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A Krohn-Rhodes Theorem for categories
JOURNAL
OF ALGEBRA
64,
37-45
(1980)
A Krohn-Rhodes
Theorem CHARLES
Department
of
Mathematics
for Categories
WELLS
and Statistics, Case Western Cleveland, Ohio 44106
Communicated Received
I.
by Saunders December
Reserve
University,
Mac Lane 8, 1978
INTRODUCTI~~V
Recently, W. Nice [5] gave a general construction for functors between small categories which generalizes the notion of the kernel of a group homoof a homomorphism (Eilenberg [l]) morphism. T&on’s “derived semigroup” is in the monoid case essentially a special case of Nice’s concept. I say “essentially” because Nice’s construction yields in general a category, not a monoid, even when the functor goes between monoids; the derived semigroup is obtained from the Nice category by adjoining a zero which is to be the product of arrows which do not compose. This situation suggests that the theory of group complexity of Rhodes, ‘I’ilson, et al., has its proper setting in finite categories instead of in finite semigroups. The first step in developing a theory of group complexity for categories has to bc the formulation and proof of a generalization of the Krohn-Rhodes Theorem. That is accomplished in this paper; after some preliminaries the main Theorem is stated and proved in Section 5, and a definition of the group complexity of a finite category is suggested there. Except for the proof of Proposition 5.1, the paper is completely selfcontained and can be read by anyone with a basic knowledge of categories and functors.
2.
THE
WREATH
PRODUCT
Let B and C be small
categories and G: C -* Set a functor. Bwr’C (or the wreath product of B by C induced by G, is a category defined as follows. An object of BwrCC is a pair (P, c) with c an object of C and P: cG 4 1 B (I B i is the set of objects of B) a function. It is useful to think of P as a set of objects of B indexed by the set cG. An arrow (h,f): (P, c) -+ (Q, d) has f: c + d an arrow of C and /\: cG - B a function with the property that for each .Y E CC, xX: .rP + .r(fG c Q). In order to define
BwrC if G is clear from context),
37 0021-8693/80j050037-09$02.00/0 Copyright 0 1980 by Academic Fress, Inc. All rights of reproduction in any form reserved.
38
CHARLES
WELLS
composition of arrows, a definition is needed. For A, A’: cGall x E cG, cod(A) = dom(xX’), define A 3 A’ by
B such that for
x . (A * A’) = 1 xh u x/i’. Then
for (h,f):
(P, c) -+ (Q, d) and (p,g): (kf)o(crtd
(2.1)
(Q, d) + (R, e), set
= (A * (fGc~)vf~g);
(2.2)
since xh: XP -+ x (fG 0 Q) =- dom(x .fG)p, the condition for defining A * (fG 0 p) is satisfied. If each cG is regarded as a discrete category, then P, Q and R are functors, A: P +fG 0 Q, fG 0p: fG 0 Q + (fo g)(G c R) are natural transformations, and * is the usual horizontal composition of natural transformations. This observation is the basis for the more general definition of wreath product for Cat-valued functors given by Kelly [3] (who calls it the composite) and Wells [7]. If F: B + Set is also a functor, there is an induced functor FwrG: BwrGC ---f Set defined as follows.
(P,c).FwrG={(m,x) for any object (P, c) of BwrcC, (P, c) -+ (Q, d) in BwrGC, then
(m,x).[(h,f).FwGj
x~cG,rn~x.PF) and
if (m, x) E (P, c) . FWYG and
=(m.(x.hF),x.fG).
Here, h F: cG - Set is the set-function-valued
function
product
is associative
in the following
(h,f):
(2.4) such that for x E G,
x . AF r= (xh) F: xPF + [x . (fG o Q)] F. The wreath
(2.3)
(2.5)
precise sense:
PROPOSITION 2.1. Let F: B + Set, G: C -+ Set, H: D + Set be functors. Then there is a category isomorphism Z makingthis diagram commute:
BwrG~“IH(CwrHD)A
(BeurT) wr”D
y’/G,trTH Set
Proof. For (A, (cL,g)): (P, (Q, d))A (P’, (Q’, d’)) in Bwr(CwrD), I shall define a function T,,, with domain dH; its value at y E dH will be a function from (yQ)G to the arrows of BwroC, as follows: x .-vT~.~ : (k YN Y/J): (Pt YQ) - (P’, Y . (YPH 0 0%
(2.6)
A KROHN-RHODES
Then
THEOREM
FOR
CATEGORIES
39
set 64 (I*,g)Y
L VA,” t‘d.
