Directory UMM :Journals:Journal_of_mathematics:TAC:
Theory and Appli
ations of Categories,
Vol. 6, No. 7, 1999, pp. 83{93.
ENRICHED LAWVERE THEORIES
Dedi
ated to Jim Lambek
JOHN POWER
ABSTRACT. We de ne the notion of enri
hed Lawvere theory, for enri
hment over
a monoidal bi
losed
ategory V that is lo
ally nitely presentable as a
losed
ategory.
We prove that the
ategory of enri
hed Lawvere theories is equivalent to the
ategory
of nitary monads on V . Moreover, the V -
ategory of models of a Lawvere V -theory
is equivalent to the V -
ategory of algebras for the
orresponding V -monad. This all
extends routinely to lo
al presentability with respe
t to any regular
ardinal. We nally
onsider the spe
ial
ase where V is Cat, and explain how the
orresponden
e extends
to pseudo maps of algebras.
1. Introdu
tion
In seeking a general a
ount of what have been
alled notions of
omputation [13℄, one may
onsider a nitary 2-monad T on C at, and a T -algebra (A; a), then make a
onstru
tion of
a
ategory B and an identity on obje
ts fun
tor j : A ! B . In making that
onstru
tion,
one is allowed to use the stru
ture on A determined by the 2-monad, and that is all. For
instan
e, given the 2-monad for whi
h an algebra is a small
ategory with a monad on it,
then one possible
onstru
tion would be the Kleisli
onstru
tion. For another example,
given the 2-monad for whi
h an algebra is a small monoidal
ategory A together with a
spe
i ed obje
t S , then a possible
onstru
tion is that for whi
h B (a; b) is de ned to be
A(S
a; S
b), with the fun
tor j : A
! B sending a map h to S
h.
We need a pre
ise statement of what we mean by saying that these
onstru
tions only
use stru
ture determined by the 2-monad. In general, one may obtain one su
h de nition
by asserting that B (a; b) must be of the form A(f a; gb) for endofun
tors f and g on A
generated by the 2-monad; similarly for de ning
omposition in B . Thus we need to know
what exa
tly we mean by endofun
tors on A and natural transformations between them
that are generated by the 2-monad. This paper is motivated by a desire to make su
h
notions pre
ise.
In order to make pre
ise the notion of fun
tor generated by a nitary 2-monad T on
a T -algebra (A; a), we rst generalise from
onsideration of a nitary 2-monad on C at
to a nitary V -monad on V for any monoidal bi
losed
ategory that is lo
ally nitely
This work has been done with the support of EPSRC grants GR/J84205: Frameworks for programming language semanti
s and logi
and R34723: The Stru
ture of Programming Languages: syntax and
semanti
s.
Re
eived by the editors 1998 De
ember 14 and, in revised form, 1999 August 27.
Published on 1999 November 30.
1991 Mathemati
s Subje
t Classi
ation: 18C10, 18C15, 18D05.
Key words and phrases: Lawvere theory, monad.
John Power 1999. Permission to
opy for private use granted.
83
Theory and Appli
ations of Categories, Vol. 6, No. 7
84
presentable as a
losed
ategory: we shall make that de nition pre
ise in the next se
tion. We move to this generality primarily for simpli
ity but also be
ause the
omputing
appli
ation will need it later, for instan
e
onsidering not mere
ategories but
ategories
enri
hed in P oset or in the
ategory of ! -
po's.
To support our main de nition, we prove a theorem: we de ne the notion of a Lawvere
V -theory, and we prove that to give a Lawvere V -theory is equivalent to giving a nitary
V -monad on V . So, for a nitary V -monad T on V , there is a Lawvere V -theory L(T ) for
whi
h the V -
ategory of models of L(T ), whi
h we de ne, is equivalent to the V -
ategory
T -Alg of algebras for the monad. Conversely, given any Lawvere V -theory T , there is
a
orresponding nitary V -monad M (T ); and these
onstru
tions yield an equivalen
e
between the
ategory of nitary V -monads on V and that of Lawvere V -theories.
On
e we have su
h a
orresponden
e, we have the de nition we seek: given a 2-monad
T and a T -algebra (A; a), an endofun
tor f on A is generated by T whenever it is in the
image of L(T ) in the model of L(T ) determined by the algebra (A; a). Similarly, a natural
transformation : f ) g is is said to be generated by T whenever it is the image of a
2-
ell of L(T ) in the model of L(T ) determined by the algebra (A; a).
For
ognos
enti of enri
hed
ategories, the de nitions and results here, at least assuming our enri
hment is over a symmetri
monoidal
ategory, should
ome as no great
surprise. However, they are a little subtle. One's rst guess for a de nition of Lawvere
V -theory may be a small V -
ategory with nite produ
ts, subje
t to a
ondition asserting
single-sortedness. But that is too
rude: not only does it not yield our theorem relating
Lawvere theories and nitary monads, as in the
ase for Set (see [1℄), but it does not
agree with the known and very useful equivalen
es between nitary monads and universal
algebra as in [9℄, [6℄, and in appli
ation to
omputation in [11℄ and [10℄, with an a
ount
for
omputer s
ientists in [14℄. So we
onsider something a little more subtle: if V is
symmetri
, we de ne a Lawvere V -theory to be a small V -
ategory with nite
otensors,
subje
t to a single-sortedness
ondition. A model is therefore a nite
otensor preserving
V -fun
tor into V . In order to extend to nonsymmetri
V , as for instan
e if V is the
ategory of small lo
ally ordered
ategories with the lax Gray tensor produ
t [11℄, we need to
introdu
e a twist: so a Lawvere V -theory is a small V -
ategory with nite
otensors, subje
t to a single-sortedness
ondition, and a model is a nite
otensor preserving V -fun
tor
into V : that allows us to speak of the V -
ategory of models, and prove our equivalen
e
with nitary V -monads on V and their V -
ategories of algebras. There has been
onsiderable development of
ategories enri
hed in monoidal bi
losed
ategories [4, 5, 6, 11℄
re
ently: the main problem with extending the usual theory is that fun
tor V -
ategories
need not exist in general; but fun
tor V -
ategories with base V do, and that suÆ
es for
our purposes here. If one is only interested in the
ase of symmetri
V here, one may simply ignore all subs
ripts t, and one will have
orre
t statements; of
ourse, exponentials,
i.e., terms of the form [X; ℄ and [X; ℄ , may also be abbreviated to [X; ℄.
The spe
ial
ase that V = Cat also
ontains a mild surprise. Given a 2-monad T
on Cat, one's primary interest lies in pseudo maps of algebras, i.e., maps of algebras
in whi
h the stru
ture need only be preserved up to
oherent isomorphism rather than
t
t
t
t
l
r
Theory and Appli
ations of Categories, Vol. 6, No. 7
85
stri
tly. Su
h maps are analysed in [2℄, with an explanation for
omputer s
ientists in [12℄.
One might imagine that, extending the above
orresponden
e in this
ase, pseudo maps
of algebras would
orrespond exa
tly to pseudo natural transformations between the
orresponding models of the
orresponding Lawvere 2-theory. But the position is a little
more deli
ate than that: pseudo natural transformations allow too mu
h
exibility to
yield a bije
tion. However, given a pair of algebras, one still has an equivalen
e between
the
ategory of pseudo maps of algebras between the two algebras and that of pseudo
natural transformations between the
orresponding models of the
orresponding Lawvere
2-theory. Hen
e one has a biequivalen
e between T -Alg and the 2-
ategory of models of
the Lawvere theory, with pseudo natural transformations. We explain that in Se
tion 5.
