Embedding of Algebras

28 Pages·0.15 MB·English

by  John MacDonald

I
Embedding of Algebras
John MacDonald
July 29, 2002
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1. Abstract
We look at some general conditions for determining
when the unit of an adjunction is monic when the gener-
ated monad is de(cid:12)ned on a category of algebras. Among
other possible applications these conditions will be shown
to provide a common approach to some classic embedding
theorems in algebra.
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2. Introduction
Let T = (T;(cid:17);(cid:22)) be a monad de(cid:12)ned on a category
C. We consider the following problem. Is (cid:17) monic for
C
a given object C of C?
There is an easy answer in the case of monads de(cid:12)ned
on Set. This is given here but goes back at least to the
thesis of Manes.
In this talk we require that T be de(cid:12)ned on a cat-
egory C of Eilenberg-Moore algebras given by (cid:12)nitary
operations and equations. It is further required that T
be generated by a pair of adjoint functors (cid:12)tting into a
certain framework.
Necessary and su(cid:14)cient conditions are given for (cid:17) to
C
be monic for given object C when T (cid:12)ts into the frame-
work described. A key condition is given in terms of a
graph assigned to object C.
Additional (cid:13)exibility is attained by relating the key
condition to several other graph conditions. This results
in another theorem, which we apply in several cases.
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3. Monads over Set.
Let T = (T;(cid:17);(cid:22)) be a monad de(cid:12)ned on Set. The
category SetT of Eilenberg-Moore algebras is said to be
nontrivial if there is at least one algebra (A;(cid:11)) whose
underlying set A has more than one element.
Lemma. If the category of Eilenberg-Moore algebras
for a monad (T;(cid:17);(cid:22)) on Set is nontrivial, then (cid:17) is
X
monic for all sets X.
Proof. Let x and y be distinct elements of a set X
with more than one element and suppose that (A;(cid:11)) is
an algebra with more than one element. Then there is a
function f : X ! A with f(x) distinct from f(y). By
the universal mapping property there is a unique algebra
morphism g : (TX;(cid:22) ) ! (A;(cid:11)) such that the following
X
diagram commutes
(cid:17)
X
X // TX
OOOOOOfOOOOOOOOg (cid:15)(cid:15)
’’ A
(cid:3)
thus (cid:17) (x) is not equal to (cid:17) (y).
X X
Thus for example, in the case of semigroups, this shows
that any set may be embedded in the free semigroup.
It is clearly rare for the category of Eilenberg-Moore
algebras to be trivial since, if trivial, each TX is either
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empty or consists of one element since it is the underlying
object of the algebra (TX;(cid:22) ). Furthermore, TX is the
X
codomain of (cid:17) , hence cannot be empty unless X is.
X
Now, we remark that the unique function T1 ! 1 is a
T-algebra, where 1 is the one point set, and the empty set
has a T-structure if and only if T(;) = ;. Accordingly,
up to isomorphism, there are only two monads (T ;(cid:17) ;(cid:22) )
1 1 1
and (T ;(cid:17) ;(cid:22) ) yielding just trivial algebras. These are
2 2 2
given by T X = 1 for each set X and T (;) = ; otherwise
1 2
T (X) = 1
2
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4. Reduction and Graph Components.
Let G be a small subcategory of a category C and let
P(G) be the power category of G. The objects of P(G)
are the subclasses of objects of G and the morphisms are
the inclusions.
The reduction functor R : Cop ! P(G) is de(cid:12)ned
G
by
R X = fAjA is inG andX ! A exists inCg
G
with the obvious de(cid:12)nition on morphisms.
An object A is reduced in C if the only C-morphism
with domain A is the identity. The subcategory G is
reduced in C if each of its objects is reduced in C. An
object X of C is G-reducible if R X is nonempty.
G
The component class [X] of an object X of a category
C (or a graph C) is the class of all objects Y which can
be connected to X by a (cid:12)nite sequence of morphisms
(e.g. X ! X (cid:0)X ! Y ). We let CompC denote the
1 2
collection of component classes.
Strong Embedding Principle for G If [X] = [Y ]
in CompC, then R X = R Y . Furthermore there is
G G
at most one morphism X ! A for each pair (X;A)
consisting of an object X of C and an object A of G.
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5. A Framework for Embeddings.
Let Alg((cid:10);E) denote the category of algebras de(cid:12)ned