(2.7)
A verification that I fulfills the claims of the Proposition is painful but straightforward. \?a the second projection, BwrGC is the discrete split normal fibration corresponding to the functor Set((-)G, 1 B I) where 1 B 1 is the set of objects of B. This statement reduces in the case that B and C are semigroups to the well-known fact (see Wells [6]) that the wreath product is a certain semidirect product. If B and C are groupoids the construction is both more and less genera1 than that of Houghton [2]. His untwisted wreath product BwrCC of groupoids is in my terminology the wreath product obtained when G: C -+ Set is the “global horn functor” defined by setting CC = (J C( -, c) (union over all objects of C) for c an object, and on arrows by composition. In Wells [7] I propose a genera1 definition for the twisted wreath product with respect to Cat-valued functors.
3. HOLONOMY
GROUPS
Let F.: C - Set be a finite-set-valued functor where C is a finite category. There is no harm in assuming that F is separated, that is, that cF n dF is empty if c and d are distinct objects of C. In fact, every set-valued functor is naturally equivalent to a separated one. Define a functor a,: C + Set as follows: if c is an object of C, c@~ is the subset of the powerset of cF consisting of all singletons {x} for x E CF and all sets im(fF) for all arrowsf: b + c of C (a fortiori, CF E cm,). If g: c -+ d in C, gGYF: caC, + daF is the image function induced by gF. Suppose A E caC, , B E dLTF. Following Eilenberg [I], B < .4 means that there is some arrow /: c -+ d for which B C A .fa;. The relation 5: is a preorder (transitive and reflexive relation, called a quasiorder by some). Like any preorder, >< induces an equivalence relation b defined bv requiring A -u B if .4 -( B and B 5: .4. PROPOSITION 3.1 (Eilenberg [ 1, p. 441). If A E cGT~ , B E dGTF, A h B and f: c -+ d has the property that B C A .fLTF , then B .-: A . fLTF and there is an arrow j: d --+ c of C such that A =- B fCn, , (f nJ)F restricted to .4 is id, , and (j”’ f)F restricted to B is idR .
Proof. By definition there is an arrow g: d -+ c such that A C B . g6TF. Since B C AfLZF C B gfOTF and #(B . gf6TF) ,< #B (where # denotes cardinality), one has B = A . fLTF = B .gfaF and analogously .4 =: B .g12?~ = A .fgC!lF. Thus (f =g)F restricted to A is a permutation of A, so for some integer n :> I, (f ?g)“F restricted to A is id, . Define f 7 g c (f cg)” 1 and the Proposition follows.
40
CHARLES
WELLS
It follows from Proposition 3.1 that if A -B then #A .--: #B. For any object c of C and set A E ~67~) let aA denote the set of proper subsets B of A which are in ~02~ and are maximal in that respect; so aA = {B ( B C A, B#A,BEC~!‘~,~~~(CCA,C#A,BCC,CEC~~)=XB=C}.N~’~~~~~~~ A = UBEdA B since every singleton is in caC, . If f: c -+ c in C is an arrow such that A . flF = A, then by Proposition 3.1 f induces a permutation of A and therefore of aB, . The set of permutations of gA obtained in this way is a (not necessarily transitive) permutation group, denoted ,%A and called the holonomy group of A. Observe for later use that & is a quotient of a submonoid of the C-endomorphism monoid of c. It is straightforward to see that if A -B, then .#‘A e & . Let A, , A, ,..., A, be a set of representatives of those equivalence classes (mod -) whose members have cardinality greater than one, indexed in such a way that if i
4.
COWIUNC
A functor F: C ---+ D which is surjective on arrows /ifrs composition if for all t:w~x,u:x-,y,v:y---,zofD,ifg:c-,dinCandgF=u,thenthereare f: b ---+ c and h: d -+ e in C with fF = t, hF = V. (I called this triangle-rejecting in Wells [7] but that is a misuse of the word “reflect.“) Now let F: C + Set, G: D + Set be functors. Then G cozws F if there arc
D’ C D,
(a)
a subcategory
(b)
a functor
G,,: D’ + Set,
(c)
a functor
H: D’ -+ C, and
(d)
a natural
transformation
6: G, 4 H 0F
for which G, is a subfunctor H lifts composition
of G ’ D’,
(hence is surjective 0 is surjectivc.
on arrows),
(4.1) and
(4.2) (4.3)
A
KROHN-RHODES
THEOREM
It follows that if f: c -+ c’ is any arrow of D such that gH = f for which dG, -%
FOR
41
CATEGORIES
of C, then there is an arrow g: d -+ d’
d’G, (4.4)
commutes. Moreover, if f ‘: c’ + c” is an arrow of C, then there is R’: d’ + d” of D (starting from the same d’!) with g’H = f’, and similarly for an arrow going into c. It is this sense that G simulates F. In Rhodes’ language, F diwides G. .4s an aid to proving the Main Theorem, it is necessary to introduce a weaker notion of covering. Given F: C + Set, G: D + Set, G weakly covers F if there is a subcategory D” CD, a surjective function K from the objects of D” onto the objects of C, and for each object d of D a function do: dG - dKDF satisfying this condition: (C) If f: c - C” is an arrow of C, and dK = c, d,K - co, then there are objects P, e, and arrows s: d + e,, , g’: e - l d,, of D” such that eh’ --- c,
e,K = co
(43)
u x . do = cF ZedG (x . d8)(faF)
C x . gG . e,8
(4.6)
(all x E dG),
and
(all y E eG).
