I ntrnat. J. lath. & ath. Sci.
(1982) 61-85
Vol. 5 No.
61
ALGEBRAS WITH ACTIONS AND AUTOMATA
W. KIHNEL and M. PFENDER
Fachbereich Mathematik
TU Berlin, Str. d. !7. Juni 135
iOOO Berlin !2, Germany
J. MESEGUER and I. SOLS
Mathematics Department
University of California
Berkeley, California 94720 U.S.A.
(Received February ]8,
|980)
In the present paper we want to give a common structure
ABSTRACT.
theory of left action,
group operations, R-modules and automata of
different types defined over various kinds of carrier objects:
graphs, presheaves,
sheaves,
sets,
topological spaces (in particular:
compactly generated Hausdorff spaces). The first section gives an
axiomatic approach to algebraic structures relative to a base
category B,
functors.
slightly more powerful than that of monadic (tripleable)
In section 2 we generalize Lawveres functorial semantics
to many-sorted algebras over cartesian closed
categories.
In section 3 we treat the structures mentioned in the beginning as
many-sorted algebras with fixed "scalar" or "input" object and show
that
they still have an algebraic (or monadic)
forgetful functor
(theorem 3.3) and hence the general theory of algebraic structures
applies. These structures were usually treated as one-sorted in the
Lawvere-setting, the action being expressed by a family of unary
operations indexed over the scalars. But this approach cannot, as
the one developed here,
describe continuity of the action (more
62
W. KUHNEL, M. PFENDER, J. MESEGUER, AND I. SOLS
general: the action to be a B-morphism), which is essential for the
e.g. modules for a sheaf of rings or
structures mentioned above,
topological automata. Finally we discuss consequences of
theorem 3.3 for the structure theory of various types of automata.
The particular case of algebras with fixed "natural numbers object"
has been studied by the authors in
KEY WORDS AND PHRASES.
[23].
Algebras, actions, automata, algebraic
functor.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. O8AO5, 18C15,
(ALGEBRAIC FUNCTORS)
INTRODUCTION
The following paper is directed to specialists.
hand
68DOSp
Customary short-
theorems, and arguments are freely used.
Triple theory (cf.
[], [2]VI, [3]3) showed
that the notion of
"algebraic structures" should be viewed as a relative notion:
structures of a given species are algebraic over some other category,
and this is best described in terms of the forgetful functor
from some category K of "algebras" into a "base" cate-
gory B.
We say U is tripleable or monadic (cf. references above).
We will use only the axiomatic properties of monadic functors.
].|
PAR-CRITERION
(cf.
[2]VI.7)
i.e.
U creates absolute coequalizers,
U has a left adjoint F, and
coequalizers preserved by any
functor.
We will use as our basic notion the following slightly stronger one:
.2 DEFINITION AND CHARACTERIZATION.
A functor U
K
B is
algebraic, if
(i)
U has a left adjoint F
(ii)
U creates
(iii)
U creates coequalizers of U-kernel pairs
(inverse) limits
into kernel pairs),
i.e.
(K-pairs mapped by U
quotients by congruences can be
63
ALGEBRAS WITH ACTIONS AND AUTOMATA
calculated downstairs.
For B finitely complete, algebraic functors are just monadic ones
which satisfy (iii). This follows from the characterization of
monadic functors in [4]3.3.
U is of finite rank if
For B
1.3 REMARK.
the
(iv) U creates filtered colimits.
Sets, "monadic" and "algebraic" are
same, again by [4], 3.3,
since in Sets coequalizers are
retractions. As we will see in section 2,
tures,
i.e.
"usual" algebraic struc-
those with operations of finite arity (in particular
L a w v e r e
finite rank,
type algebras) have algebraic forgetful
functors of
if B has enough properties of Sets.
We prefer "algebraic" to "monadic" because of the following quite
important structure theorems and because generation of congruences,
isomorphism theorems, Z a s s e n h a u s
H 8 1 d e r
lemma, J o r d a n-
complete well-powered and cocomplete,
K
Let U
LIFTING THEOREM.
complete, well-powered,and cocomplete,
see
[5], [6].
B be algebraic.
then so
is K.
kernel pairs and a coequalizer-mono factorization,
PROOF.
B into B
K
theorem hold for K with algebraic U
If B is
If B has
then so has K.
Since U creates limits, completeness of B implies
completeness of K.
For a B-monomorphism
phism
K’
m’ =K
B
m
such that
UK there is at most one K-(mono)mor-
Um’
m:
If such an
m’ exists, then
the
left column in
KP(U(gKoFm))
I
UFB
UK =B
UgK’
U(KP(gKoFm))
UFm
m
--
UFUK
is exact,
i.e.
a
kernel pair- co-
equalizer diagram.
UK
is uniquel determined by B since m is mono, and, being a
retraction, UgK’ is a coequalizer of its kernel pair.Since U creates
W.