by a set of operators (cid:10) and identities E. Suppose
U V
A B D (1)
// //
is a diagram such that
0 0
(a) A, B and D are the categories of ((cid:10);E), ((cid:10) ;E )
00 00 00 0
and ((cid:10) ;E ) algebras, respectively, with (cid:10) (cid:18) (cid:10)
00 0
and E (cid:18) E , and
0 00
(b) V is the forgetful functor on operators (cid:10) (cid:0) (cid:10) and
0 00
identities E (cid:0)E and U is a functor commuting with
the underlying set functors on A and B. Note that
U is not necessarily a functor forgetting part of (cid:10)
and E.
We next describe a functor C : B ! Grph associ-
V
ated to each pair consisting of a diagram (1) of algebras
0
and an adjunction (L;V U;’ ) : D ! A, where Grph
is the category of directed graphs.
Given an object G of B let the objects of the graph
C (G) be the elements of the underlying set jLV Gj of
V
LV G.
Recursive de(cid:12)nition of the arrows of C (G):
V
0 0 0
! (j(cid:17) jx ;(cid:1)(cid:1)(cid:1) ;j(cid:17) jx ) ! j(cid:17) j! (x ;(cid:1)(cid:1)(cid:1) ;x )
ULVG VG 1 VG n VG G 1 n
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0 00
is an arrow if ! is in the set (cid:10) (cid:0)(cid:10) of operators forgotten
by V and (x ;(cid:1)(cid:1)(cid:1) ;x ) is an n-tuple of elements of jGj
1 n
for which ! (x ;(cid:1)(cid:1)(cid:1) ;x ) is de(cid:12)ned.
G 1 n
If d ! e is an arrow of C (G), then so is
V
(cid:26) (d ;(cid:1)(cid:1)(cid:1) ;d;(cid:1)(cid:1)(cid:1) ;d ) ! (cid:26) (d ;(cid:1)(cid:1)(cid:1) ;e;(cid:1)(cid:1)(cid:1) ;d )
LVG 1 q LVG 1 q
for (cid:26) an operator of arity q in (cid:10) and d ;(cid:1)(cid:1)(cid:1) ;d ;d ;(cid:1)(cid:1)(cid:1) ;d
1 i(cid:0)1 i+1 q
arbitrary elements of jLV Gj.
0 0
If (cid:12) : G ! G 2 B, then C ((cid:12)) : C (G) ! C (G )
V V V
is the graph morphism which is just the function jLV (cid:12)j :
0
jLV Gj ! jLV G j on objects and de(cid:12)ned recursively on
arrows in the obvious way.
In the following proposition note that if, in the dia-
gram (1),D is the category of sets, then an adjunction
0
(L;V U;’ ) is given by letting LX be the free ((cid:10);E)
algebra on the set X.
Proposition. Suppose
U V
A B D
// //
ff
L
is a diagram of algebras as in (1), with given adjunc-
0
tion (L;V U;’ ) : D ! A. Then there is an adjunc-
tion (F;U;’) : B ! A with the following speci(cid:12)c
properties:
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(a) The underlying set of FG is CompC (G).
V
(b) If (cid:26) is an operator of arity n in (cid:10), then (cid:26) is
FG
de(cid:12)ned by
(cid:26) ([c ];(cid:1)(cid:1)(cid:1) ;[c ]) = [(cid:26) (c ;(cid:1)(cid:1)(cid:1) ;c )]
FG 1 n LVG 1 n
where c ;(cid:1)(cid:1)(cid:1) ;c are members of the set jLV Gj of
1 n
objects of the graph C (G).
V
(c) The unit morphism (cid:17) : G ! UFG of (F;U;(cid:30))
G
has an underlying set map which is the compo-
0 0
sition [(cid:0)]: j (cid:17) j, where j(cid:17) j : jGj ! jLV Gj =
VG VG
0
ObjC (G) is the set map underlying the unit (cid:17) :
V VG
0
V G ! V ULV G of the adjunction (L;V U;(cid:30) ) and
[(cid:0)] : ObjC (G) ! CompC (G) is the component
V V
function.
Suppose the hypotheses of the proposition hold. Let
S be a subgraph of C (G) having the same objects
V V
jLV Gj and the same components as C (G). Then the
V
proposition remains valid under substitution of S for
V
C (G) throughout. This allows us to \picture" the ad-
V
joint using a possibly smaller set of arrows than those
present in C (G). Accordingly, we de(cid:12)ne a V picture
V
of the adjoint F to U at G 2 jBj to be any quotient
category C(= C(S )) of the free category generated by
V
such a subgraph S of C (G). This proposition is then
V V
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valid upon substitution of the underlying graph of a V
picture C for C (G) throughout.
V
We now present the (cid:12)rst embedding theorem.
Theorem. Let
U V
A B D
// //
hhff F
L
0
be given with adjunctions (L;V U;’ ) : D ! A and
(F;U;’) : B ! A as described in the Proposition.
Given G 2 jBj let C(S ) be any V picture of the
V
adjoint F to U at G.
Then the unit morphism (cid:17) : G ! UFG of the ad-
G
junction (F;U;’) is monic if and only if the following
hold
0
(a) The discrete subcategory G = (cid:17) (jGj) is reduced
VG
0 0
in C(S ) for (cid:17) the unit of (L;V U;(cid:30) ).
V VG
(b) If [A] = [B] in CompC(S ) with A;B 2 jGj, then
V
R A = R B, where R : C(S )op ! P(G) is
G G G V
the reduction functor.
0
(c) The unit morphism (cid:17) is monic.
VG
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