( y . e8)( fMF) C y . g’G . d,,e
(4.7) (4.7‘)
G weakly covers F with height i if the function t? in the preceding definition satisfies the requirement that for x E dG the set x . dt? is of height
if Homc(c,
Homc& c’) = 1>.’ ,-. (,therwise. Let /: C# -+ Set he the unique
functor
which
c’) f
::
on objects takes c to {c).
PROPOSITIOS 4.1. Let C he a finite category and F: C --, Set ajnite-set-calued functor, Let N be the maximum value of the height functionfor F. Then J: C# + Set weakly covers I; with height !Y.
Proof. In the notation of the definition of “weakly covers,” take D -.- D” : -C#, K the identity function, G /, ~8: c J -+ cflF the function taking {c) to {cF}. Then (4.5) (4.7’) follow easily (take e - c, e,, - c,, in (4.5)).
42
CHARLES PROPOSITION
jinite-set-valued covers F.
WELLS
4.2. Let C be a finite category and F: C + Set a faithful functor. If G: D - Set weakly covers F with hezght 0, then G
Proof. By hypothesis there is a subcategory D” CD, a surjective function K from the objects of D” to the objects of C, and for each object d of D” a function do: dG + dKflF that satisfy (4.5) through (4.7’). Let D’ be the subcategory of D whose objects are the objects of D” and whose arrows consist of all the g, g’ given by the definitions of “weakly covers with height 0” for all arrows f of C. (It is easy to check that D’ is closed under composition and has the requisite identity arrows.) For each object d of D’ and each x E dG, x . d# :: {x . de}. Then d8 is surjective by (4.6). Define a functor H: D’ + C as follows. For each object d, dH = dK. If g: d -+ d’ in D’, suppose dH = c, d’H = c’. By construction there is an arrow f: c -+ c’ such that for all .r E dG, (x . d@(foc,)
C x . gG . d’&
but since df? and d’8 are singleton-valued,
this means
(x . d0) f F = x . gG . d’0.
(4.8)
There cannot be another f making (4.8) true because F is faithful and d0 is surjective. Let gH 7.: f. Then if g’: d’ + d” is in I)’ with g’H = f ‘, we have (x.dO)fFcf’F
= [x.gG.d’B]f’F = x . gg’G . da0 = (x . de) . ff’F
so again because d0 is surjective and F is faithful H must preserve composition. H preserves identity arrows by a similar argument. Thus H is a functor. It follows from (4.8) that fJ is a surjective natural transformation from G 1D’ to H c:F. Taking G, - G i D’, I have already verified (4. I) and (4.3). The assumption (C) forces H to lift composition, so (4.2) is true. This proves Proposition 4.2.
5. THE
MAIN
THEOREM
A functor F: C 4 Set is a constant-function functor, or c.f. functor if for every arrow f: c + d of C, f F is a constant function.
for short,
THEOREM. Let C be a finite category and F: C + Set a faithful jinite-setvalued functor. Then for some integer n there are categories D, , D, ,..., D, and functors Fi: D, + Set such that
A KROHK-RHODES
THEOREM
(i)
F,WY F,wr ‘. . wrF,, covers F, and
(ii)
for
fumtor,
(a) or
each i, one of the following
FOR
43
CATEGORIES
two possiblities
hold:
Di is a finite category with no more objects than C has and F, is a c.f.
(b) Di is a group which is a homomorphic image of a submonoid of the endomorphism monoid of some object of C, and Fi is the right regular representation ofD;. Note. The Theorem is stated in a form intended to suggest the possibility of defining the “group complexity” of a finite category C: that would presumably be the least number of groups appearing in any iterated wreath product W of finite categories D, ,..., D, and functors Fi: Di + Set where (a) for each i either F, is a c.f. functor or Di is a finite group and (b) there is a subcategory DC W and a composition lifting functor H: D-C. The iterated wreath product is well-defined by Proposition 2.1. Proof of the Theorem. The theorem follows immediately from Propositions 5.1 and 5.2 below and Proposition 2.1. Some definitions are needed. If X is a set, X denotes the monoid consisting of the identity function and all constant transformations of X. If G is a permutation group on X, G = G U X is a monoid of transformations of X. PROPOSITION 5.1. Let G be a permutation group on a set X. Let R be the action of G on X, K the action of X on X, and p the right regular representation of G. Then R is covered by Kwrp: XwrG + Set.