64
KHNEL,
M. PFENDER, J. MESEGUER, AND I. SOLS
coequalizers of U-kernel pairs, especially of KP(gKFm), there is
a unique FB
FB
gK’
K’ mapped
K’
w"
UgK’.B,
by U into UFB
So m admits at most one
m’
K’
namely
K with Um’
m,
hence B well-powered implies K well powered.
Cocompleteness: Since U creates coequalizers of
(U-)kernel pairs,
and B has coequalizers of kernel pairs, K has them too.
A general theorem (cf.
trary pairs
A
_K
g
[7] 3.53) then gives coequalizers of arbi-
in K:
take the coequalizer of
the intersection
over the set (K well-powered) of all kernel pairs on K through
which
factors, this intersection being a kernel pair.
g
Coproducts in K may now be constructed as follows (cf.
the well-
known "free product"-construction for groups):
KP K
i
i
FUK.
K.
i
F in
R
]
lji
i
F( UKi) -F(_UKi) /
I
I
F in.
R is the kernel pair generated by the pairs
KP(Ki)
By construction of R, there is a unique j. making the right hand
rectangle commute. The
Ji
F(IUKi
I
R
make
into a coproduct of the
verification being a simple diagram chase. This shows that K
K.,
i
is cocomplete.
ments
show:
By replacing
by other colimits,
the same argu-
For B complete and well-powered, K is eocomplete at
least as far as B is.
Factorization: For K
f
K’
UK
let
q
D>
m
UK
be the coequalizer-mono factorization of Uf.
q
coequ(KP(Uf))
coequ(U(KPf)), since U creates kernel pairs.
By U algebraic, there exists a coequalizer
and hence a factorization K
C
m--
K
i
of KPf over q
K over that of Uf. m is mono,
65
ALGEBRAS WITH ACTIONS AND AUTOMATA
since monos are characterized by the pair of identities to be
q.e.d.
kernel pair, and U creates kernel pairs,
L i n t o n
There is no assumption on K in the theorem.
in [8], corollary 2 cocompleteness of K,
already has coequalizers of reflexive
1.5
PRESENTATION
generators) together with
K-object,
[8]).
For B complete and well-
K---- B algebraic, any B E B ("set" of
G----UFB
P2
the coequalizer
i.e.
if U is monadic and K
pairs.
OF ALGEBRAS.
powered with coequalizers, U
shows
("relations") presents a
FB/G of
(in K relative U,
PI’P2
KEK has a presentation, namely
gK
K.
UKP(EK)EE FUK
cf.
Each
PROOF.
The intersection
P!
P2
2
of all kernel pairs on FB
FB
(in B) exists (K well-powered and
through which G--------UFB factors
complete by ;.4) and is a (U-)kernel pair [7],
monadic,
ey.:qts and has
FB/G =: FB/G
FB
Second part of the assertion:
erty.
U
3.5). Hence, by U
the desired prop-
K is a retraction
(for NUK), hence coequalizer of its kernel pair UKP(gK), so
a coequalizer relative U of UKP(gK),
1.6
COMPOSITION THEOREM.
because U is
If in K
B
faithful,
C
gK
is
q.e.d.
U and V are
algebraic [of finite rank] then so is their composition V U
PROOF’.
VU is right adjoint as composition of right adjoints,
and creates
limits, because U and V do (create them in two steps).
Now let
VUK
be a VU-kernel pair and
C
coequ(VUPo
VUp
(Up o ,Up
being a V-kernel pair
(trivial), V algebraic implies unique existence of UK
with
V
q,
and
coequ(UPo, Up
q
). Since V creates limits,
in B
W. KUHNEL, M. PFENDER, J. MESEGUER, AND I. SOLS
66
especially kernel pairs,
so
(po,pl)
q
VU
,
=q), and
VU
If U and V create filtered colimits,
them in two
create
(hence
with respect to
Uniqueness of
verified stepwise.
does:
U
C in K such that
coequ(p o,p| ).
is a kernel pair in B of
By U algebraic, there exists a
is a U-kernel pair.
K
unique
(Up o ,Up
steps.
So VU is of finite rank.
q is
then VU
q.e.d.
Composition of monadic functors is not monadic in general. The
Graph and Graph
Cat
forgetful functors
their composition is not,
Sets
2
are monadic,
since images of functors need not to be
(sub)categories.
TRIANGLE THEOREM.
If
K
U
B.V_ C, K has coequal-
VU, then if V and W are monadic, so is U.
izers, and W
If C is complete,
cocomplete and well-powered and if V and W are
algebraic [of finite rank], then so is U.
U right adjoint follows from [9]
PROOF.
theorem
(adjoint
triangles)
U
K
B
is a derivable triangle,
K
has
since K and therefore
the needed coequalizers
C
FV
and
"-
FVe
id
coequ(FVFV
B
FV
FV)
by V being monadic.