I am surprised I have not been able to nd our main results, in the
ase of symmetri
V , in the literature on enri
hed
ategories. There has been work on enri
hed monads, for
instan
e [9℄ and [6℄, and there has been work on enri
hed nite limit theories, primarily [8℄
and [5℄. But the
losest work to this of whi
h I am aware is by Dubu
[3℄, and that does not
a
ount for the nitary nature of Lawvere theories, whi
h is
entral for us. Of
ourse, what
we say here is not expli
itly restri
ted to nitariness: one
ould easily extend everything
to
ardinality < for any regular .
The paper is organized as follows. In Se
tion 2, we re
all the basi
fa
ts about enri
hed
ategories we shall use to de ne our terms. In Se
tion 3, we de ne Lawvere V -theories
and their models, and prove that every nitary V -monad on V gives rise to a Lawvere
V -theory with the same V -
ategory of models. And in Se
tion 4, we give the
onverse,
i.e., to ea
h Lawvere V -theory, we dis
over a nitary V -monad with the same V -
ategory
of models. We also prove the equivalen
e between the
ategories of Lawvere V -theories
and nitary V -monads on V . Finally, in Se
tion 5, we explain the more fundamental
pseudo maps of algebras when V = Cat, and show how they relate to maps of Lawvere
2-theories.
For symmetri
V , the standard referen
e for all basi
stru
tures other than monads
is Kelly's book [7℄. For nonsymmetri
V , a reasonable referen
e is [11℄, but the
entral
results were proved in the more general setting of
ategories enri
hed in bi
ategories as
in [4, 5, 6℄. For 2-
ategories, the best relevant referen
e is [2℄, and for an explanation
dire
ted towards
omputer s
ientists, see [12℄.
p
2. Ba
kground on enri
hed
ategories
A monoidal
ategory V is
alled bi
losed if for every obje
t X of V , both
X : VÆ ! VÆ
and X
: VÆ ! VÆ have right adjoints, denoted [X; ℄ and [X; ℄ respe
tively. For
monoidal bi
losed lo
ally small V , it is evident how to de ne V -
ategories , V -fun
tors
and V -natural transformations , yielding the 2-
ategory V -Cat of small V -
ategories. The
ategory VÆ enri
hes to a V -
ategory with hom given by [X; Y ℄ . Note that [X; Y ℄
annot
be repla
ed by [X; Y ℄ here, using the usual
onventions of Kelly's book [7℄. One
an
speak of representable V -fun
tors and there is an elegant theory of V -adjun
tions, see for
instan
e [4℄. If VÆ is
omplete, then it is shown in [7℄ that V Cat is
omplete. So we
an
r
l
r
l
r
Theory and Appli
ations of Categories, Vol. 6, No. 7
86
speak of the Eilenberg-Moore V -
ategory for a V -monad. If V is also
o
omplete, there
is an elegant theory of limits and
olimits in V -
ategories generalising the situation for
symmetri
monoidal
losed V , see for instan
e [5℄. Note that for a V -
ategory C , there is
a
onstru
tion of what should
learly be
alled C , but C is a V -
ategory, not a priori
a V -
ategory.
In general, there is no de nition of a fun
tor V -
ategory. However, if V is
omplete,
for a small V -
ategory C , one does have a fun
tor V -
ategory of the form [C ; V ℄, whose
obje
ts are V -fun
tors from C into V , and with homs given by the usual
onstru
tion
for symmetri
V . Details appear in [5℄, where it is denoted P C . This also agrees with
Street's
onstru
tion [16℄, but the latter is formulated in terms of modules.
A V -monad on a V -
ategory C
onsists of a V -fun
tor T : C ! C and V -natural
transformations : 1 ) T and : T 2 ) T satisfying the usual three axioms. The
Eilenberg-Moore V -
ategory T -Alg has as obje
ts the TÆ -algebras, where TÆ is the ordinary monad underlying T , on the ordinary
ategory CÆ underlying C . Given algebras
(A; a) and (B; b), the hom obje
t T -Alg ((A; a); (B; b)) is the equaliser in V of the diagram
op
op
t
op
op
t
C (A; B )
t
t
C (a; B )
T
R
- C (T (A); B )
(1)
C (T (A); b)
C (T (A); T (B ))
Composition in T -Alg is determined by that in C . Moreover, there is a forgetful V fun
tor U : T -Alg ! C , whi
h has a left adjoint. If C is
omplete, then so is T -Alg , and
U preserves limits.
Given monads T and S on C , a map of monads from T to S is a V -natural transformation : T ) S that
ommutes with the unit and multipli
ation data of the monads.
With the usual
omposition of V -natural transformations, this yields the ordinary
ategory Mnd(C ) of monads on C .
A monoidal bi
losed
ategory V is
alled lo
ally nitely presentable as a
losed
ategory
if VÆ is lo
ally nitely presentable, if the unit I of V is nitely presentable, and if x
y
is nitely presentable whenever x and y are. Hen
eforth, all monoidal bi
losed
ategories
to whi
h we refer will be assumed to be lo
ally nitely presentable as
losed
ategories.
A V -
ategory C is said to have nite tensors if for every nitely presentable obje
t x of
V , and for every obje
t A of C , [x; C (A; )℄ : C ! V is representable, i.e., if there exists
an obje
t x
A of C together with a natural isomorphism [x; C (A; )℄
= C (x
A; ).
The V -
ategory C has nite
otensors if it satis es the dual
ondition that for every
nitely presentable x in V and every A in C , the V -fun
tor [x; C ( ; A)℄ : C
! V is
representable, where V is the dual of V ; i.e., there exists an obje
t A of C together
with a natural isomorphism [ x; C ( ; A) ℄
= C ( ; A ). A V -
ategory C is
o
omplete
whenever CÆ is
o
omplete, C has nite tensors, and x
: CÆ ! CÆ preserves
olimits
for all nitely presentable x. A
o
omplete V -
ategory C is lo
ally nitely presentable if
r
r
t
l
x
t
l
x
op
t
Theory and Appli
ations of Categories, Vol. 6, No. 7
87
is lo
ally nitely presentable, C has nite
otensors, and ( )x : CÆ ! CÆ preserves
ltered
olimits for all nitely presentable x. In general, a V -fun
tor is
alled nitary if
the underlying ordinary fun
tor is so, i.e., if it preserves ltered
olimits. We therefore
denote the full sub
ategory of M nd(C ) determined by the nitary monads on C by
M ndf (C ). In [6℄, nitary monads on an lfp V -
ategory are
hara
terized in terms of
algebrai
stru
ture.
For any V that is lo
ally nitely presentable as a
losed
ategory, V is ne
essarily
a lo
ally nitely presentable V -
ategory. The
otensor AX is given by [X; A℄l . Given a
small nitely
otensored Vt -
ategory C , we write F C (C; Vt) for the full sub-V -
ategory
of [C; Vt ℄ determined by those V -fun
tors that preserve nite
otensors. It follows from
Freyd's adjoint fun
tor theorem, and from the fa
t that the in
lusion preserves
otensors,
that F C (C; Vt ) is a full re
e
tive sub-V-
ategory of [C; Vt ℄, i.e., the in
lusion has a left
adjoint. It follows immediately from the de nitions that the in
lusion is also nitary.
Note that, in all that follows, if V is symmetri
, one may simply disregard all subs
ripts
t, and one will have
orre
t statements, with
orre
t proofs: if V is symmetri
, then the
symmetri
monoidal
ategory Vt is isomorphi
to V , and of
ourse [X; ℄l and [X; ℄r
agree.
CÆ
3. Enri
hed Lawvere theories
We seek a de nition of a Lawvere V -theory. In order to validate that de nition, we shall
prove that the V -
ategory of models of a Lawvere V -theory is nitarily monadi
over
V ; and for any nitary V -monad T on V , the V -
ategory T -Alg is the V -
ategory of
models of a Lawvere theory. More elegantly, we shall establish an equivalen
e between
the ordinary
ategory of nitary V -monads on V and the
ategory of Lawvere V -theories.
