What is concrete equivalence?

Applied Categorical Structures 2: 57-70, 1994. 57 (~) 1994 Kluwer Academic Publishers. Printed in the Netherlands.

What Is Concrete Equivalence? For the sixtieth birthday o f Nico Pumpliin

HANS-E. PORST * Fachbereich Mathematik und Informatik, Universitiit Bremen, 28334 Bremen, Germany

(Received: 9 February 1993; accepted: 23 November 1993)

Abstract. The notion of concrete equivalence is introduced, based on a modification of the traditional notion of concrete functor. The discussion of examples includes a direct (i.e. not referring to any monadicity theorem) proof of the fact that monadicity is stable under concrete equivalence.

Mathematics Subject Classifications (1991). Primary 18A05, Secondary 18C15.

Key words: Concrete functor, concrete equivalence, monadic functors.

1. Introduction

In the recent book "Abstract and Concrete Categories" [2] the authors regretfully state that, "though it makes sense to say that two concrete categories over (a category) X are concretely isomorphic, it makes little sense to say that they are concretely equivalent" [2, 5.13]. The argument here is that "concrete equivalence" fails to be a symmetric relation since the equivalence-inverse of a concrete functor, which happens to be an equivalence, might fail to be a concrete functor again. This really being true would be a serious drawback to the otherwise fruitful concept of concrete category since clearly equivalence is a more important concept than isomorphism when comparing categories. Recall that, traditionally (see [2]), for concrete categories (A, U) and (B, V) over some base category X, i.e., for faithful functors U: A --+ X and V: B --+ X a functor F : A ~ B is called a concrete functor if F is compatible with the underlying functors U and V in the sense that the diagram

F A ~ B

X

commutes, while any such F is called a concrete isomorphism provided F is an isomorphism in addition, which is equivalent to say that F has a concrete inverse.

Hospitality of the Department of Mathematics, Applied Mathematics and Astronomy at UNISA is gratefully acknowledged, where this note was completed during an extended visit.

58 HANS-E. PORST

One might argue that both of these concepts are too narrow: - shouldn't a functor naturally isomorphic to a concrete functor be a concrete

functor, too (which it isn't necessarily)? - shouldn't concrete categories as (Ab, U) and (Ab, Ab(Z , - ) ) be concretely

isomorphic, or at least concretely equivalent (here U: A b > Se t denotes the obvious underlying functor of the category A b of Abelian groups and A b ( Z , - ) the obvious hom-functor)?

In the sequel we will present a concept of concrete functor - based on the categorical imperative that "the actual identity of the objects is irrelevant" (see, e.g. [3, p. 99]) - which in a natural way allows to define concrete equivalences and concrete isomorphisms respectively remedying the situation described above.

2. C o n c r e t e F u n c t o r s and C o n c r e t e T r a n s f o r m a t i o n s

DEFINITION 2.1. Let (A, U) and (B, V) be concrete categories over some category X. A concretefunctor (F, ~): (A, U) > (B, V) from (A, U) to (B, V) is apair (F, ~), where F: A -~ B is a functor and ~: V F ~-~ U is a natural isomorphism. Concrete functors of the form (F, 1) will be called strictly concrete 1.

Composition of concrete functors (F, ~): (A, U) ~ (B, V) and (G, ~y): (B, V) > (C, W) is defined by (G, ~/) o (F, p) : (GF, cy o ~yF).

Hence, the difference between concrete functors in the sense of this definition and strictly concrete functors is, that the latter make concrete categories "live" in the (strict) 2-category Cat/X, while, in this note, we think of Ca t /x as a pseudo 2-category; the importance of this difference has been stressed before (see, e.g. [4, p. 5]).

REMARK 2.2. Which of these concrete functors are to be seen as the "right" morphisms between concrete categories clearly is debatable. Considering the concept of concrete category (A, U) primarily as a concept for describing how categorical properties of A are determined by U one probably would prefer concrete functors. Considering concrete categories however as a structure in its own right, where besides its categorical properties the additional structure on A given by its decom- position into the fibres of U is of interest, one might prefer strictly concrete functors, since these respect fibres (in the strongest sense). One then could, however, also think of the following concept of fibre respecting concrete functor, i.e., of concrete functors (F, ~) satisfying in addition

U A = U A ~ '," ~A = ~A'.

This could be seen as more natural (and strictly weaker - see 2.5 examples l(a) and 2 and the example in 4.7) concept than strictly concrete functors. We will, however, not address this aspect in the sequel.

LEMMA 2.3. Concrete functors are faithful.