By
P a r
’s
monadicity criterion we need to show (for the first
part of our assertion)
that U creates absolute coequalizers.
But since V preserves absolute coequalizers (as such, any functor
does), U creates them.
Second assertion:
It is easy to verify that a functor (e.g. V,W)
into a category having limits,colimits of a given kind which creates
these
limits, colimits respectively, preserves them. This proves
the second part of the assertion"
U creates "things" because
67
ALGEBRAS WITH ACTIONS AND AUTOMATA
V preserves them, W creates them, and U maps them into the
original things, because V creates (uniqueness). U right adjoint
holds by the first assertion since the lifting theorem provides
all coequalizers in K via W.
q.e.d.
(full) subcategory is equationally defined, we will
Each time a
use the following theorem for proving the inclusion functor to be
algebraic.
1.8
BIRKHOFF-INCLUSION THEOREM.
For complete, well-
powered K having coequalizers and a coequalizer-mono factorization,
a full
I
inclusion
L
---->
K
is reflective with coequalizers as
reflections (regular epi-reflective), if L is closed under products
and
(mono-)subobjects. It is algebraic,
if, in addition, L is
closed under coequalizers of kernel pairs,
B i r k h o f f
M
subcategory..
Apply the non-numerated "Satz" on page 96 of [10]
PROOF.
taking
it is a
i.e.
class of all K-monos,
class of all regular epis
(coequalizers). K is E-cowellpowered, since it is well-powered
(bijective correspondence between kernel pairs and (regular)
quotients) and satisfies the diagonal-condition for
E,M
since
every regular epi is coequalizer of its kernel pair.
Explicit construc.
of the reflector:
take out of a representative family
Given KEK,
(one representant
f.
for each iso-class) of regular quotients the family K
of
those quotients
rization
K
(fi)
h L.EL.
I
Lo
Then tile regular epi-mono
I
Li
facto-
gives the regular epi reflection
RK
K
K
RK.
The second assertion is obvious by definition of
"algebraic".
68
2.
W. KUHNEL, M. PFENDER, J. MESEGUER, AND I. SOLS
MANY
SORTED ALGEBRAS
This section generalizes two principal results of
sorted
[11]
to many-
(heterogeneous) algebras over cartesian closed categories:
"theories are algebras" (theorem 2.3) and "models of algebraic
theories have algebraic forgetful functors of finite rank"
(theorem 2.6).
The
case of Sets as base
and
[14]. A further generalization
category has been treated in
to
monoidal categories is given in [15]
in the unpublished [16]
2.1
the case of
I-sorted algebraic type
(closed)
(one-sorted algebras) and
(many-sorted algebras).
For a set I, YESets
DEFINITIONS:
[12], [13],
or
Ix I
scheme of operators
is called an
(cf.
I may be seen as a directed graph with node set I*;
12],
13]).
YEGraphl.
which has only arrows with codomain in I (coarity I).
A
l-model in a category B with
is a graph-morphism M: l--- B
I*A
i
I...i n
MA
and given binary product
which preserves products:
Mi1...Mi n
0
Example:
I
{R,A},
l:
(...(MiMi2)..)xMi n
(C)R =
A
(where we wrote RxA instead of
is the scheme of left modules
RAffI*)
+
RR
AxA
RxA
(over varying rings). Modules are
l-models satisfying the usual equations.
ALGEBRAS WITH ACTIONS AND AUTOMATA
f
6 Sets
(fi)i61
M’
M
f
l-homomorphism
A
I
((Mi),
69
is a family
(M’i))
compatible with the operations. For each A
i
in Y
i
n
fi.
J=!
MA
M’A
()
Mi
M’i
fi
This defines a category Mod(/,B) with forgetful functor
Mod(Y,B)
U
B
I
M
(Mi)
i61
2.2 THEOREM:
(i)
U
Mod(Y,B)
B
I
has a left adjoint,
B- preserves coproducts
(and hence
if B has and if
distributes over
coproducts)
(ii)
U
(iii)
U creates absolute coequalizers
Mod(l,B)
B
creates
limits
(B arbitrary)
(B arbitrary)
(iv) U creates coequalizers of U-kernel pairs, if
simulta-
neously preserves coequalizers of kernel pairs,
if B is cartesian closed and regular epis
in particular
(coequalizers) are
closed under composition.
(v)
U creates filtered colimits,
if B- preserves them (in
particular if B is cartesian closed).
The proof is a purely formal extension of
15]. For B
PROOF.
of
that of
Sets, (i) and (iv) are proved in
For B =Sets,
these word algebras
general case is suggested.
12] and
13].
[]2] and [13] prove (i) by construction
(free) word algebras over given (Bi) 6 Sets I
structure of
theorem .4 in
By analysis of the
the following proof
in the
W. KUHNEL, M. PFENDER, J. MESEGUER, AND I. SOLS
70
(composite) operations of
Clone of
Define Z E Sets
I*xi
Graph
recursively as follows:
I*
id.