First we shall give our de nition of Lawvere V -theory. Observe that the full sub
ategory Vf of V determined by (the isomorphism
lasses of) the nitary obje
ts of V has
nite tensors given as in V and is small.
A Lawvere V -theory
onsists of a small Vt -
ategory T with nite
otensors, together with an identity on obje
ts stri
tly nite
otensor preserving Vt -fun
tor
op
: Vf
!T.
3.1. Definition.
It is immediate that if V = Set, this de nition agrees with the
lassi
al one. We
extend the usual
onvention for Set by informally referring to T as a Lawvere V -theory,
leaving the identity on obje
ts Vt -fun
tor impli
it. A map of Lawvere V -theories from T
to S is a stri
t nite
otensor preserving Vt -fun
tor from T to S that
ommutes with the
op
Vt -fun
tors from Vf . Together with the usual
omposition of Vt -fun
tors, this yields a
ategory we denote by LawV .
We now extend the usual de nition of a model of a Lawvere theory.
A model of a Lawvere V -theory T is a nite
otensor preserving Vt fun
tor from T to Vt . The V-
ategory of models of T is F C (T ; Vt ), the full sub-V -
ategory
of [T ; Vt ℄ determined by the nite
otensor preserving Vt -fun
tors.
3.2. Definition.
Theory and Appli
ations of Categories, Vol. 6, No. 7
88
Let T be a V -monad on V . Then, one
an speak of the Kleisli V -
ategory
K l (T ) for T , and there is a V -fun
tor j : V
! K l(T ) that is the identity on obje
ts,
has a right adjoint, and satis es the usual universal property of Kleisli
onstru
tions, as
explained in Street's arti
le [15℄. Sin
e j has a right adjoint, it preserves tensors. So
if we restri
t j to the nitely presentable obje
ts of V , we have an identity on obje
ts
(stri
tly) nite tensor preserving V -fun
tor from Vf to the full sub-V -
ategory K l(T )f of
K l (T ) determined by the obje
ts of Vf . Dualising, we have a Lawvere V -theory, whi
h
we denote by L(T ). This
onstru
tion extends to a fun
tor L : M nd(V ) ! LawV . Sin
e
our primary interest is in nitary V -monads, we also use L to denote its restri
tion to
M ndf (V ).
3.3. Example.
3.4. Theorem.
-
T Alg
.
Let T be a nitary V -monad on V . Then F C (L(T ); Vt) is equivalent to
roof. There is a
omparison V -fun
tor from K l(T ) to T -Alg , and it preserves tensors
sin
e the
anoni
al fun
tors from V into ea
h of K l(T ) and T -Alg do. By
omposition
with the in
lusion of K l(T )f into K l(T ), we have a V -fun
tor
: K l(T )f ! T -Alg ,
and this yields a V -fun
tor
~ : T -Alg ! [K l(T )op
f ; Vt ℄ sending a T -algebra (A; a) to the
op
Vt -fun
tor T -Alg (
( ); (A; a)). This V -fun
tor fa
tors through F C (K l (T )f ; Vt ) sin
e
preserves nite tensors and sin
e representables preserve nite
otensors. Moreover, it is
fully faithful sin
e every obje
t of T -Alg is a
anoni
al
olimit of a diagram in K l(T )f .
So it remains to show that
~ is essentially surje
tive.
Suppose h : K l(T )op
! Vt preserves nite
otensors. Let A = h(I ), where I is the
f
unit of V . The behaviour of h on all obje
ts is fully determined by its behaviour on I
op
sin
e h preserves nite
otensors and every obje
t of K l(T )op
f , i.e., every obje
t of Vf , is
a nite
otensor of I . The behaviour of h on homs gives, for ea
h nitely presentable x,
a map in V from K l(T )(I ; x) to [Ax ; A℄l , or equivalently, sin
e
otensors in V are given
by a left exponential, and by the usual properties of maps in Kleisli
ategories and maps
from units, a map in V from [x; A℄l
T x to A. This must all be natural in x, thus, by
nitariness of T , we have a map a : T A ! A. Fun
toriality of h, together with the
op
Vt -fun
tor from Vf
into K l(T )op , for
e a to be an algebra map. It is routine to verify
that
~((A; a)) is isomorphi
to h.
P
4. The
onverse
In this se
tion, we start by proving a
onverse to Theorem 3.4. This involves
onstru
ting
a nitary monad M (T ) from a Lawvere V -theory T . Having proved that result, we
establish that L together with M form an equivalen
e of
ategories between the
ategories
of nitary V -monads on V and Lawvere V -theories, for any monoidal bi
losed V that is
lo
ally nitely presentable as a
losed
ategory.
First we re
all Be
k's monadi
ity theorem. Given a fun
tor U : C ! D, a pair of
arrows h1 ; h2 : X ! Y in C is
alled a U -split
oequaliser pair if there exist arrows
h : UY
! Z , i : Z ! U Y , and j : U Y ! U X in D su
h that h U h1 = h U h2 , i
Theory and Appli
ations of Categories, Vol. 6, No. 7
89
splits h, j splits h1 , and U h2 j = i h. It follows that h is a
oequaliser of U h1 and U h2 ,
and that
oequaliser is preserved by all fun
tors. Be
k's theorem says
4.1. Theorem.
A fun
tor
U
:C
U
has a left adjoint
U
re
e
ts isomorphisms, and
C
has and
U
!
D
is monadi
if and only if
preserves
oequalisers of
U
-split
oequaliser pairs.
For a detailed a
ount of Be
k's theorem, see Barr and Wells' book [1℄. Now we are
ready to prove our main result.
Let T be an arbitrary Lawvere V -theory. Then there is a nitary monad
su
h that M (T ) Alg is equivalent to F C (T ; Vt ).
4.2. Theorem.
M
(T ) on
V
roof. Re
all from the de nition of Lawvere V -theory, we have a nite
otensor preserving
op
Vt -fun
tor from Vf
to T . By
omposition, this yields a V -fun
tor U : F C (T ; Vt ) !
op
F C (Vf ; Vt ), but the latter is equivalent to V : one proof of this is by applying Theorem 3.4
to the identity monad. The V -fun
tor U is given by evaluation at I . Moreover, it has a
left adjoint sin
e the in
lusion of F C (T ; Vt ) into [T ; Vt ℄ has a left adjoint, as mentioned
in Se
tion 2, and sin
e evaluation fun
tors from V -presheaf
ategories have left adjoints
given by tensors. Moreover, it is nitary sin
e ( )x preserves ltered
olimits for nitely
presentable x. It remains to show that U is monadi
.
We shall apply Be
k's monadi
ity theorem. It is routine to verify that U re
e
ts
isomorphisms, and one
an readily
al
ulate that U preserves the
oequalisers of U -split
oequaliser pairs: sin
e F C (T ; Vt ) is a full re
e
tive sub-V -
ategory of [T ; Vt ℄, it is
o
omplete; and the required
oequaliser lifts to all obje
ts as they are all nite
otensors
of the unit I , so one need merely apply the asso
iated
otensor to the given map and its
splitting. That proves monadi
ity of the underlying ordinary fun
tor of U ; extending that
to the enri
hed fun
tor follows immediately from the fa
t that U respe
ts
otensors.
P
Our
onstru
tion M extends routinely to a fun
tor M : LawV ! M ndf (V ). Re
all
from the previous se
tion that we also have a fun
tor L : M ndf (V ) ! LawV . In fa
t
we have
The fun
tors M :
form an equivalen
e of
ategories.
4.3. Theorem.
LawV
!
( ) and
M ndf V
L:
( )
M ndf V
!