Proof. Follows Recent expositions
immediately from “Method II” of Meyer and Thompson [4]. are in Eilenberg [1, II, Cor. 3.21 and Wells [6, Theorem 13.21.
PROPOSITION 5.2. Let F: C -+ Set be a faithful fkite-set-valued functor, C ajnite category. Let A, ,..., A, be representatives of equivalence classes (Mod -) and let e. = z?“~ be the corresponding holonomy group. Then F is covered by
R,wrR,wr
.‘. wrR,wrJ:
where Ri is the action of x
4wrgzwr
.‘* wr.@VwrC#
4 Set,
on aA, .
Proof. Suppose F: C + Set satisfies and suppose G: D + Set weakly covers functor G: XwroD - Set which weakly Proposition then follows from Propositions Since x is a category with only one identified with an object of D; however
the hypotheses of the Proposition, F with height i. I shall construct a covers F with height i - 1. The 4.1 and 4.2. object, an object of XwrD may be I shall write such an object d as di
44
CHARLES
WELLS
when I regard it as an object of .~wYD. An arrow (A,f): d’+ ei has f: d + e in D and X: dG - z any function. It is given that G: D + Set weakly covers F with height i. Thus there is a subcategory D” CD, a function K from the objects of D” to the objects of C, and functions do: dG 4 dKoZ, satisfying (4.5) through (4.7’). To prove the Proposition it is necessary to construct a subcategory E of gwz,D, a functor G: zwr,D + Set, a function L from the objects of E to the objects of C, and arrows di$: diG-+ diLtZF satisfying the analogs of (4.5)-(4.7’) with all d4+h of height
if x * d8 has height if x * do has height
(5.1)
Observe that (B, x) . d’t,b has height
(5.2)
and (B, , y) . ei$ . fOZpC (B,, , y) . (h’, g’) R,wrG
. $v
for (B, x) E diRiwyG, (B, , y) E eiRiwrG). The arrows of E will (h, g) and (h’, g’) satisfying (5.2) and (5.3). Observe that (B, x)(‘\, g) R,wrG
= (B . ml, x . gG).
(5.3) be all arrows
(5.4)
Case 1. If x . dt? has height
A KROHN-RHODES
THEOREM
FOR
CATEGORIES
45
subset of Ai by (4.7); it is a proper subset because z’c+ is a bijection and x . df? .f6YF is a proper subset of x . gG . e,8 (because its height is less). Therefore there is an element B, of gi which contains x * df? . f6TF. v6’lF. Let xh be the constant function from ,%‘i to gi with value B, , and let V: ai + c be the “inverse” to v given by Proposition 3.1. Then for any B cSfi , (B: x) . d’# . fCTF x . do. f& = x ’ d0 . fn, . zaF . %ZF C B, . ?XC!~ = (B, x)(/\, g) . R,wrG . e,,$b, the last equality by (5.4) and (5.1). Thus (5.2) holds in this case. Case 3. xd8 has height i; by (4.7) and the assumption that G weakly covers F with height i, x . gG . e,f? also has height i and moreover xdo = s . gG . e,,8. Let u: c + ai be an arrow taking x . dg to Ai, I: a, --f c its “inverse” as in Proposition 3.1, and v be as in Case 2. Let xh be the element of ,$ induced by tifc (it is easy to check that Ai . zifvUZF = Ai). Then for any B c.gi , (B, x) . d’4/1 . f& = B . iibl, . fC& = B . tifwiX& = B . x/\ . ~62’~ = (B . x/i, x .gG) . e,i#, as required. This verifies (5.2) and (5.3) is verified analogously.
ACKNOWLEDGMEHT The proving
author is grateful this theorem.
to S. Eilenberg
for suggesting
the use of holonomy
groups
in
REFERRNCF~ 1. S. EILF-NBERC, “Automata, Languages, and Machines, Vol. B,” Academic Press, 1976. (Some chapters by B. Tilson.) 2. C. HOUGHTON. The wreath product of groupoids, /. London Math. Sot. (2) IO (l975), 179-188. 3. G. KELLY, On clubs and doctrines, in “Category Seminar Sydney 1972-73,” Lecture Notes in Mathematics No. 420, Springer-Verlag. Sevv York/Berlin, 1974. 4. A. R. ~MEYER AND C. THOMPSON. Remarks on algebraic decomposition of automata, J. Moth. Sys. Theory 3 (1969), I IO-I IS. 5. W. NICO. Congruences and Extensions for small categories and monoids, preprint, Tulane Univ., 1978. 6. C. WELLS, Some applications of the wreath product construction. Amer. Math. Monthly 83 (1976), 317-338. 7. C. WELLS, Wreath product decomposition of categories, unpublished manuscript, 1976.