For
i
i
arrows
A
i
|’’"
i
i
(iEl).
J
in I and A.
J
i
n
Y
in
are
j
w(w
x A
j<n
(l_<j<_n)
w
.A.
A
in Y
i.
n
i
is in l
(and that is all).
Mod(/,B)
For each A
there
compatible with substitution, namely A idoi
A W(w_.)
Ao
Aw.,
id
B
Bi
A’ in Mod(Y,B) is
A
and each f
J
j_<n
Z
is a unique extension A
compatible also with all composite operations w in I (proof by
induction).
Construction of FB
W
Abbreviations: dom(A
cod(A
B
A
W
-
B i
I
(Bi)i I B_
for B
i)
:= i ("codomain"),
i)
...i
n
:=
.
:= A ("domain"),
{w
:
B i..
x
j<n
Then define
the i-th carrier object FBi of FB by
FBi
Bdom
wEl.
i},
cod w
injections
w
Bdom
in w
w
FBi.
i
How to define FB(A
i
i) ?
l...i n
Consider the following diagram:
Bdo m
,(w.)
x
x
j<n
Bdom
w
in
j<n
in wj
FBio
FB(A)
j_<n
’(Wl
’Wn
By distributivity of x with respect to coproducts,
FB
()
FBi
the rows above
ALGEBRAS WITH ACTIONS AND AUTOMATA
x
((wj)
-.
71
constitute a coproduct, hence the in
U(wj)
j<n
induce a morphism FBW. This clearly defines FB Mod(l,B).
Front adjunction qB
(in
Universal property of
qB
i
---I
FB
f
(FBi).
FB:
(Bi)
For a homomorphic extension
i
(Bi)
idi)i I
A
of
UA,
B
f
in I consider the diagram
fi
FBi
Ai
(2)
Aw
in w
f i.
Bw
J
x A i.
BiN
/in idij=x(nB) ij
f i.
FB i
J
The lower triangle commutes,
since f extends f;
since f is compatible with any w
il’’’in----i
the
frame commutes,
in Y.
An induction on I shows the left triangle to commute. Hence fi
is necessarily induced out of the coproduct by the Aw o
fi..
jn
This f extends f
(take w=id.) and is a l-homomorphism
as
is shown by
the
commutative
i
using the "coproduct-row" and commutativity of (I),
upper rectangle in (2),
proves
and
the recursive definition of A.
This
(i).
(ii) is proved straight forward by using the universal property of
im
in the i-th component.
For (iii) and (iv), consider a pair
R==fM
g
in Mod(l,B) which
gives rise to an absolute coequalizer diagram or a kernel pair
coequalizer diagram
(hence -case (iv)-- Ri
UR
Ug
gl
UM
C
Mi
Ci
in
I
respectively
is such a diagram).
KHNEL,
W.
72
M. PFENDER, J. MESEGUER, AND I. SOLS
Then, by hypothesis, for each A
.i
i!
n
in Y,
-----i
the upper
row in
.’.
fi
J---C(A)
M(A)
I
IRW
Ri
q i.
qA
fA
gA
R(A)
M
Cw
qi
-_ Mi
gl
:= C i.
=
Ci
is a coequalizer diagram, hence induces a unique C compatible
with q.
q
So there exists a unique C E Mod(Y,B) over C such that
M
C is a homomorphism. A simple diagram chase shows q to be
a coequalizer of
f and g in Mod(Y,B).
(v) is proved similarly using the fact that a binary functor
which preserves filtered colimits in each place, commutes with them,
[5],
see
6.5.
Let us now turn to many-sorted algebraic theories and their models.
DEFINITIONS:
2.3
An 1-sorted algebraic theory is a category
I* (free monoid over I), given products
ITi
T with object set
p
[A
i!
..i
A morphism
n
"J----i.].
,n’ i
j=!
----T’
t :T
I terminal.
and
of such theories
is a functor which is
identity on objects and preserves the given projections. This
defines a category Th with obvious forgetful functor
I
V
Thl----
Graphl,.
is a functor M
For B with given finite product, a T-model in B
T
B which preserves the given finite products.
M’ is
A T-homomorphism f: M
i.e.
a family f
for all A-
(fi)i
E
a
transformation compatible with
BI((Mi),
(Mi))
i in T. This defines the category Funct
T-models in B with forgetful functor U
2.4
Sets
satisfying (I) in 2.!
THEOREM:
The
forgetful functor
(T,B) of
Funct (T,B)----B
V
I.
Thl----Graphl.
is algebraic of finite rank (cf. ;.2).