LawV
roof. Given a nitary V -monad T on V , it follows from Theorem 3.4 and the
onstru
tion of M that M L(T ) is isomorphi
to T . For the
onverse, given a Lawvere V -theory T ,
we need to study the
onstru
tion of M (T ). It is given by taking the left adjoint to the
fun
tor from F C (T ; Vt ) to V given by evaluation at the unit I . The left adjoint, applied
to a nitely presentable obje
t x of V , yields the representable fun
tor T (x; ) : T ! Vt :
this follows sin
e x is a nite tensor of I , and by preservation of nite
otensors. Now
applying Yoneda yields the result that LM (T ) is isomorphi
to T . Thus we are done.
P
Theory and Appli
ations of Categories, Vol. 6, No. 7
5. 2-Monads on
90
Cat
We now restri
t our attention to the
ase that V = Cat. So we
onsider nitary 2-monads
on Cat. An example is that there is a nitary 2-monad for whi
h the algebras are small
monoidal
ategories with a spe
i ed obje
t S , as mentioned in the introdu
tion. Another
nitary 2-monad on Cat has an algebra given by a small
ategory with a monad on it,
again as in the introdu
tion. There are many variants of these examples: see [2℄ or [12℄
for many more examples and more detail.
When one does have a 2-monad T , the maps of primary interest are the pseudo maps
of algebras. These
orrespond to the usual notion of stru
ture preserving fun
tor. They
are de ned as follows.
T -algebras (A; a) and (B; b), a pseudo map of algebras
(A; a) to (B; b)
onsists of a fun
tor f : A ! B and a natural isomorphism
5.1. Definition. Given
TA
a
?
A
Tf
- TB
+f
b
f
- B?
su
h that
T 2A
Ta
?
TA
a
?
A
T 2f -
+T f
TB
Tb
Tf
- T B?
+f
b
f
T 2A
- B?
A
=
?
TA
a
?
A
T 2f -
TB
B
Tf
- T B?
+f
b
f
- B?
from
Theory and Appli
ations of Categories, Vol. 6, No. 7
91
and
A
f
-B
A
A
idA
idB =
?
A
A
and
g
2-
ell
f
between pseudo maps
that respe
ts
f and g.
a
- B?
(f; f)
?
TA
?
A
and
(g; g)
f
-B
B
Tf
- T B?
+f
b
f
- B?
is a natural transformation between
f
By using the
omposition of Cat, it follows that T -algebras, pseudo maps of T -algebras,
and 2-
ells form a 2-
ategory, whi
h we denote by T -Algp. This agrees with the notation
in some of the relevant literature, and this situation has undergone extensive study, in
parti
ular in [2℄; and for an a
ount dire
ted towards
omputer s
ientists, see [12℄.
Similarly, given a small 2-
ategory C , one has the notion of pseudo natural transformation between 2-fun
tors h; k : C ! Cat: one has isomorphisms where the de nition
of natural transformation has
ommuting squares, and those isomorphisms are subje
t to
two
oheren
e
onditions, expressing
oheren
e with respe
t to
omposition and identities in C . Thus, for any Lawvere 2-theory T , we have the 2-
ategory F Cp(T ; Cat), given
by nite
otensor preserving 2-fun
tors, pseudo natural transformations, and the evident
2-
ells. If a nitary 2-monad T
orresponds to the Lawvere 2-theory T , one might guess
that the 2-equivalen
e between T -Alg and F C (T ; Cat) would extend to a 2-equivalen
e
between T -Algp and F Cp(T ; Cat), but it does not!
Let T be the identity 2-monad on Cat. Then T -Algp is 2-equivalent
to Cat. But F Cp(T ; Cat) is not, as it
ontains more maps. First, all fun
tors lie in
F Cp(T ; Cat) via the in
lusion of Cat, whi
h is 2-equivalent to F C (T ; Cat), in F Cp(T ; Cat).
But also, one may vary any
omponent of a pseudo natural transformation , say 2 , by an
isomorphism, leaving every other
omponent xed, and vary the stru
tural isomorphisms
of by
onjugation, and one still has a pseudo natural transformation.
5.2. Example.
So the
orresponden
e we seek is a little more subtle. The notion of 2-equivalen
e between 2-
ategories is quite rare. More
ommonly, one has a biequivalen
e . This amounts
to relaxing fully faithfulness to an equivalen
e on hom
ategories, and similarly for essential surje
tivity. In our
ase, we already have essential surje
tivity. So we
ould ask
whether our 2-fun
tor
~ : T -Alg ! F C (T ; Cat) extends to giving, for ea
h pair of algebras ((A; a); (B; b)), an equivalen
e between T -Algp((A; a); (B; b)) and the
ategory of
Theory and Appli
ations of Categories, Vol. 6, No. 7
92
pseudo natural transformations between
~((A; a)) and
~((B; b)), hen
e a biequivalen
e of
2-
ategories. In fa
t, we have
~:
(T ; C at).
5.3. Theorem. The 2-fun
tor
between
T -Algp
and
F Cp
T -Alg
!
FC
(T ; C at)
extends to a biequivalen
e
Proof. If one routinely follows the argument for essential surje
tivity of
~ as in the proof
of Theorem 3.4, and ones uses the 2-dimensional property of the
olimit de ning T A,
one obtains a bije
tion between pseudo maps of T -algebras from (A; a) to (B; b), and
those pseudo natural transformations between
~((A; a)) and
~((B; b)) that respe
t nite
otensors stri
tly, i.e., up to equality, not just isomorphism. Now, sin
e
otensors are
pseudo limits, they are automati
ally bilimits, and so every pseudo natural transformation
is isomorphi
to one that respe
ts nite
otensors stri
tly. Lo
al fully faithfulness is
straightforward.
Referen
es
[1℄ M. Barr and C. Wells, Toposes, Triples, and Theories, Grundlehren der math.
Wissens
haften 278, Springer-Verlag (1985).
[2℄ R. Bla
kwell, G.M. Kelly, and A.J. Power, Two-dimensional monad theory. J.
Pure Appl. Algebra 59 (1989) 1{41.
[3℄ E. Dubu
, Kan extensions in enri
hed
ategory theory, Le
ture Notes in Math.
145, Springer-Verlag (1970).
[4℄ R. Gordon and A.J. Power, Enri
hment through variation, J. Pure Appl. Algebra
120 (1997) 167{185.
[5℄ R. Gordon and A.J. Power, Gabriel Ulmer duality in
ategories enri
hed in
bi
ategories, J. Pure Appl. Algebra 137 (1999) 29{48.
[6℄ R. Gordon and A.J. Power, Algebrai
stru
ture on bi
ategory enri
hed
ategories, J. Pure Appl. Algebra 130 (1998) 119{132.
[7℄ G.M. Kelly, Basi
on
epts of enri
hed
ategory theory, London Math. So
.
Le
ture Notes Series 64, Cambridge University Press (1982).
[8℄ G.M. Kelly, Stru
tures de ned by nite limits in the enri
hed
ontext 1, Cahiers
de Top et Geom Di erentielle 23 (1980) 3{42.
[9℄ G.M. Kelly and A.J. Power, Adjun
tions whose
ounits are
oequalizers, and
presentations of nitary enri
hed monads, J. Pure Appl. Algebra 89 (1993) 163{
179.
Theory and Appli
ations of Categories, Vol. 6, No. 7
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[10℄ Y. Kinoshita, P.W. O'Hearn, A.J. Power, M. Takeyama, and R.D. Tennent,
An axiomati
approa
h to binary logi
al relations with appli
ations to data
re nement, Pro
Theoreti
al Aspe
ts of Computer S
ien
e, Le
ture Notes in
Computer S
ien
e 1281, Springer (1997) 191{212.
[11℄ Y. Kinoshita and A.J. Power, Lax naturality through enri
hment, J. Pure Appl.
Algebra 112 (1996) 53{72.
[12℄ A.J. Power, Why tri
ategories? Information and Computation 120 (1995) 251{
262.
[13℄ A.J. Power, Modularity in Denotational Semanti
s, Pro
. Mathemati
al Foundations of Programming Semanti
s 97, Ele
troni
Notes in Theoreti
al Computer
S
ien
e 6 (1997).
