
8.10. Example. We close our set of examples with a wreath from [Lack & Street, 2002]. Take a short exact sequence in the category of groups
1→A→E →G→1.
Forg ∈Ganda ∈Adenote the action ofgonabyag =g−1ag∈A, and letρ:G×G→A be a normalized cocycle corresponding to the extension. Let A be the monad on Set
determined by the groupA(as described in example 8.7, there for a monoidM), and take the functorT =G× −:Set→Set. For the wreath defineσX:X →G×A×X such that
σA(x) = (e, e, x), and forf =hf1, f2, f3i:Y →G×A×X, define fs:A→G×A×X as
fs(a, y) = (f1(y), af1(y)·f2(y), f3(y)),
and ft:G×Y →G×A×X as
ft(g, y) = (g·f1(y), ρ(g, f1(y))·f2(y), f3(y)).
We only verify that the last condition from Theorem 8.4 is satisfied and leave the rest to the reader. We have that
(fs)t(g, a, y) = (g·f
1(y), ρ(g, f1(y))·af1(y)·f2(y), f3(y)). Thus for ℓ =hℓ1, ℓ2, ℓ3i:Z →G×A×X we have that (fs)t(ℓt(g, z)) is
(g·ℓ1(z)·f1ℓ3(z), ρ(g·ℓ1(y), f1ℓ3(z))·(ρ(g, ℓ1(z))·ℓ2(z))f1ℓ3(z)·f2ℓ3(z), f3ℓ3(z)), whereas ((fs)tℓ)t(g, z) is
(g·ℓ1(z)·f1ℓ3(z), ρ(g, ℓ1(z)·f1ℓ3(z)·ρ(ℓ1(z), f1ℓ3(z))·ℓ2(z)f1ℓ3(z)·f2ℓ3(z), f3ℓ3(z)). The fact that they are equal follows from the cocycle condition
ρ(g·h, k)·ρ(g, h)k=ρ(g, h·k)ρ(h, k)
9. Extension systems in the bicategory of profunctors
In the Introduction we also mentioned extension systems of the form (ϕ, ψ:S ⊣ H, η, ξ) on an objectCin a 2-categoryK. We recall thatη: 1C →S:C→Cand thatξ:S →HS
is to be the mate of a 2-cell µ:SS →S so that explicitly we have
ξ = (S ϕS //HS2 Hµ //HS)
and
µ= (S2 Sξ //SHS ψS //S).
We begin this section by reconciling our two usages of the term extension system. 9.1. Lemma. The data (S, η, µ) constitute a monad if and only if the data (ϕ, ψ:S ⊣ H, η, ξ) satisfy the following three equations:
1 η //
ϕ
!
!
B B B B B B B
B S
ξ
HS,
S2 Sξ //SHS
ψS
S
ηS
O
O
1S
/
/S,
S2 Sξ //SHS ψS //
ξHS
S
ξ
HSHSHψS //HS.
(25)
Proof. Apply S to the first of these equations and post-compose the result with ψS. From the definition of µ this gives the monad equation µ·Sη = 1S. Conversely, given
µ·Sη = 1S, the mate 1C → HS with respect to the given adjunction of 1S is ϕ while
that ofµ·Sη isHµ·HSη·ϕ =Hµ·ϕS·η=ξ·η, from the definition ofξ. The second of the equations given above is immediately the monad equationµ·ηS = 1S and conversely.
Starting with the third equation, apply S− and note the commutativity of each of the three squares pasted west or south of the result, as shown in the diagram below. From the definition of µ, the outer square gives the associativity equation µ·Sµ=µ·µS.
S3 S
2ξ
/
/
SξS
S2HS SψS //
SξHS
S2
Sξ
SHS2 SHSξ//
ψS2
SHSHSSHψS //
ψSHS
SHS
ψS
Finally, starting with µ·Sµ =µ·µS, get the third equation of the statement from
Now recall, from [Borceux, 1994] say, the bicategory Pro of categories, profunctors,
and equivariant 2-cells. There is a pseudofunctor (−)∗:Cat→Pro which is the identity
on objects and takes a functor F:X→A to the profunctor with F∗(A, X) = A(A, F X).
For any parallel pair of functors F, G:X → A, there is a bijection between equivariant
2-cells F∗ → G∗ and natural transformations F → G. That is, (−)∗ is locally fully
faithful. Most importantly, for any functor F, there is an adjunction F∗ ⊣ F∗ in Pro,
whereF∗(X, A) = A(F X, A). It follows that to describe a monad (S, η, µ) in Catvia an
extension system, we can either use the general theory of the bulk of this paper or avail ourselves of the adjunction S∗ ⊣ S∗ in Pro and proceed using the elementary definition
provided by the data of Lemma 9.1, subject to the equations (25).
Now, note that in Prothe compositeS∗S∗ :C→C has S∗S∗(B, A) =C(SB, SA) so which is just the blister equation (3) of Section 2. Equivariance of ξ in the variable A means that, for all u:A →X, we have (Su·f)S
=Su·fS
which follows easily from (6). The point here is that when the data for the elementary definition is expanded in Pro,
it amounts to the same data required for the non-elementary definition in Cat. This
reconciles our usage of the termextension system in both cases.