OF ALGEBRA
64,
37-45
(1980)
A Krohn-Rhodes
Theorem CHARLES
Department
of
Mathematics
for Categories
WELLS
and Statistics, Case Western Cleveland, Ohio 44106
Communicated Received
I.
by Saunders December
Reserve
University,
Mac Lane 8, 1978
INTRODUCTI~~V
Recently, W. Nice [5] gave a general construction for functors between small categories which generalizes the notion of the kernel of a group homoof a homomorphism (Eilenberg [l]) morphism. T&on’s “derived semigroup” is in the monoid case essentially a special case of Nice’s concept. I say “essentially” because Nice’s construction yields in general a category, not a monoid, even when the functor goes between monoids; the derived semigroup is obtained from the Nice category by adjoining a zero which is to be the product of arrows which do not compose. This situation suggests that the theory of group complexity of Rhodes, ‘I’ilson, et al., has its proper setting in finite categories instead of in finite semigroups. The first step in developing a theory of group complexity for categories has to bc the formulation and proof of a generalization of the Krohn-Rhodes Theorem. That is accomplished in this paper; after some preliminaries the main Theorem is stated and proved in Section 5, and a definition of the group complexity of a finite category is suggested there. Except for the proof of Proposition 5.1, the paper is completely selfcontained and can be read by anyone with a basic knowledge of categories and functors.
2.
THE
WREATH
PRODUCT
Let B and C be small
categories and G: C -* Set a functor. Bwr’C (or the wreath product of B by C induced by G, is a category defined as follows. An object of BwrCC is a pair (P, c) with c an object of C and P: cG 4 1 B (I B i is the set of objects of B) a function. It is useful to think of P as a set of objects of B indexed by the set cG. An arrow (h,f): (P, c) -+ (Q, d) has f: c + d an arrow of C and /\: cG - B a function with the property that for each .Y E CC, xX: .rP + .r(fG c Q). In order to define
BwrC if G is clear from context),
37 0021-8693/80j050037-09$02.00/0 Copyright 0 1980 by Academic Fress, Inc. All rights of reproduction in any form reserved.
38
CHARLES
WELLS
composition of arrows, a definition is needed. For A, A’: cGall x E cG, cod(A) = dom(xX’), define A 3 A’ by
B such that for
x . (A * A’) = 1 xh u x/i’. Then
for (h,f):
(P, c) -+ (Q, d) and (p,g): (kf)o(crtd
(2.1)
(Q, d) + (R, e), set
= (A * (fGc~)vf~g);
(2.2)
since xh: XP -+ x (fG 0 Q) =- dom(x .fG)p, the condition for defining A * (fG 0 p) is satisfied. If each cG is regarded as a discrete category, then P, Q and R are functors, A: P +fG 0 Q, fG 0p: fG 0 Q + (fo g)(G c R) are natural transformations, and * is the usual horizontal composition of natural transformations. This observation is the basis for the more general definition of wreath product for Cat-valued functors given by Kelly [3] (who calls it the composite) and Wells [7]. If F: B + Set is also a functor, there is an induced functor FwrG: BwrGC ---f Set defined as follows.
(P,c).FwrG={(m,x) for any object (P, c) of BwrcC, (P, c) -+ (Q, d) in BwrGC, then
(m,x).[(h,f).FwGj
x~cG,rn~x.PF) and
if (m, x) E (P, c) . FWYG and
=(m.(x.hF),x.fG).
Here, h F: cG - Set is the set-function-valued
function
product
is associative
in the following
(h,f):
(2.4) such that for x E G,
x . AF r= (xh) F: xPF + [x . (fG o Q)] F. The wreath
(2.3)
(2.5)
precise sense:
PROPOSITION 2.1. Let F: B + Set, G: C -+ Set, H: D + Set be functors. Then there is a category isomorphism Z makingthis diagram commute:
BwrG~“IH(CwrHD)A
(BeurT) wr”D
y’/G,trTH Set
Proof. For (A, (cL,g)): (P, (Q, d))A (P’, (Q’, d’)) in Bwr(CwrD), I shall define a function T,,, with domain dH; its value at y E dH will be a function from (yQ)G to the arrows of BwroC, as follows: x .-vT~.~ : (k YN Y/J): (Pt YQ) - (P’, Y . (YPH 0 0%
(2.6)
A KROHN-RHODES
Then
THEOREM
FOR
CATEGORIES
39
set 64 (I*,g)Y
L VA,” t‘d.