[14℄ Edmund Robinson, Variations on algebra: monadi
ity and generalisations of
algebrai
theories, Math. Stru
tures in Computer S
ien
e (to appear).
[15℄ Ross Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972) 149{
168.
[16℄ Ross Street, Enri
hed
ategories and
ohomology, Quaestione Math. 6 (1983)
265{283.
Department of Computer S
ien
e
University of Edinburgh
King's Buildings
Edinburgh, EH9 3JZ
S
otland
Email: ajpd
s.ed.a
.uk
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a
ations of Categories,
Vol. 6, No. 7, 1999, pp. 83{93.
ENRICHED LAWVERE THEORIES
Dedi
ated to Jim Lambek
JOHN POWER
ABSTRACT. We de ne the notion of enri
hed Lawvere theory, for enri
hment over
a monoidal bi
losed
ategory V that is lo
ally nitely presentable as a
losed
ategory.
We prove that the
ategory of enri
hed Lawvere theories is equivalent to the
ategory
of nitary monads on V . Moreover, the V -
ategory of models of a Lawvere V -theory
is equivalent to the V -
ategory of algebras for the
orresponding V -monad. This all
extends routinely to lo
al presentability with respe
t to any regular
ardinal. We nally
onsider the spe
ial
ase where V is Cat, and explain how the
orresponden
e extends
to pseudo maps of algebras.
1. Introdu
tion
In seeking a general a
ount of what have been
alled notions of
omputation [13℄, one may
onsider a nitary 2-monad T on C at, and a T -algebra (A; a), then make a
onstru
tion of
a
ategory B and an identity on obje
ts fun
tor j : A ! B . In making that
onstru
tion,
one is allowed to use the stru
ture on A determined by the 2-monad, and that is all. For
instan
e, given the 2-monad for whi
h an algebra is a small
ategory with a monad on it,
then one possible
onstru
tion would be the Kleisli
onstru
tion. For another example,
given the 2-monad for whi
h an algebra is a small monoidal
ategory A together with a
spe
i ed obje
t S , then a possible
onstru
tion is that for whi
h B (a; b) is de ned to be
A(S
a; S
b), with the fun
tor j : A
! B sending a map h to S
h.
We need a pre
ise statement of what we mean by saying that these
onstru
tions only
use stru
ture determined by the 2-monad. In general, one may obtain one su
h de nition
by asserting that B (a; b) must be of the form A(f a; gb) for endofun
tors f and g on A
generated by the 2-monad; similarly for de ning
omposition in B . Thus we need to know
what exa
tly we mean by endofun
tors on A and natural transformations between them
that are generated by the 2-monad. This paper is motivated by a desire to make su
h
notions pre
ise.
In order to make pre
ise the notion of fun
tor generated by a nitary 2-monad T on
a T -algebra (A; a), we rst generalise from
onsideration of a nitary 2-monad on C at
to a nitary V -monad on V for any monoidal bi
losed
ategory that is lo
ally nitely
This work has been done with the support of EPSRC grants GR/J84205: Frameworks for programming language semanti
s and logi
and R34723: The Stru
ture of Programming Languages: syntax and
semanti
s.
Re
eived by the editors 1998 De
ember 14 and, in revised form, 1999 August 27.
Published on 1999 November 30.
1991 Mathemati
s Subje
t Classi
ation: 18C10, 18C15, 18D05.
Key words and phrases: Lawvere theory, monad.
John Power 1999. Permission to
opy for private use granted.
83
Theory and Appli
ations of Categories, Vol. 6, No. 7
84
presentable as a
losed
ategory: we shall make that de nition pre
ise in the next se
tion. We move to this generality primarily for simpli
ity but also be
ause the
omputing
appli
ation will need it later, for instan
e
onsidering not mere
ategories but
ategories
enri
hed in P oset or in the
ategory of ! -
po's.
To support our main de nition, we prove a theorem: we de ne the notion of a Lawvere
V -theory, and we prove that to give a Lawvere V -theory is equivalent to giving a nitary
V -monad on V . So, for a nitary V -monad T on V , there is a Lawvere V -theory L(T ) for
whi
h the V -
ategory of models of L(T ), whi
h we de ne, is equivalent to the V -
ategory
T -Alg of algebras for the monad. Conversely, given any Lawvere V -theory T , there is
a
orresponding nitary V -monad M (T ); and these
onstru
tions yield an equivalen
e
between the
ategory of nitary V -monads on V and that of Lawvere V -theories.
On
e we have su
h a
orresponden
e, we have the de nition we seek: given a 2-monad
T and a T -algebra (A; a), an endofun
tor f on A is generated by T whenever it is in the
image of L(T ) in the model of L(T ) determined by the algebra (A; a). Similarly, a natural
transformation : f ) g is is said to be generated by T whenever it is the image of a
2-
ell of L(T ) in the model of L(T ) determined by the algebra (A; a).
For
ognos
enti of enri
hed
ategories, the de nitions and results here, at least assuming our enri
hment is over a symmetri
monoidal
ategory, should
ome as no great
surprise. However, they are a little subtle. One's rst guess for a de nition of Lawvere
V -theory may be a small V -
ategory with nite produ
ts, subje
t to a
ondition asserting
single-sortedness. But that is too
rude: not only does it not yield our theorem relating
Lawvere theories and nitary monads, as in the
ase for Set (see [1℄), but it does not
agree with the known and very useful equivalen
es between nitary monads and universal
algebra as in [9℄, [6℄, and in appli
ation to
omputation in [11℄ and [10℄, with an a
ount
for
omputer s
ientists in [14℄. So we
onsider something a little more subtle: if V is
symmetri
, we de ne a Lawvere V -theory to be a small V -
ategory with nite
otensors,
subje
t to a single-sortedness
ondition. A model is therefore a nite
otensor preserving
V -fun
tor into V . In order to extend to nonsymmetri
V , as for instan
e if V is the
ategory of small lo
ally ordered
ategories with the lax Gray tensor produ
t [11℄, we need to
introdu
e a twist: so a Lawvere V -theory is a small V -
ategory with nite
otensors, subje
t to a single-sortedness
ondition, and a model is a nite
otensor preserving V -fun
tor
into V : that allows us to speak of the V -
ategory of models, and prove our equivalen
e
with nitary V -monads on V and their V -
ategories of algebras. There has been
onsiderable development of
ategories enri
hed in monoidal bi
losed
ategories [4, 5, 6, 11℄
re
ently: the main problem with extending the usual theory is that fun
tor V -
ategories
need not exist in general; but fun
tor V -
ategories with base V do, and that suÆ
es for
our purposes here. If one is only interested in the
ase of symmetri
V here, one may simply ignore all subs
ripts t, and one will have
orre
t statements; of
ourse, exponentials,
i.e., terms of the form [X; ℄ and [X; ℄ , may also be abbreviated to [X; ℄.
The spe
ial
ase that V = Cat also
ontains a mild surprise. Given a 2-monad T
on Cat, one's primary interest lies in pseudo maps of algebras, i.e., maps of algebras
in whi
h the stru
ture need only be preserved up to
oherent isomorphism rather than
t
t
t
t
l
r
Theory and Appli
ations of Categories, Vol. 6, No. 7
85
stri
tly. Su
h maps are analysed in [2℄, with an explanation for
omputer s
ientists in [12℄.
One might imagine that, extending the above
orresponden
e in this
ase, pseudo maps
of algebras would
orrespond exa
tly to pseudo natural transformations between the
orresponding models of the
orresponding Lawvere 2-theory. But the position is a little
more deli
ate than that: pseudo natural transformations allow too mu
h
exibility to
yield a bije
tion. However, given a pair of algebras, one still has an equivalen
e between
the
ategory of pseudo maps of algebras between the two algebras and that of pseudo
natural transformations between the
orresponding models of the
orresponding Lawvere
2-theory. Hen
e one has a biequivalen
e between T -Alg and the 2-
ategory of models of
the Lawvere theory, with pseudo natural transformations. We explain that in Se
tion 5.