,
73
ALGEBRAS WITH ACTIONS AND AUTOMATA
1-sorted theories are O-algebras
PROOF:
in Sets for the
following l*xl*-sorted scheme of operators
id
@:
A E I*
,(A,A),
(A,B)(B,C)-----,(A,C),
("identities")
A,B,C E I*
("composition")
pr
[I
...in,ij)]j=l
(i
(A’il)’’’(A’in)
n’
n 61N,i,
,’i (A,il ..in )’
("
6 I
ji.
n
("projections")
6IN, A 6 I*,
6 I
i.j
("induced arrows")
T 6 Mod(@,Sets)
iff the following hold"
is a theory,
associativity of o,
neutrality of id,
A
I
c
hence
f"
n
Prnf)
f
(uniqueness of the induced arrow)
il "’’in
By theorem 2.2, Mod(@,Sets)-----*Sets
Th
f
(prlf
(defining equation for induced arrows),
for f
(f
prj
cf.
[24i.
l*xl* is algebraic of finite
rank
Mod(@, Sets) is a full subcategory defined by equations,
(easy verification) closed under subobjects, products,
quotients, and filtered colimits. By lifting and
inclusion theorem,
Th
I
B i r k h o f f
-----Mod(@, Sets) is algebraic of finite rank,
hence by the composition theorem also
ThI
V
Graph
Th
I
I
Mod(@, Sets)
Sets
l*xl*
Graphi..
We draw now some conclusions from this theorem which will be needed
for the proof of
2.5
(i)
Each M
the main theorem below.
EXTENSION OF MODELS AND PRESENTATION OF THEORIES.
Mod(l,B) extends uniquely
being the free theory generated by
an isomorphism
Mod(l,B)
to M
I E
Funct (FI,B)
FI
Th
This defines
Graphl..
Funct (FI,B) of categories compatible
with the forgetful functors
(ii) Each s.pecies (I,G), l,G
presents the theory T
I
Graphi.
Sets
l*xl* G
_c
Ixl,
FI/G, and Funct (T,B) is isomorphic
KHNEL,
W.
74
to
Mod((Y,G),B) the full subcategory of Mod(Y,B) consisting of
those M
_pr!
G
(iii)
M. PFENDER, J. MESEGUER, AND I. SOLS
Pr 2
Each T
B satisfying G,
FK
i.e. M
B equalizes
=FZ
E Th
has an
I
@lebraic
presentation (I,G),
i.e.
Y, G E Sets I*I
PROOF.
M
l
o
TM______
composition,
The
(i): M admits a full image factorization
M
with T
of M.
I
I
defined by
TM(A,B
(MA,MB)
with
identities, and projections inherited from B.
M
Graphl.-morphism
in Th
Th
M
by the theorem
o
extends uniquely to M :FI
o
M
M
T
M
is then the unique extension
o
The statement now follows from the observation that homo-
morphisms f
M’
M
are just models in B
2
by the evaluations
a bijection between
B at 0 and
the Hom-sets,
B
Ol
which followed
give M and M’. This yields
compatible with the forgetful
functors, hence functorial.
(ii) follows from 1.5 and FZ/G being a coequalizer of
pr
Pr2
G-’-FK
relative V; for the homomorphisms cf.
(iii) follows from the fact that each T
arrows with codomain in I,
2.6
THEOREM.
Th
I
proof of (i).
is generated by its
since any arrow is induced by those.
For an 1-sorted algebraic theory T
Th
I
[finitely presented] and [B an elementary topos with NNO or] B
a well-powered cartesian closed category with coequalizer-mono
factorization and coequalizers closed under composition,
the care-
gory Funct (T,B) of T-models in B has an algebraic forgetful
I
functor U: Funct (T,B)
B of finite rank
PROOF.
Mod((Y,G),B)
By 2.5,
full
(ii) and (iii) it is sufficient to show that
> Mod(l,B)
B
I
is algebraic of finite rank
75
ALGEBRAS WITH ACTIONS AND AUTOMATA
for Z,G E Sets
G c FZ
FZ. The second factor has this property
[for B being an elementary topos with NNO the
by theorem 2.2,
J o h n s t o n e’s
existence of a left adjoint is guaranteed by
theorem]
Fulfillment of equations
A
(u,v)
i in G by Z-models
is clearly stable under subobjects and products and also under
quotients and filtered colimits, since x simultaneously preserves
regular epis and commutes with filtered colimits. This
implies the
theorem by Lifting theorem, Birkhoff-inclusion theorem and
composition theorem.
By the triangle-theorem we get immediately:
2.7
COROLLARY.
For B as above and a morphism t
the "algebraic" functor Funct
x
T’-T in
Th
Funct (T’,B)
x
(t,B): Funct x (T,B)
is algebraic of finite rank.
2.8
Sets
EXAMPLES.
The conditions for B are satisfied by Sets,
(C small, "presheaves"), Grothendieck-topoi(full subcate-
gories of Sets
Graph(= Sets
of j-sheaves for a topology j
-===e
MxM
), Graph M (= Sets
), Cat (cf.