WHAT IS CONCRETE EQUIVALENCE? 59

LEMMA 2.4. Le t (G, 7): (A, U) ~ (B, V ) be a concrete functor. I f F : A --+ B

is a f u n c t o r a n d 7r: F - ~G a natural isomorphism, then (F, 7 o VTc): (A, U) , (B, V ) is a concrete functor.

EXAMPLES 2.5.

1. Let U: Ab ~ Set denote the canonical underlying functor of the category of Abelian groups; let ~5: Ab(Z, - ) - >U be the canonical representation of U, and let [Z, - ] : Ab > Ab be the inner hom-functor of Ab with U o [Z, - ] = A b ( Z , - ) . Then (a) (1, ~5) : (Ab, U) ~ (Ab, Ab(g , - ) ) is a concrete functor which is not

strictly concrete, as is (1 ,a -1) : (Ab, A b ( Z , - ) ) , (Ab, U). (1,a - t ) respects fibres, while (1, a) fails to do so.

(b) [Z, - ] : (Ab, Ab(Z, - ) ) ~ (Ab, U) is a strictly concrete functor. (c) ( [ g , - ] , 1)o (1,c5)= ([Z,-],c5) : (Ab, U) ~ (Ab, U) is aconcretefunc-

tor; (d) (1, ~) o ([7/,, - ] , 1) =

([Z, - ] , ~ [Z, - ] ) : (Ab, Ab(g , - ) ) , (Ab, Ab(Z, - ) ) is a concrete functor.

2. Let S be a skeleton of the category A with embedding E : S ~ A; let [: A ~ S be an equivalence inverse with natural isomorphisms 0: E ! ~ >1, 1 = or: I E > 1. Consider (A, 1) and (S, E) as concrete categories over A. (a) (E, 1): (S, E) > (A, 1) is a strictly concrete functor; (b) (I, 6) : (A, 1) > (S, E) is a concrete functor, which is strictly concrete

only if A is skeletal. In fact, only for A skeletal there exists a strictly conrete functor (A, 1) > (S, E) at all.

Observe that, in this example, both concrete functors are fibre respecting in the sense of Remark 2.2.

3. Let U: POS ~ Set be the canonical underlying set functor of the category of partially ordered sets and/~: POS --+ Set the re la t ion func tor with/~(M, _< ) = _< and R f = f 2 I<_.

(a) There exists no 6 such that (1, 6): (POS, U) > (POS, R) is a concrete functor (for any such 6 one would have by naturality 62a = 62 o R1 with the monotone map 1::2 d > 2 from the two-element discrete set into the two-element chain; but R1 fails to an injection).

(b) If Cat denotes the category of small categories and M: Cat ~ Set the obvious morphism functor, the interpretation of a partially ordered set as a small category yields a concrete functor (F, ~) : (POS, R) ---9 (Cat, M) with

h 9~(M,<):MF(M,<-) ~ <_, x ~ y , ~ ( x , y ) .

Next we define concrete natural transformations in the obvious way.

60 HANS-E. PORST

DEFINITION 2.6. Let (F, 9~), (G, 7): (A, U) , (B, V) be concrete functors. A natural transformation 7r: F > G is called a concrete natural transformation provided that

V F v~ VG ~ U = V F ~ U

LEMMA 2.7. (a) Concrete natural transformations are pointwise bimorphisms. (b ) Concrete natural transformations between strictly concrete functors are pre-

cisely the identity-carried 2 transformations.

LEMMA 2.8. (a) Given concrete functors (F, 9~), (G, 7), (H, X): (A, U) , (B, V) and concrete natural transformations 7r: (F, qo) , (G, 7), 6: (G, 7)

, (H, X), then 6o 7r: (F, qo) , (H, X) is a concrete natural transformation. (b) 1F: (F, qo) ~ (F, ~) is always concrete.

Using horizontal composition of natural transformations one also gets

LEMMA2.9 . Given concrete functors (F ,~) , (G, 7 ) : (A ,U) , (B,V) anda concrete natural transformation 7r: (F, ~) , (G, 7), then (a) HTr: (H, X) o (F, qo) , (H, X) o (G, 7) is a concrete natural transformation

for every concretefunctor (H, X): (B, V) , (C, W); (b) 7rH: (F, ~) o (H, X) > (G, 7) o (H, X) is a concrete natural transformation

for every concretefunctor (H, X): (C, W) > (A, U); (c) Horizontal composition of concrete natural transformations yields concrete

natural transformations.

EXAMPLES 2.10.