(2.7)
A verification that I fulfills the claims of the Proposition is painful but straightforward. \?a the second projection, BwrGC is the discrete split normal fibration corresponding to the functor Set((-)G, 1 B I) where 1 B 1 is the set of objects of B. This statement reduces in the case that B and C are semigroups to the well-known fact (see Wells [6]) that the wreath product is a certain semidirect product. If B and C are groupoids the construction is both more and less genera1 than that of Houghton [2]. His untwisted wreath product BwrCC of groupoids is in my terminology the wreath product obtained when G: C -+ Set is the “global horn functor” defined by setting CC = (J C( -, c) (union over all objects of C) for c an object, and on arrows by composition. In Wells [7] I propose a genera1 definition for the twisted wreath product with respect to Cat-valued functors.
3. HOLONOMY
GROUPS
Let F.: C - Set be a finite-set-valued functor where C is a finite category. There is no harm in assuming that F is separated, that is, that cF n dF is empty if c and d are distinct objects of C. In fact, every set-valued functor is naturally equivalent to a separated one. Define a functor a,: C + Set as follows: if c is an object of C, c@~ is the subset of the powerset of cF consisting of all singletons {x} for x E CF and all sets im(fF) for all arrowsf: b + c of C (a fortiori, CF E cm,). If g: c -+ d in C, gGYF: caC, + daF is the image function induced by gF. Suppose A E caC, , B E dLTF. Following Eilenberg [I], B < .4 means that there is some arrow /: c -+ d for which B C A .fa;. The relation 5: is a preorder (transitive and reflexive relation, called a quasiorder by some). Like any preorder, >< induces an equivalence relation b defined bv requiring A -u B if .4 -( B and B 5: .4. PROPOSITION 3.1 (Eilenberg [ 1, p. 441). If A E cGT~ , B E dGTF, A h B and f: c -+ d has the property that B C A .fLTF , then B .-: A . fLTF and there is an arrow j: d --+ c of C such that A =- B fCn, , (f nJ)F restricted to .4 is id, , and (j”’ f)F restricted to B is idR .
Proof. By definition there is an arrow g: d -+ c such that A C B . g6TF. Since B C AfLZF C B gfOTF and #(B . gf6TF) ,< #B (where # denotes cardinality), one has B = A . fLTF = B .gfaF and analogously .4 =: B .g12?~ = A .fgC!lF. Thus (f =g)F restricted to A is a permutation of A, so for some integer n :> I, (f ?g)“F restricted to A is id, . Define f 7 g c (f cg)” 1 and the Proposition follows.
40
CHARLES
WELLS
It follows from Proposition 3.1 that if A -B then #A .--: #B. For any object c of C and set A E ~67~) let aA denote the set of proper subsets B of A which are in ~02~ and are maximal in that respect; so aA = {B ( B C A, B#A,BEC~!‘~,~~~(CCA,C#A,BCC,CEC~~)=XB=C}.N~’~~~~~~~ A = UBEdA B since every singleton is in caC, . If f: c -+ c in C is an arrow such that A . flF = A, then by Proposition 3.1 f induces a permutation of A and therefore of aB, . The set of permutations of gA obtained in this way is a (not necessarily transitive) permutation group, denoted ,%A and called the holonomy group of A. Observe for later use that & is a quotient of a submonoid of the C-endomorphism monoid of c. It is straightforward to see that if A -B, then .#‘A e & . Let A, , A, ,..., A, be a set of representatives of those equivalence classes (mod -) whose members have cardinality greater than one, indexed in such a way that if i
4.
COWIUNC
A functor F: C ---+ D which is surjective on arrows /ifrs composition if for all t:w~x,u:x-,y,v:y---,zofD,ifg:c-,dinCandgF=u,thenthereare f: b ---+ c and h: d -+ e in C with fF = t, hF = V. (I called this triangle-rejecting in Wells [7] but that is a misuse of the word “reflect.“) Now let F: C + Set, G: D + Set be functors. Then G cozws F if there arc
D’ C D,
(a)
a subcategory
(b)
a functor
G,,: D’ + Set,
(c)
a functor
H: D’ -+ C, and
(d)
a natural
transformation
6: G, 4 H 0F
for which G, is a subfunctor H lifts composition
of G ’ D’,
(hence is surjective 0 is surjectivc.
on arrows),
(4.1) and
(4.2) (4.3)
A
KROHN-RHODES
THEOREM
It follows that if f: c -+ c’ is any arrow of D such that gH = f for which dG, -%
FOR
41
CATEGORIES
of C, then there is an arrow g: d -+ d’
d’G, (4.4)
commutes. Moreover, if f ‘: c’ + c” is an arrow of C, then there is R’: d’ + d” of D (starting from the same d’!) with g’H = f’, and similarly for an arrow going into c. It is this sense that G simulates F. In Rhodes’ language, F diwides G. .4s an aid to proving the Main Theorem, it is necessary to introduce a weaker notion of covering. Given F: C + Set, G: D + Set, G weakly covers F if there is a subcategory D” CD, a surjective function K from the objects of D” onto the objects of C, and for each object d of D a function do: dG - dKDF satisfying this condition: (C) If f: c - C” is an arrow of C, and dK = c, d,K - co, then there are objects P, e, and arrows s: d + e,, , g’: e - l d,, of D” such that eh’ --- c,
e,K = co
(43)
u x . do = cF ZedG (x . d8)(faF)
C x . gG . e,8
(4.6)
(all x E dG),
and
(all y E eG).