I am surprised I have not been able to nd our main results, in the
ase of symmetri
V , in the literature on enri
hed
ategories. There has been work on enri
hed monads, for
instan
e [9℄ and [6℄, and there has been work on enri
hed nite limit theories, primarily [8℄
and [5℄. But the
losest work to this of whi
h I am aware is by Dubu
[3℄, and that does not
a
ount for the nitary nature of Lawvere theories, whi
h is
entral for us. Of
ourse, what
we say here is not expli
itly restri
ted to nitariness: one
ould easily extend everything
to
ardinality < for any regular .
The paper is organized as follows. In Se
tion 2, we re
all the basi
fa
ts about enri
hed
ategories we shall use to de ne our terms. In Se
tion 3, we de ne Lawvere V -theories
and their models, and prove that every nitary V -monad on V gives rise to a Lawvere
V -theory with the same V -
ategory of models. And in Se
tion 4, we give the
onverse,
i.e., to ea
h Lawvere V -theory, we dis
over a nitary V -monad with the same V -
ategory
of models. We also prove the equivalen
e between the
ategories of Lawvere V -theories
and nitary V -monads on V . Finally, in Se
tion 5, we explain the more fundamental
pseudo maps of algebras when V = Cat, and show how they relate to maps of Lawvere
2-theories.
For symmetri
V , the standard referen
e for all basi
stru
tures other than monads
is Kelly's book [7℄. For nonsymmetri
V , a reasonable referen
e is [11℄, but the
entral
results were proved in the more general setting of
ategories enri
hed in bi
ategories as
in [4, 5, 6℄. For 2-
ategories, the best relevant referen
e is [2℄, and for an explanation
dire
ted towards
omputer s
ientists, see [12℄.
p
2. Ba
kground on enri
hed
ategories
A monoidal
ategory V is
alled bi
losed if for every obje
t X of V , both
X : VÆ ! VÆ
and X
: VÆ ! VÆ have right adjoints, denoted [X; ℄ and [X; ℄ respe
tively. For
monoidal bi
losed lo
ally small V , it is evident how to de ne V -
ategories , V -fun
tors
and V -natural transformations , yielding the 2-
ategory V -Cat of small V -
ategories. The
ategory VÆ enri
hes to a V -
ategory with hom given by [X; Y ℄ . Note that [X; Y ℄
annot
be repla
ed by [X; Y ℄ here, using the usual
onventions of Kelly's book [7℄. One
an
speak of representable V -fun
tors and there is an elegant theory of V -adjun
tions, see for
instan
e [4℄. If VÆ is
omplete, then it is shown in [7℄ that V Cat is
omplete. So we
an
r
l
r
l
r
Theory and Appli
ations of Categories, Vol. 6, No. 7
86
speak of the Eilenberg-Moore V -
ategory for a V -monad. If V is also
o
omplete, there
is an elegant theory of limits and
olimits in V -
ategories generalising the situation for
symmetri
monoidal
losed V , see for instan
e [5℄. Note that for a V -
ategory C , there is
a
onstru
tion of what should
learly be
alled C , but C is a V -
ategory, not a priori
a V -
ategory.
In general, there is no de nition of a fun
tor V -
ategory. However, if V is
omplete,
for a small V -
ategory C , one does have a fun
tor V -
ategory of the form [C ; V ℄, whose
obje
ts are V -fun
tors from C into V , and with homs given by the usual
onstru
tion
for symmetri
V . Details appear in [5℄, where it is denoted P C . This also agrees with
Street's
onstru
tion [16℄, but the latter is formulated in terms of modules.
A V -monad on a V -
ategory C
onsists of a V -fun
tor T : C ! C and V -natural
transformations : 1 ) T and : T 2 ) T satisfying the usual three axioms. The
Eilenberg-Moore V -
ategory T -Alg has as obje
ts the TÆ -algebras, where TÆ is the ordinary monad underlying T , on the ordinary
ategory CÆ underlying C . Given algebras
(A; a) and (B; b), the hom obje
t T -Alg ((A; a); (B; b)) is the equaliser in V of the diagram
op
op
t
op
op
t
C (A; B )
t
t
C (a; B )
T
R
- C (T (A); B )
(1)
C (T (A); b)
C (T (A); T (B ))
Composition in T -Alg is determined by that in C . Moreover, there is a forgetful V fun
tor U : T -Alg ! C , whi
h has a left adjoint. If C is
omplete, then so is T -Alg , and
U preserves limits.
Given monads T and S on C , a map of monads from T to S is a V -natural transformation : T ) S that
ommutes with the unit and multipli
ation data of the monads.
With the usual
omposition of V -natural transformations, this yields the ordinary
ategory Mnd(C ) of monads on C .
A monoidal bi
losed
ategory V is
alled lo
ally nitely presentable as a
losed
ategory
if VÆ is lo
ally nitely presentable, if the unit I of V is nitely presentable, and if x
y
is nitely presentable whenever x and y are. Hen
eforth, all monoidal bi
losed
ategories
to whi
h we refer will be assumed to be lo
ally nitely presentable as
losed
ategories.
A V -
ategory C is said to have nite tensors if for every nitely presentable obje
t x of
V , and for every obje
t A of C , [x; C (A; )℄ : C ! V is representable, i.e., if there exists
an obje
t x
A of C together with a natural isomorphism [x; C (A; )℄
= C (x
A; ).
The V -
ategory C has nite
otensors if it satis es the dual
ondition that for every
nitely presentable x in V and every A in C , the V -fun
tor [x; C ( ; A)℄ : C
! V is
representable, where V is the dual of V ; i.e., there exists an obje
t A of C together
with a natural isomorphism [ x; C ( ; A) ℄
= C ( ; A ). A V -
ategory C is
o
omplete
whenever CÆ is
o
omplete, C has nite tensors, and x
: CÆ ! CÆ preserves
olimits
for all nitely presentable x. A
o
omplete V -
ategory C is lo
ally nitely presentable if
r
r
t
l
x
t
l
x
op
t
Theory and Appli
ations of Categories, Vol. 6, No. 7
87
is lo
ally nitely presentable, C has nite
otensors, and ( )x : CÆ ! CÆ preserves
ltered
olimits for all nitely presentable x. In general, a V -fun
tor is
alled nitary if
the underlying ordinary fun
tor is so, i.e., if it preserves ltered
olimits. We therefore
denote the full sub
ategory of M nd(C ) determined by the nitary monads on C by
M ndf (C ). In [6℄, nitary monads on an lfp V -
ategory are
hara
terized in terms of
algebrai
stru
ture.
For any V that is lo
ally nitely presentable as a
losed
ategory, V is ne
essarily
a lo
ally nitely presentable V -
ategory. The
otensor AX is given by [X; A℄l . Given a
small nitely
otensored Vt -
ategory C , we write F C (C; Vt) for the full sub-V -
ategory
of [C; Vt ℄ determined by those V -fun
tors that preserve nite
otensors. It follows from
Freyd's adjoint fun
tor theorem, and from the fa
t that the in
lusion preserves
otensors,
that F C (C; Vt ) is a full re
e
tive sub-V-
ategory of [C; Vt ℄, i.e., the in
lusion has a left
adjoint. It follows immediately from the de nitions that the in
lusion is also nitary.
Note that, in all that follows, if V is symmetri
, one may simply disregard all subs
ripts
t, and one will have
orre
t statements, with
orre
t proofs: if V is symmetri
, then the
symmetri
monoidal
ategory Vt is isomorphi
to V , and of
ourse [X; ℄l and [X; ℄r
agree.