(compactly generated Hausdorff spaces (cf.
spaces (cf.
on C),
[17]:4.9,7)). An analysis
1613.5), CGHaus
[18])), Tol(tolerance
of the proofs shows that the
theorem is still valid for countable T (i.e.
able Z) and the category of countable sets,
generated by a countsince B x- still
preserves colimits although it has no right adjoint.
A long list of l-sorted algebraic theories (presented by operations
and equations)
is given in [3] :3.
Examples for the many-sorted case:
W. KUHNEL, M. PFENDER, J. MESEGUER, AND I. SOLS
76
I. LEFT ACTIONS (called monoid automata in [17]: 5.6):
1_
Z
R
RxA
RxR
A
RxRxA
monoid eeuations for R and
G
RxA
models: Lact(B)
"xA..
*
RxA
(1 ,A)
A
-
(left actions in B (with varying scalar monoid R)).
MODULES:
)-
in 2.
as
abelian group equations for A,
G(unitary ring equations for R,
and the above equations
*).
for
Similarly: R,S-bimodules and group operations for R a group.
3. AUTOMATA:
Medvedev-au toma t a
IxS
S
Mealy-automata
IxS
S
IxS
S
Y
Moore-automata
2.9
REMARK:
other categories,
0
no
equations.
The conclusion in 2.6 remains valid for many
too,
if we replace "algebraic" by "monadic".
Especially this holds for every category B with an INS-functor
V
B
Sets in the sense of
category in the sense of
creates
[19]
or equivalently B being a Top-
[20]. In this case U
absolute coequalizers,
Mod(Y,B)---B
and we have the following commutative
diagram of functors:
Mod(Y,B)
Mod(Y,Sets)
U
U’
B
I
Sets
U’ has a left adjoint by 2.6 and Mod(Y,V) has
a
left adjoint,
too
(take the "discrete structure" on every component of a model in
Sets yielding a model in B).
77
ALGEBRAS WITH ACTIONS AND AUTOMATA
V
I
is an INS-functor and has also a left adjoint
-
("discrete
structure"), hence by the triangle-theorem ([9]) U has
adjoint,
too.
monadic.
This holds e.g.
P a r
It follows from the
a left
criterion that U is
for B being the category of
topological,
19] for
uniform, measurable, compactly generated, limit spaces (cf.
other examples.
Mod(Y,Sets) is
Note that the induced functor Mod(Y,V): Mod(Y,B)
again an INS-functor. The initial object can be defined in each
component of the model,
a model
is complete and cocomplete
will be used later
[19]: 1.6) which
(cf.
(cf.3.4).
ALGEBRAS WITH ACTIONS
The examples at
the end of
the applications,
cf.
this defines
In the one-sorted case this is mentioned in [19].
in B.
Hence Mod(Y,B)
3.
and it follows directly that
section 2 become more interesting for
if the "scalar" or "input" components are fixed,
[7] and the examples in 3.4 and 3.5. We show here that under
certain conditions
or algebraic
3.2) fixing components still gives monadic
forgetful functors.
A theory with J-action (J E Sets)
ALGEBRAS WITH ACTIONS
3.
is an I
(3.
0
J-sorted theory T having the following factorization
property:
A’
Every T-arrow AB
factorization AB
Now let
T]
(Example:
Tj-model
Thj
Tj
over
be
(A,A’ E J*, B
I *) has a
pr
the
full subcategory of T with object set J*
theory of unitary rings) and R
some base category
Then a T-model M with
MiT J
Together with homomorphisms f
_B
Mod(Tj,B)_
with (given)
finite products.
R is called an R-model
(fi)i I j:M
be a fixed
(for T).
M’ with fj
idMj
KHNEL,
W.
78
M. PFENDER, J. MESEGUER, AND I. SOLS
E J, these form
for
forgetful functor U
MOdR(T,B)_ ----B_
R
MOdR(T,B)_
subcategory
a
I
of
Mod(T,B)_
with
Now we give a sufficient
condition for the factorization property, which is trivially satisfied in all
examples.
o,r
If T E
LEMMA:
3.2
Thi0 J
A -----i (i
generated by arrows
i.e.
is generated by J-actions,
I
0
J* for i
J) with A
J, then
T has the factorization property of 3.1.
By hypothesis the arrows of a generating graph
PROOF:
I
Graph(ioj).
of T have
the factorization property.
show that it is preserved under inducing morphisms
and under composition.
into a product
trivial.
AB A’B’-----A"B" be in T with
Now let
pr
A’ B
AxB
The former is
We have to
A’
pr
AB
A
and similarly for W’
A
Then
i.e.