1. In Lemma 2.4, the natural transformation :r: (F, 7oV:r) , (G, 7) is concrete. 2. Not every natural transformation between concrete functors is concrete: con-

sider the category (pSet, U) of pointed sets as a concrete category over Set; let 1 be the identity functor on pSet (considered as a strictly concrete functor) and ~r: 1 > 1 the natural transformation which, on a pointed set (X, p), is the constant map r-p7 with value p. ~r is not concrete by Lemma 2.7(a).

3. In 2.5 example 1 the natural isomorphism 7r: [Z, - ] ~ )lAb is a concrete natural transformation between the concrete functors ( [Z , - ] , 1) and (1,8-I) : (Ab, Ab(Z, - ) ) , (Ab, U).

4. In 2.5 example 2 the natural isomorphism 6: E I " ~1 is a concrete natural transformation (E, 1) o (I, 6) = (EI, 6) , (1, 1), while a : I E - , 1 is a" concrete natural transformation (I, 6) o (E, 1) = (IE, 1) , (1, 1).

DEFINITION 2.11. Concrete functors (F, ~), (G, 7): (A, U) , (B, V) are called concretely isomorphic if there exists a concrete natural isomorphism 7r: (F, ~) ( a , 7 ) .

LEMMA 2.12. I f 7r: (F, ~) ~ (G, 7) is a concrete natural isomorphism, then : r - l : (G, 7) > (F, (p) is a concrete natural isomorphism, too.

WHAT lS CONCRETE EQUIVALENCE? 61

PROPOSITION 2.13. The following are equivalent for any (B, V ) :

(i) (B, V) is uniquely transportable, (ii) for every concrete functor (F, ~): (A, U)

strictly concrete functor ( F' , 1): (A, U)

concrete category

(B, V) there exists a unique > (B, V) concretely isomorphic to

Proof Since (B, V) is uniquely transportable, for every A in A there exists a unique B-morphism rCA: F A ~ F ' A with VrcA = ~A: V F A ~ UA, and 7rA is an isomorphism. Putting

F ' ( A f> B ) : = F ' A ~ F A Ff F B ~B F ' B

one gets a strictly concrete functor (F ' , 1) : (A, U) ~ (B, V), and re: (F, ~) > (F ' , 1) is a concrete natural isomorphism. Clearly, F t (as well as rr) is unique.

k Conversely, if V B , X is an X-isomorphism, consider the category I as a

concrete category over X by means of the functor rX-~ with value X. The functor rBT: 1 --+ B with value B becomes a concrete functor (rBT, k) : (I, r X 7) > (B, V), which by assumption has a unique strict modification; it follows that there is a unique B-object B ~ with V B ' = X such that k becomes a B-isomorphism B -+ B'.

F ~ as in (ii) above will be called the strict modification of (F, cp).

PROPOSITION 2.14, Let (F, V)), (G, 7): (A, U) , (B, V) be concretefunctors, whith V uniquely transportable. Let 7r: F ~ F' and 6: G ---+ G' be the strict modifications of(F, ~o) and ( G, 7) respectively. For any concrete natural transformat ion#: (F ,~) , (G,7) thenatural transformat ionoo#oTr- l :F ~ , G' is identity-carried.

3. Concrete Equivalences and Concrete Isomorphisms

The following definitions now are natural, though redundant as will be shown in the sequel.

DEFINITION 3.1. A concrete functor (F, ~) : (A, U) > (B, V) is called a - concrete equivalence, if there exist a concrete functor

(a, 2/): (B, V) , (A, U)

and a pair of concrete natural isomorphisms

r/: (G, 2/) o (F, ~) ~> (1A,1U) and e:(F,w) o(G,2/) ~ (1B,1V);

((F, ~), (G, 2/), r/, e) will then be called a concrete equivalence situation.

6 2 H A N S - E . P O R S T

- concrete isomorphism, if there exists a concrete functor (G, 7) as above such that (G, 7) o (F, 9)) = (1A, 1U) and (F, 9)) o (G, 7) = (1B, 1v).

Concrete categories (A, U) and (B, V) over X are called concretely equivalent (respectively concretely isomorphic), if there exists a concrete equivalence (respectively a concrete isomorphism) (F, 9)): (A, U) ~ (B, V). (G, 7) as above will be called concrete equivalence (respectively isomorphism) inverse of (F, 9)).

The following result shows that,

- for (F, 9)) to be a concrete equivalence F only has to be an (abstract) equivalence;

- for (G, 7) to be a concrete equivalence inverse of (F, 9)) not only means a com- patibility condition on 7 but also, that G, r/, e have to be chosen canonically, i.e., as an adjunction for F;

- in order to get a concrete equivalence inverse for (F, 9)) with F an equivalence, one can choose any adjoint equivalence inverse G and make it a concrete functor by means of a unique natural isomorphism.