( y . e8)( fMF) C y . g’G . d,,e
(4.7) (4.7‘)
G weakly covers F with height i if the function t? in the preceding definition satisfies the requirement that for x E dG the set x . dt? is of height
if Homc(c,
Homc& c’) = 1>.’ ,-. (,therwise. Let /: C# -+ Set he the unique
functor
which
c’) f
::
on objects takes c to {c).
PROPOSITIOS 4.1. Let C he a finite category and F: C --, Set ajnite-set-calued functor, Let N be the maximum value of the height functionfor F. Then J: C# + Set weakly covers I; with height !Y.
Proof. In the notation of the definition of “weakly covers,” take D -.- D” : -C#, K the identity function, G /, ~8: c J -+ cflF the function taking {c) to {cF}. Then (4.5) (4.7’) follow easily (take e - c, e,, - c,, in (4.5)).
42
CHARLES PROPOSITION
jinite-set-valued covers F.
WELLS
4.2. Let C be a finite category and F: C + Set a faithful functor. If G: D - Set weakly covers F with hezght 0, then G
Proof. By hypothesis there is a subcategory D” CD, a surjective function K from the objects of D” to the objects of C, and for each object d of D” a function do: dG + dKflF that satisfy (4.5) through (4.7’). Let D’ be the subcategory of D whose objects are the objects of D” and whose arrows consist of all the g, g’ given by the definitions of “weakly covers with height 0” for all arrows f of C. (It is easy to check that D’ is closed under composition and has the requisite identity arrows.) For each object d of D’ and each x E dG, x . d# :: {x . de}. Then d8 is surjective by (4.6). Define a functor H: D’ + C as follows. For each object d, dH = dK. If g: d -+ d’ in D’, suppose dH = c, d’H = c’. By construction there is an arrow f: c -+ c’ such that for all .r E dG, (x . d@(foc,)
C x . gG . d’&
but since df? and d’8 are singleton-valued,
this means
(x . d0) f F = x . gG . d’0.
(4.8)
There cannot be another f making (4.8) true because F is faithful and d0 is surjective. Let gH 7.: f. Then if g’: d’ + d” is in I)’ with g’H = f ‘, we have (x.dO)fFcf’F
= [x.gG.d’B]f’F = x . gg’G . da0 = (x . de) . ff’F
so again because d0 is surjective and F is faithful H must preserve composition. H preserves identity arrows by a similar argument. Thus H is a functor. It follows from (4.8) that fJ is a surjective natural transformation from G 1D’ to H c:F. Taking G, - G i D’, I have already verified (4. I) and (4.3). The assumption (C) forces H to lift composition, so (4.2) is true. This proves Proposition 4.2.
5. THE
MAIN
THEOREM
A functor F: C 4 Set is a constant-function functor, or c.f. functor if for every arrow f: c + d of C, f F is a constant function.
for short,
THEOREM. Let C be a finite category and F: C + Set a faithful jinite-setvalued functor. Then for some integer n there are categories D, , D, ,..., D, and functors Fi: D, + Set such that
A KROHK-RHODES
THEOREM
(i)
F,WY F,wr ‘. . wrF,, covers F, and
(ii)
for
fumtor,
(a) or
each i, one of the following
FOR
43
CATEGORIES
two possiblities
hold:
Di is a finite category with no more objects than C has and F, is a c.f.
(b) Di is a group which is a homomorphic image of a submonoid of the endomorphism monoid of some object of C, and Fi is the right regular representation ofD;. Note. The Theorem is stated in a form intended to suggest the possibility of defining the “group complexity” of a finite category C: that would presumably be the least number of groups appearing in any iterated wreath product W of finite categories D, ,..., D, and functors Fi: Di + Set where (a) for each i either F, is a c.f. functor or Di is a finite group and (b) there is a subcategory DC W and a composition lifting functor H: D-C. The iterated wreath product is well-defined by Proposition 2.1. Proof of the Theorem. The theorem follows immediately from Propositions 5.1 and 5.2 below and Proposition 2.1. Some definitions are needed. If X is a set, X denotes the monoid consisting of the identity function and all constant transformations of X. If G is a permutation group on X, G = G U X is a monoid of transformations of X. PROPOSITION 5.1. Let G be a permutation group on a set X. Let R be the action of G on X, K the action of X on X, and p the right regular representation of G. Then R is covered by Kwrp: XwrG + Set.