AB
A’xB’
A
A
such that for any T 6
equalizers and
finite rank]
T 6 Th I
0
J
PROOF:
I 6 Sets arbitrary,
Thi,
B
Mod(T,B)
U
q.e.d.
l
Mod(T,B)_
has
co-
is monadic or algebraic
[of
(See 2.6 for sufficient conditions for B).
UR MOdR(T,B)_
Then
A.
Let B be a category with given finite products
THEOREM:
3.3
’’"
pr
again factorizes through AB
’o
the J-part of
A"xB"
_B
I
has
the same property for any
with J-action (see 3.1).
Construction of a left adjoint
We will construct
FR---@
U
R
using
F ---4 U
F
R
for
U
R
Mod(T,B)--- B
theory
ALGEBRAS WITH ACTIONS AND AUTOMATA
Fj --q Uj
Mod(T j,B)
Let B
(Bi)iEl
(Rj,Bi)iEl, jJ
or F(Rj,Bi)
B
J
F
and
79
B
UI Mod(TI,B
I
I
be a given family of B-objects.
For the family
there exists the free T-algebra
F((Rj,Bi)iI,jj),
for short.
We first prove the following
LEMMA:
F(Rj,Bi)
PROOF:
We show that
FjUjR.
IT J
universal problem corresponding to
Let R’ be a
Tj-algebra
and f
a family of B-morphisms.
(AxB
R’ (W)
A)
Uj
Rj
for the object
UjR’
R’j)jEj: UjR
Define an extension
(Rj)jtj.
UjR
R’ Mod(T,B) of R’
(terminal object) for all i t I and
uniquely by R’i
’
(fj
solves the
F(Rj,Bi)IT J
F(Rj,Bi)Ij
o pr
for A
J* where W is given by the
factorization property 3.1.
So by our assumptions there exists a unique T-homomorphism
f
R’
F(Rj,Bi)
such that diagram (1) commutes:
(Rj,Bi)il,j J
(1)
(fj,Bil)i1,.jEj
’i)i10j
UR’
Uf
rl
UF(Rj ,Bi)
R’
The lemma follows now from R’
and by considering the J-
J
components of (I)
(existence of an extension of
and by the observation that any f
extends to an f
extension)
(fj)jj)
F(Rj,Bi)ITj-----R’
in
Mod(Tj,)
F(Rj,Bi)----- R’ in Mod(T,B) (uniqueness of the
q.e.d.
80
KHNEL,
W.
Now let
g
M. PFENDER, J. MESEGUER, AND I. SOLS
be the back-adjunction for
id
Fj
i.e.
Uj,
UjR
U jR
UjR
UjgR
Uj (FjUjR)
p-
and
Pl
P2
Define the
FjUjR
B
the kernel pair of R in
10J -morphism
pair
UjP---UjPl
Ujp2
(rl,r2)
Uj Fj Uj R
Mod(Tj,B).
by
(F(Rj
J
Bi)(j))jEj
r
(F(Rj,Bi)(i))iE I
Define
FR((Bi)i I)
Coeq(rl,r2)
(F(Rj,Bi)
I
(i))il
(rl,r 2
to be the coequalizer of
relative U in Mod(T,B), and the front adjunction
: (Bi)i I
URFR((Bi)i I
as the composition in
(Bi)i I
(Hi (Rj ,Bi) il
(2)
fi((Bi)iEl
U I(F(Rj,Bi)
U (coeq (r
I
Ul.(coeq(rl,r2
I
UR (ceq(r l,r2))
I
’r2)l
URFR((Bi)iI)
It remains to prove
(i)
Coeq(rl,r2)
MOdR(T,B
(ii) the universal property of a free construction with respect
the forgetful functor U
R.
ALGEBRAS WITH ACTIONS AND AUTOMATA
(i) We have to verify
We will show that
g,
g
p!
coeq(rl,r2)
Let g
Mod(Tj,).
P2"
Coeq(rl,r2)
FjUjR----
81
R.
J
(pl,P2)
is a coequalizer of
J
M
be a
in
satisfying
Tj-homomorphism
By an argument used in the lemma, M can be extended
in a unique way to a T-model M by Mi
for all iEl, and g can be
extended uniquely to a T-homomorphism g
F(Rj,Bi)
U
r! --U r 2
toeq(rl,r 2) relative
that
h
Coeq(r!,r2)
U there exists a unique T-homomorphism
M
satisfying
h o
which implies that the restriction
morphism satisfying h
o
J
Coeq(rl,r2)
to R because
,
coeq(r!,r 2)
coeq(r!,r2)
,
is the unique
hlj
IJ
J
and the
of course,
Tj-homom-
g. This implies
(pl,p2)
is a coequalizer of
J
(pl,p2)
It follows
and hence by definition of the coequalizer
1-components of h are the terminal morphisms into
that
M.
and hence isomorphic
was the kernel pair of the coequalizer R.
By changing the representant of
coeq(r,r2)
we get equality.