LEMMA 3.2. Let (F, 9)): (A, U) given

(1) a functor G: B --+ A

(2) natural isomorphisms rl: G F ~ ~ 1A and e: F G ~ ~ 1B

(3) a natural isomorphism 7: UG ~ ~ V,

the following are equivalent:

(i) ( ( F, 9)), (G, 7), r/, e) is a concrete equivalence situation.

(ii) c~ )U(GF) g~ U = ( U G ) F 7~ V F ~, U

/3) v ( F a ) v = ( v F ) a u a v

(iii) (F, G, ~1-1 , e) is an adjunction and 7 = Ve o 9)-1G

, (B, V) be a concrete functor. I f there are

Proof The equivalence of (i) and (ii) follows by direct application of the definitions of composition of concrete functors and of concrete natural transformation. In order to prove that (ii) implies (iii), it is enough to show that eF = F~] (Gc = ~/G then will follow dually). For this, consider the following diagram where (1) commutes by/3), (2) commutes by c~), and (3) commutes since 9) is natural. Since 9) is monic and V faithful, the equality eF = F~/follows.

To prove the converse, only oz) has to be checked: 9) o 7 F = 9) o V e F o 9 ) - I G F

= 9) o VF~I o 9 ) - I G F (since eF = F~/by assumption) = U~7 (by naturality of 9))


V F G F ~ V F

V c F

(3)

V F ~

(1) U G F

V F U

The following is now immediate:

PROPOSITION 3.3. A concretefunctor (F, ~)" (A, U) equivalence iff the functor F: A ~ B is an equivalence.

, (B, V) is a concrete

The next corollary deals with a frequent situation.

COROLLARY 3.4. Let (F, 1): (A, U) , (B, V) be a strictly concrete equivalence, (t 7', G, % e) an adjoint situation. Then (G, Ve) is a concrete equivalence inverse o f (F , 1).

By straightforward calculation one gets moreover the following

COROLLARY 3.5. Let (F, 9)" (A, U) ~ (B, V) be a concrete equivalence with concrete equivalence inverse ( G , 7); assume that U is uniquely transportable. Then the strict modification o f ( G , 7) is a concrete equivalence inverse of ( F, 9), too.

As a corollary one gets alternative descriptions of adjoint equivalences:

COROLLARY 3.6. Let F: A -+ B and G: B --~ A be functors with natural isomorphisms ~7:1 ~ ~ G F and e: F G ~ ~ 1. Then the following are equivalent: (i) (F, G, % e) is an adjunction

(ii) The concretefunctors (F, 1) : (A, F) ~ (B, 1) and (G, 4): (B, 1) ~ (A, F) over B form a concrete equivalence situation ( ( F, 1), (G, 4), ~7-1,4).

(iii) The concretefunctors (F, ~7-1): (A, 1) ~ (B, G) and (G, 1) : (B, G) (A, 1) over A form a concrete equivalence situation ( ( F, 7-1), (G, 1), q - l , 4).

REMARK 3.7. on strictly concrete equivalences: 1. It follows from Lemma 3.2 that, for a strictly concrete isomorphism F, the

inverse F -1 is a strictly concrete functor (see [2, 5.13]).

64 HANS-E. PORST

2. The above proof shows what prevents a (concrete) equivalence inverse (G, ~/) of a strictly concrete eqmvalence F from being strict again: the canonical choice of 7/and e has to allow for an identity-carried e (i.e. Ve = 1). Because of U~ = VF~7 = VeF this then implies that ~/will be identity-carried, too. The converse, however, does not hold: ~ might be identity-carried without such a choice for e being possible. 2.5 example 2 provides a counterexample.

3. A strictly concrete equivalence, the concrete equivalence inverse of which is strict, too, need not be a strictly concrete isomorphism. A simple counterexample is the following modification of 2.5 example 2: consider the strictly

concrete functors (S, ls) E (A, I ) z (S, ls) over S as base category. 4. A strictly concrete equivalence with strictly concrete equivalence inverse is

a strictly concrete isomorphism, however, provided the underlying functors involved are amnestic in the sense of [2].

EXAMPLES 3.8•

1. With notation as in 2.5 example 1 we have

(a) (1,6): (Ab, U) ~ (Ab, Ab(Z, - ) ) is a concrete isomorphism; its concrete inverse is (1,5-1); none of these functors is strictly concrete•

(b) [Z, - ] : (Ab, Ab(Z, - ) ) , (Ab, U) is a strictly concrete equivalence; its concrete equivalence inverse is (Z ® - , Ue) with CA: [Z, Z ® A] -~ A the canonical isomorphism, e is not identity-carried; Z ® - is not strictly concrete• This is not a concrete isomorphism. Observe that, though all underlying functors in this example are amnestic, 3.7 remark 4 does not apply, since Ab(Z, - ) fails to create isomorphisms.