CÆ
3. Enri
hed Lawvere theories
We seek a de nition of a Lawvere V -theory. In order to validate that de nition, we shall
prove that the V -
ategory of models of a Lawvere V -theory is nitarily monadi
over
V ; and for any nitary V -monad T on V , the V -
ategory T -Alg is the V -
ategory of
models of a Lawvere theory. More elegantly, we shall establish an equivalen
e between
the ordinary
ategory of nitary V -monads on V and the
ategory of Lawvere V -theories.
First we shall give our de nition of Lawvere V -theory. Observe that the full sub
ategory Vf of V determined by (the isomorphism
lasses of) the nitary obje
ts of V has
nite tensors given as in V and is small.
A Lawvere V -theory
onsists of a small Vt -
ategory T with nite
otensors, together with an identity on obje
ts stri
tly nite
otensor preserving Vt -fun
tor
op
: Vf
!T.
3.1. Definition.
It is immediate that if V = Set, this de nition agrees with the
lassi
al one. We
extend the usual
onvention for Set by informally referring to T as a Lawvere V -theory,
leaving the identity on obje
ts Vt -fun
tor impli
it. A map of Lawvere V -theories from T
to S is a stri
t nite
otensor preserving Vt -fun
tor from T to S that
ommutes with the
op
Vt -fun
tors from Vf . Together with the usual
omposition of Vt -fun
tors, this yields a
ategory we denote by LawV .
We now extend the usual de nition of a model of a Lawvere theory.
A model of a Lawvere V -theory T is a nite
otensor preserving Vt fun
tor from T to Vt . The V-
ategory of models of T is F C (T ; Vt ), the full sub-V -
ategory
of [T ; Vt ℄ determined by the nite
otensor preserving Vt -fun
tors.
3.2. Definition.
Theory and Appli
ations of Categories, Vol. 6, No. 7
88
Let T be a V -monad on V . Then, one
an speak of the Kleisli V -
ategory
K l (T ) for T , and there is a V -fun
tor j : V
! K l(T ) that is the identity on obje
ts,
has a right adjoint, and satis es the usual universal property of Kleisli
onstru
tions, as
explained in Street's arti
le [15℄. Sin
e j has a right adjoint, it preserves tensors. So
if we restri
t j to the nitely presentable obje
ts of V , we have an identity on obje
ts
(stri
tly) nite tensor preserving V -fun
tor from Vf to the full sub-V -
ategory K l(T )f of
K l (T ) determined by the obje
ts of Vf . Dualising, we have a Lawvere V -theory, whi
h
we denote by L(T ). This
onstru
tion extends to a fun
tor L : M nd(V ) ! LawV . Sin
e
our primary interest is in nitary V -monads, we also use L to denote its restri
tion to
M ndf (V ).
3.3. Example.
3.4. Theorem.
-
T Alg
.
Let T be a nitary V -monad on V . Then F C (L(T ); Vt) is equivalent to
roof. There is a
omparison V -fun
tor from K l(T ) to T -Alg , and it preserves tensors
sin
e the
anoni
al fun
tors from V into ea
h of K l(T ) and T -Alg do. By
omposition
with the in
lusion of K l(T )f into K l(T ), we have a V -fun
tor
: K l(T )f ! T -Alg ,
and this yields a V -fun
tor
~ : T -Alg ! [K l(T )op
f ; Vt ℄ sending a T -algebra (A; a) to the
op
Vt -fun
tor T -Alg (
( ); (A; a)). This V -fun
tor fa
tors through F C (K l (T )f ; Vt ) sin
e
preserves nite tensors and sin
e representables preserve nite
otensors. Moreover, it is
fully faithful sin
e every obje
t of T -Alg is a
anoni
al
olimit of a diagram in K l(T )f .
So it remains to show that
~ is essentially surje
tive.
Suppose h : K l(T )op
! Vt preserves nite
otensors. Let A = h(I ), where I is the
f
unit of V . The behaviour of h on all obje
ts is fully determined by its behaviour on I
op
sin
e h preserves nite
otensors and every obje
t of K l(T )op
f , i.e., every obje
t of Vf , is
a nite
otensor of I . The behaviour of h on homs gives, for ea
h nitely presentable x,
a map in V from K l(T )(I ; x) to [Ax ; A℄l , or equivalently, sin
e
otensors in V are given
by a left exponential, and by the usual properties of maps in Kleisli
ategories and maps
from units, a map in V from [x; A℄l
T x to A. This must all be natural in x, thus, by
nitariness of T , we have a map a : T A ! A. Fun
toriality of h, together with the
op
Vt -fun
tor from Vf
into K l(T )op , for
e a to be an algebra map. It is routine to verify
that
~((A; a)) is isomorphi
to h.
P
4. The
onverse
In this se
tion, we start by proving a
onverse to Theorem 3.4. This involves
onstru
ting
a nitary monad M (T ) from a Lawvere V -theory T . Having proved that result, we
establish that L together with M form an equivalen
e of
ategories between the
ategories
of nitary V -monads on V and Lawvere V -theories, for any monoidal bi
losed V that is
lo
ally nitely presentable as a
losed
ategory.
First we re
all Be
k's monadi
ity theorem. Given a fun
tor U : C ! D, a pair of
arrows h1 ; h2 : X ! Y in C is
alled a U -split
oequaliser pair if there exist arrows
h : UY
! Z , i : Z ! U Y , and j : U Y ! U X in D su
h that h U h1 = h U h2 , i
Theory and Appli
ations of Categories, Vol. 6, No. 7
89
splits h, j splits h1 , and U h2 j = i h. It follows that h is a
oequaliser of U h1 and U h2 ,
and that
oequaliser is preserved by all fun
tors. Be
k's theorem says
4.1. Theorem.
A fun
tor
U
:C
U
has a left adjoint
U
re
e
ts isomorphisms, and
C
has and
U
!
D
is monadi
if and only if
preserves
oequalisers of
U
-split
oequaliser pairs.
For a detailed a
ount of Be
k's theorem, see Barr and Wells' book [1℄. Now we are
ready to prove our main result.
Let T be an arbitrary Lawvere V -theory. Then there is a nitary monad
su
h that M (T ) Alg is equivalent to F C (T ; Vt ).
4.2. Theorem.
M
(T ) on
V
roof. Re
all from the de nition of Lawvere V -theory, we have a nite
otensor preserving
op
Vt -fun
tor from Vf
to T . By
omposition, this yields a V -fun
tor U : F C (T ; Vt ) !
op
F C (Vf ; Vt ), but the latter is equivalent to V : one proof of this is by applying Theorem 3.4
to the identity monad. The V -fun
tor U is given by evaluation at I . Moreover, it has a
left adjoint sin
e the in
lusion of F C (T ; Vt ) into [T ; Vt ℄ has a left adjoint, as mentioned
in Se
tion 2, and sin
e evaluation fun
tors from V -presheaf
ategories have left adjoints
given by tensors. Moreover, it is nitary sin
e ( )x preserves ltered
olimits for nitely
presentable x. It remains to show that U is monadi
.
We shall apply Be
k's monadi
ity theorem. It is routine to verify that U re
e
ts
isomorphisms, and one
an readily
al
ulate that U preserves the
oequalisers of U -split
oequaliser pairs: sin
e F C (T ; Vt ) is a full re
e
tive sub-V -
ategory of [T ; Vt ℄, it is
o
omplete; and the required
oequaliser lifts to all obje
ts as they are all nite
otensors
of the unit I , so one need merely apply the asso
iated
otensor to the given map and its
splitting. That proves monadi
ity of the underlying ordinary fun
tor of U ; extending that
to the enri
hed fun
tor follows immediately from the fa
t that U respe
ts
otensors.
P
Our
onstru
tion M extends routinely to a fun
tor M : LawV ! M ndf (V ). Re
all
from the previous se
tion that we also have a fun
tor L : M ndf (V ) ! LawV . In fa
t
we have
The fun
tors M :
form an equivalen
e of
ategories.