In Section 8, we remarked that Manes’ original presentation of monads in Cat does
not require thatS:C→Cbe given as a functor nor thatη: 1C→S be given as a natural
transformation. It is interesting to note that the bicategory Pro also provides a venue
to discuss this simplification, in terms of the formal theory of monads. For any category
C, write|C| for the set of objects ofC regarded as a discrete category. In Pro we have,
objects, functor K:|C| →C admits an (op)action of C that we callκ:K∗C →K∗. The
(C, A) component of this action, PB∈|C|K∗(C, B)×C(B, A) → K∗(C, A) is determined
by the C(C, B)×C(B, A)→C(C, A) which are given by the composition of C.
The profunctor K∗, together with κ, exhibits C as the Kleisli object in Pro for the
monad C. Moreover, the equivalence mediated by this data restricts to functors, in the
sense that to give a functor F :C→D is to give a functor Fb:|C| →D together with an
action Fb∗C → Fb∗. Similarly, to give a natural transformation τ:F → G:C → D is to
give a natural transformation bτ:Fb → Gb:|C| → D that respects the actions. An action
b
F∗C→Fb∗ has a mateC→Fb∗Fb∗. The two action equations translate to makeC→Fb∗Fb∗
a morphism of monads on |C| and conversely.
In the case of the data given originally by Manes, we have in these terms: η:K →
b
S:|C| → C and now the requisite morphism of monads C → Sb∗Sb∗ is given by the
composite
C=K∗K∗ K ∗η
∗
/
/K∗Sb
∗
(−)S
/
/Sb∗Sb
∗
— the monad morphism equations arising from two of those satisfied by the extension operator (−)S
.
References
Jon Beck. Distributive laws. Lecture Notes in Mathematics 80, (1969), 119-140. F. Borceux. Handbook of Categorical Algebra. Cambridge University Press, 1994.
S. Eilenberg and J. Moore, Adjoint functors and triples. Illinois J. Math. 9 (1965), 381-398. Stephen Lack and Ross Street, The formal theory of monads II. Journal of pure and
applied algebra 175 (2002) 234-265.
E. G. Manes. Algebraic Theories. Springer-Verlag, 1976.
Ernie Manes and Philip Mulry, Monad compositions I: general constructions and recursive distributive laws. Theory and Applications of Categories, Vol. 18, 2007, No. 7, pp 172-208.
F. Marmolejo, R. D. Rosebrugh and R. J. Wood, A basic distributive law. J. Pure and Appl. Algebra 168 (2002) 209-226.
F. Marmolejo, Doctrines whose structure forms a fully faithful adjoint string. Theory and Applications of Categories, Vol 3, No. 2, 1999, pp. 24-44.
D. Varacca. Probability, nondeterminism and concurrency: two denotational models for probabilistic computation. PhD Dissertation, BRICS PhD School, 2003.
Instituto de Matem´aticas
Universidad Nacional Aut´onoma de M´exico ´
Area de la Investigaci´on Cient´ıfica, Circuito Exterior, Ciudad Universitaria Coyoac´an 04510, M´exico, D.F. M´exico
Department of Mathematics and Statistics, Dalhousie University Chase Building, Halifax, Nova Scotia, Canada B3H 3J5
Email: [email protected] [email protected]
This article may be accessed at http://www.tac.mta.ca/tac/ or by anonymous ftp at
significantly advance the study of categorical algebra or methods, or that make significant new contribu-tions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.
Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.
Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at
http://www.tac.mta.ca/tac/and by ftp. It is archived electronically and in printed paper format. Subscription information. Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail [email protected] a full name and postal address. For in-stitutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh,[email protected]. Information for authors. The typesetting language of the journal is TEX, and LATEX2e
strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files athttp://www.tac.mta.ca/tac/. Managing editor. Robert Rosebrugh, Mount Allison University: [email protected] TEXnical editor. Michael Barr, McGill University: [email protected]
Assistant TEX editor. Gavin Seal, Ecole Polytechnique F´ed´erale de Lausanne:
gavin [email protected]
Transmitting editors.
Clemens Berger, Universit´e de Nice-Sophia Antipolis,[email protected]
Richard Blute, Universit´e d’ Ottawa: [email protected]
Lawrence Breen, Universit´e de Paris 13: [email protected]
Ronald Brown, University of North Wales: ronnie.profbrown (at) btinternet.com
Aurelio Carboni, Universit`a dell Insubria: [email protected]
Valeria de Paiva, Cuill Inc.: [email protected]
Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu
Martin Hyland, University of Cambridge: [email protected]
P. T. Johnstone, University of Cambridge: [email protected]
Anders Kock, University of Aarhus: [email protected]
Stephen Lack, University of Western Sydney: [email protected]
F. William Lawvere, State University of New York at Buffalo: [email protected]
Tom Leinster, University of Glasgow,[email protected]
Jean-Louis Loday, Universit´e de Strasbourg: [email protected]
Ieke Moerdijk, University of Utrecht: [email protected]
Susan Niefield, Union College: [email protected]
Robert Par´e, Dalhousie University: [email protected]
Jiri Rosicky, Masaryk University: [email protected]
Brooke Shipley, University of Illinois at Chicago: [email protected]
James Stasheff, University of North Carolina: [email protected]
Ross Street, Macquarie University: [email protected]
Walter Tholen, York University: [email protected]
Myles Tierney, Rutgers University: [email protected]
Robert F. C. Walters, University of Insubria: [email protected]