Proof. Follows Recent expositions
immediately from “Method II” of Meyer and Thompson [4]. are in Eilenberg [1, II, Cor. 3.21 and Wells [6, Theorem 13.21.
PROPOSITION 5.2. Let F: C -+ Set be a faithful fkite-set-valued functor, C ajnite category. Let A, ,..., A, be representatives of equivalence classes (Mod -) and let e. = z?“~ be the corresponding holonomy group. Then F is covered by
R,wrR,wr
.‘. wrR,wrJ:
where Ri is the action of x
4wrgzwr
.‘* wr.@VwrC#
4 Set,
on aA, .
Proof. Suppose F: C + Set satisfies and suppose G: D + Set weakly covers functor G: XwroD - Set which weakly Proposition then follows from Propositions Since x is a category with only one identified with an object of D; however
the hypotheses of the Proposition, F with height i. I shall construct a covers F with height i - 1. The 4.1 and 4.2. object, an object of XwrD may be I shall write such an object d as di
44
CHARLES
WELLS
when I regard it as an object of .~wYD. An arrow (A,f): d’+ ei has f: d + e in D and X: dG - z any function. It is given that G: D + Set weakly covers F with height i. Thus there is a subcategory D” CD, a function K from the objects of D” to the objects of C, and functions do: dG 4 dKoZ, satisfying (4.5) through (4.7’). To prove the Proposition it is necessary to construct a subcategory E of gwz,D, a functor G: zwr,D + Set, a function L from the objects of E to the objects of C, and arrows di$: diG-+ diLtZF satisfying the analogs of (4.5)-(4.7’) with all d4+h of height
if x * d8 has height if x * do has height
(5.1)
Observe that (B, x) . d’t,b has height
(5.2)
and (B, , y) . ei$ . fOZpC (B,, , y) . (h’, g’) R,wrG
. $v
for (B, x) E diRiwyG, (B, , y) E eiRiwrG). The arrows of E will (h, g) and (h’, g’) satisfying (5.2) and (5.3). Observe that (B, x)(‘\, g) R,wrG
= (B . ml, x . gG).
(5.3) be all arrows
(5.4)
Case 1. If x . dt? has height
A KROHN-RHODES
THEOREM
FOR
CATEGORIES
45
subset of Ai by (4.7); it is a proper subset because z’c+ is a bijection and x . df? .f6YF is a proper subset of x . gG . e,8 (because its height is less). Therefore there is an element B, of gi which contains x * df? . f6TF. v6’lF. Let xh be the constant function from ,%‘i to gi with value B, , and let V: ai + c be the “inverse” to v given by Proposition 3.1. Then for any B cSfi , (B: x) . d’# . fCTF x . do. f& = x ’ d0 . fn, . zaF . %ZF C B, . ?XC!~ = (B, x)(/\, g) . R,wrG . e,,$b, the last equality by (5.4) and (5.1). Thus (5.2) holds in this case. Case 3. xd8 has height i; by (4.7) and the assumption that G weakly covers F with height i, x . gG . e,f? also has height i and moreover xdo = s . gG . e,,8. Let u: c + ai be an arrow taking x . dg to Ai, I: a, --f c its “inverse” as in Proposition 3.1, and v be as in Case 2. Let xh be the element of ,$ induced by tifc (it is easy to check that Ai . zifvUZF = Ai). Then for any B c.gi , (B, x) . d’4/1 . f& = B . iibl, . fC& = B . tifwiX& = B . x/\ . ~62’~ = (B . x/i, x .gG) . e,i#, as required. This verifies (5.2) and (5.3) is verified analogously.
ACKNOWLEDGMEHT The proving
author is grateful this theorem.
to S. Eilenberg
for suggesting
the use of holonomy
groups
in
REFERRNCF~ 1. S. EILF-NBERC, “Automata, Languages, and Machines, Vol. B,” Academic Press, 1976. (Some chapters by B. Tilson.) 2. C. HOUGHTON. The wreath product of groupoids, /. London Math. Sot. (2) IO (l975), 179-188. 3. G. KELLY, On clubs and doctrines, in “Category Seminar Sydney 1972-73,” Lecture Notes in Mathematics No. 420, Springer-Verlag. Sevv York/Berlin, 1974. 4. A. R. ~MEYER AND C. THOMPSON. Remarks on algebraic decomposition of automata, J. Moth. Sys. Theory 3 (1969), I IO-I IS. 5. W. NICO. Congruences and Extensions for small categories and monoids, preprint, Tulane Univ., 1978. 6. C. WELLS, Some applications of the wreath product construction. Amer. Math. Monthly 83 (1976), 317-338. 7. C. WELLS, Wreath product decomposition of categories, unpublished manuscript, 1976.