4.3. Theorem.
LawV
!
( ) and
M ndf V
L:
( )
M ndf V
!
LawV
roof. Given a nitary V -monad T on V , it follows from Theorem 3.4 and the
onstru
tion of M that M L(T ) is isomorphi
to T . For the
onverse, given a Lawvere V -theory T ,
we need to study the
onstru
tion of M (T ). It is given by taking the left adjoint to the
fun
tor from F C (T ; Vt ) to V given by evaluation at the unit I . The left adjoint, applied
to a nitely presentable obje
t x of V , yields the representable fun
tor T (x; ) : T ! Vt :
this follows sin
e x is a nite tensor of I , and by preservation of nite
otensors. Now
applying Yoneda yields the result that LM (T ) is isomorphi
to T . Thus we are done.
P
Theory and Appli
ations of Categories, Vol. 6, No. 7
5. 2-Monads on
90
Cat
We now restri
t our attention to the
ase that V = Cat. So we
onsider nitary 2-monads
on Cat. An example is that there is a nitary 2-monad for whi
h the algebras are small
monoidal
ategories with a spe
i ed obje
t S , as mentioned in the introdu
tion. Another
nitary 2-monad on Cat has an algebra given by a small
ategory with a monad on it,
again as in the introdu
tion. There are many variants of these examples: see [2℄ or [12℄
for many more examples and more detail.
When one does have a 2-monad T , the maps of primary interest are the pseudo maps
of algebras. These
orrespond to the usual notion of stru
ture preserving fun
tor. They
are de ned as follows.
T -algebras (A; a) and (B; b), a pseudo map of algebras
(A; a) to (B; b)
onsists of a fun
tor f : A ! B and a natural isomorphism
5.1. Definition. Given
TA
a
?
A
Tf
- TB
+f
b
f
- B?
su
h that
T 2A
Ta
?
TA
a
?
A
T 2f -
+T f
TB
Tb
Tf
- T B?
+f
b
f
T 2A
- B?
A
=
?
TA
a
?
A
T 2f -
TB
B
Tf
- T B?
+f
b
f
- B?
from
Theory and Appli
ations of Categories, Vol. 6, No. 7
91
and
A
f
-B
A
A
idA
idB =
?
A
A
and
g
2-
ell
f
between pseudo maps
that respe
ts
f and g.
a
- B?
(f; f)
?
TA
?
A
and
(g; g)
f
-B
B
Tf
- T B?
+f
b
f
- B?
is a natural transformation between
f
By using the
omposition of Cat, it follows that T -algebras, pseudo maps of T -algebras,
and 2-
ells form a 2-
ategory, whi
h we denote by T -Algp. This agrees with the notation
in some of the relevant literature, and this situation has undergone extensive study, in
parti
ular in [2℄; and for an a
ount dire
ted towards
omputer s
ientists, see [12℄.
Similarly, given a small 2-
ategory C , one has the notion of pseudo natural transformation between 2-fun
tors h; k : C ! Cat: one has isomorphisms where the de nition
of natural transformation has
ommuting squares, and those isomorphisms are subje
t to
two
oheren
e
onditions, expressing
oheren
e with respe
t to
omposition and identities in C . Thus, for any Lawvere 2-theory T , we have the 2-
ategory F Cp(T ; Cat), given
by nite
otensor preserving 2-fun
tors, pseudo natural transformations, and the evident
2-
ells. If a nitary 2-monad T
orresponds to the Lawvere 2-theory T , one might guess
that the 2-equivalen
e between T -Alg and F C (T ; Cat) would extend to a 2-equivalen
e
between T -Algp and F Cp(T ; Cat), but it does not!
Let T be the identity 2-monad on Cat. Then T -Algp is 2-equivalent
to Cat. But F Cp(T ; Cat) is not, as it
ontains more maps. First, all fun
tors lie in
F Cp(T ; Cat) via the in
lusion of Cat, whi
h is 2-equivalent to F C (T ; Cat), in F Cp(T ; Cat).
But also, one may vary any
omponent of a pseudo natural transformation , say 2 , by an
isomorphism, leaving every other
omponent xed, and vary the stru
tural isomorphisms
of by
onjugation, and one still has a pseudo natural transformation.
5.2. Example.
So the
orresponden
e we seek is a little more subtle. The notion of 2-equivalen
e between 2-
ategories is quite rare. More
ommonly, one has a biequivalen
e . This amounts
to relaxing fully faithfulness to an equivalen
e on hom
ategories, and similarly for essential surje
tivity. In our
ase, we already have essential surje
tivity. So we
ould ask
whether our 2-fun
tor
~ : T -Alg ! F C (T ; Cat) extends to giving, for ea
h pair of algebras ((A; a); (B; b)), an equivalen
e between T -Algp((A; a); (B; b)) and the
ategory of
Theory and Appli
ations of Categories, Vol. 6, No. 7
92
pseudo natural transformations between
~((A; a)) and
~((B; b)), hen
e a biequivalen
e of
2-
ategories. In fa
t, we have
~:
(T ; C at).
5.3. Theorem. The 2-fun
tor
between
T -Algp
and
F Cp
T -Alg
!
FC
(T ; C at)
extends to a biequivalen
e
Proof. If one routinely follows the argument for essential surje
tivity of
~ as in the proof
of Theorem 3.4, and ones uses the 2-dimensional property of the
olimit de ning T A,
one obtains a bije
tion between pseudo maps of T -algebras from (A; a) to (B; b), and
those pseudo natural transformations between
~((A; a)) and
~((B; b)) that respe
t nite
otensors stri
tly, i.e., up to equality, not just isomorphism. Now, sin
e
otensors are
pseudo limits, they are automati
ally bilimits, and so every pseudo natural transformation
is isomorphi
to one that respe
ts nite
otensors stri
tly. Lo
al fully faithfulness is
straightforward.
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es
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Wissens
haften 278, Springer-Verlag (1985).
[2℄ R. Bla
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Pure Appl. Algebra 59 (1989) 1{41.
[3℄ E. Dubu
, Kan extensions in enri
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ategory theory, Le
ture Notes in Math.
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[4℄ R. Gordon and A.J. Power, Enri
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120 (1997) 167{185.
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bi
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ategory theory, London Math. So
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[10℄ Y. Kinoshita, P.W. O'Hearn, A.J. Power, M. Takeyama, and R.D. Tennent,
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Department of Computer S
ien
e
University of Edinburgh
King's Buildings
Edinburgh, EH9 3JZ
S
otland
Email: ajpd
s.ed.a
.uk
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am.a
.uk
G. Max Kelly, University of Sydney: maxkmaths.usyd.edu.au
Anders Ko
k, University of Aarhus: ko
kimf.au.dk
F. William Lawvere, State University of New York at Bu alo: wlawverea
su.buffalo.edu
Jean-Louis Loday, Universite de Strasbourg: lodaymath.u-strasbg.fr
Ieke Moerdijk, University of Utre
ht: moerdijkmath.ruu.nl
Susan Nie eld, Union College: niefielsunion.edu
Robert Pare, Dalhousie University: parems
s.dal.
a
Andrew Pitts, University of Cambridge: ap
l.
am.a
.uk
Robert Rosebrugh, Mount Allison University: rrosebrughmta.
a
Jiri Rosi
ky, Masaryk University: rosi
kymath.muni.
z
James Stashe , University of North Carolina: jds
harlie.math.un
.edu
Ross Street, Ma
quarie University: streetmath.mq.edu.au
Walter Tholen, York University: tholenmathstat.yorku.
a
Myles Tierney, Rutgers University: tierneymath.rutgers.edu
Robert F. C. Walters, University of Sydney: walters bmaths.usyd.edu.au
R. J. Wood, Dalhousie University: rjwoodms
s.dal.
a