Mod(T,B)
1OJ
B
B be an R-model and f
(fi:Bi
precisely: use the fact that U
creates
(More
iso-
morphisms)
(ii)
Let M
T
a family of B-morphisms.
unique T-homomorphism
With f.
j
f
F(Rj,Bi)
id
for each jJ there exists a
Rj
M
(fi)i10 J
(Rj,Bi)i1,j J
Mi)iE I
in Mod(T,B) satisfying
(Rj,Mi)iEI,jj
j
UF(Rj,Bi)
Considering only the J-components we get the equation
f
gR
J
UM
W. KUHNEL, M. PFENDER, J. MESEGUER, AND I. SOLS
82
Uf o r
and hence Uf o r
which yields a unique f satisfying
F(Rj ,Bi)
M
coeq(r ,r
2
Coeq(r! ,r2)
FR(Bi
which in fact is a
gR
because coeq(r
MOdR(T,B)-morphism_
i,r2)i J
f
The uniqueness of f follows stepwise.
This proves U
U
R
creates
R
to have a
left adjoint F
absolute coequalizers,
R.
since U does and identities can
be taken as the J-components of this coequalizer.
Hence U
R
is
monadic by the P a r &-criterion 1.1. The same argument shows that
U
R
creates
coequalizers of
colimits] if U does.
(UR-)kernel
pairs [and filtered
q.e.d.
3.4 APPLICATIONS AND EXAMPLES:
For applying theorem 3.3 the
conditions concerning the base category B are satisfied by all
C
categories mentioned in 2 8 and 2 9: Sets, Sets --, Countable Sets,
Graph,
Cat, CGHaus, Tol and all Top-Categories over Set, e.g. Top,
CG, Unif, Meas, Lim...
Furthermore the factorization property of the theory T
is satisfied in all the examples in 2.8,
(cf.3.
if we fix the monoid,
3.2)
the
ring, or the input object as needed.
Left R-actions and
(left) R-modules over B for a fixed R:
Of particular interest are topological R-modules for a topological
unitary ring R and R-modules for a ring-object R in a category
Sho(C) of set valued sheaves, i.e. R a sheaf of rings. By the
theorem 3.3,
U
ModR----
this category has an algebraic forgetful functor
Sho(C). By the triangle theorem we get an algebraic
j
83
ALGEBRAS WITH ACTIONS AND AUTOMATA
forgetful functor
Mod R
Ab
into the category of abelian group
objects in Sh.(C), i.e. into the category of sheaves with values in
j
abelian groups. The same is true for R-S-bimodules in Sh.(C).
Automata with fixed input I:
automata mentioned
Corresponding to the three types of
in 2.8 Mod I(T,B)_ for a fixed I
E_B
becomes the
category of Medvedev-, Mealy- or Moore-automata with fixed input
respectively (the category of Medvedev-automata with fixed input is
denoted by B-Medv in [|7]).
In each case the forgetful functor to B resp. B
monadic depending on the base category (cf.
Of special interest are deterministic,
is algebraic or
2.9).
topological, compactly
generated, and tolerance automata (cf.[|7],
B
2
[2|]).
Automata over
Cat are useful for the theory of formal languages
(cf.[22]). By
algebraicity or monadicity of the forgetful functor there follow
most of
sect.
the structural properties of automata mentioned in
[|7]
||.
Note that in case of fixed output O the forgetful functor is not
monadic, in fact the factorization property is not fulfilled.
3.5
The results of
GENERALIZATION TO MONOIDAL CATEGORIES.
section 2 are preserved, if the 1-sorted L a w v e r e
theories are
replaced by [symmetric] monoidal theories and B in theorem 2.6 is
monoidal closed instead of cartesian closed (and
(R)
preserves
simultaneously coequalizers of kernel pairs), see [|5], and [16]
for the heterogeneous case. The same is true for theorem 3.3 because
of its "relative" formulation and since we did not use the universal
properties of
in the proof.
A list of [symmetric] monoidal categories which are "nice" in the
above sense is to be found in [15]:3.|.
Theorem 3.3 then gives
particular the following examples of action-categories with
in
K’HNEL,
W.
84
M. PFENDER, J. MESEGUER, AND I. SOLS
algebraic forgetful functors: Left action over abelian groups
(R-modules), partial and bilinear automata, and topological automata with
6, % continuous in each component separately (using
17], [21 ]).
(Top,@)) (cf.
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PAREIGIS, B. Categories and Functors, New York-London 1970.
PFENDER, M. Kongruenzen, Konstruktion von Limiten und
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MAHR, B.
Erzeugung von Kongruenzen, Diplom Arbeit TU Berlin
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HANSEN, H.
Kategorielle Betrachtungen zu den Stzen von
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EHRIG, H./KIERMEIER, K.-D./KREOWSKI, H.-J./KUHNEL, W.
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