2. The concrete category (POS, R) is concretely equivalent to the essentially equational (see [1]) concrete category (OrdAlg, U) defined as follows: - Objects of OrdAlg are 6-tuples (A, c, d, % ~, ~) consisting of a set A, unary

operations c and d, and essentially equational partial binary operations

A 2 D dom',/~--~ A with dom'~= {(x,y) lcx = d y }

• A 2 D d o m ~ A with dom~p={(x ,y) l c x = c y A d x = d y }

• A 2 D d o m ¢ ~ A with d o m ¢ = { ( x , y ) ] c x = d y A c y = d x }

subject to the equations

CC = d c ~-. c~

dT(x, y) = dx,

v ) = x ,

v ) = x ,

dd = cd = d

c~(x, y) = cy

¢(x, y) = y


- OrdAlg-morphisms (A1, ca, dl, ")/1, q01, ~31 ) maps f : A1 --+ A2 satisfying

f (c la) = c2f(a), f (d ja) = d2f(a)

for every a E A. - U ( A , c , d , % ~ , ¢ ) = d , U f = f The equivalence is given by - the strictly concrete functor F: POS --+ OrdAlg defined by

f2 F ( M , <_) = (<_, ~, d, ~, @, ~), F( (M, <_) f (N, E_)) = <_ ~

with

and

(A2, C2, d2, "~2, ~2, ~32) are

- and its concrete equivalence inverse (G, ~), defined by

G(A, c, d, % ~, ¢) = (d[d], {(da, ca) I a e d}) , Gf(da) = f(da) = dr(a)

and I~A: {(da, ca) l a E A} --+ A, (da, ca) H a;

for details see [6]. 3. In 2.5 example 2 the concrete functor (E, 1)" (S, E) ~ (A, 1) is a strictly

concrete equivalence; its concrete equivalence inverse is (I, Q). The following are equivalent: - A is skeletal - (I, 6) is strictly concrete - (E, 1) is a (strictly) concrete isomorphism.

REMARK 3.9. Concrete equivalence hence can appear in any of the following versions:

sci: strictly concrete isomorphism (~7 = 1, e = 1, ~ = 1, hence 7 = 1) These are the concrete isomorphisms in the sense of [2].

ei: concrete isomorphism Q / = 1, e = 1),

see: strictly c_oncrete e_quivalence ((y = 1,7 = 1, hence Ue = 1, U~/= 1),

e e : c_oncrete equivalence.

Here (sci)---> (ci), but not conversely (3.8 example la); ( s c i ) ~ (sce), but not conversely (3.7 remark 3); ( s c e ) ~ (ce), but not conversely (3.8 example lb); (ci)----->, (ce), but not conversely (3.8 example lb).

We don't consider (ci) and (sce) as important here; these notions would probably be of interest if combined with the property of respecting fibres in the sense of Remark 2.2.

= ( y , y ) , d ( x , y ) = ( x , x ) , v ( ( x , y ) , ( y , z ) ) =

66 HANS-E. PORST

4. M o n a d i c F u n c t o r s

According to how seriously the categorical imperative mentioned at the end of Section 1 is taken, one of the following two definitions of monadicity of a functor U: A ~ X with left adjoint F and associated monad T = (UF, 7, UeF) is prefered in the literature: U is called monadic iff, respectively

- the comparison functor C: A ~ X T is an isomorphism ([2], [5]),

- the comparison functor C: A ~ X T is an equivalence ([3]).

We will, in this note, reserve the term "monadic" for the second option, while, in the first case, we use the term "strictly monadic".

If one considers monadicity not primarily as a property of the functor U but rather of the concrete category (A, U) - as it is done in [2] for strict monadicity - one should expect monadicity to be stable under concrete equivalence. The rest of this section is devoted to this problem.

It is we l lknown (see e.g. [71, [8], [9]) that the comparison functor C: A > X ~" for any adjunction (U: A > X, F : X ~ A, 7, e) the monad of which coincides with a given monad T on X, can be regarded as the unit of an adjunction. In this context one usually considers the following (quasi) categories: M o n X , having as objects the monads % = (T, 7, #) on X and as morphisms a: T ~ ~ ' those natural transformations a: T ' ~ T fulfilling

ct. 7' = 7 (1)

o~. # ' = # . (c~ o c~) = # . c~T. T'c~; (2)

and A d 0 X , having as objects adjunctions (U: A , X, F : X , A, 7, e) based on X ( A arbitrary), and as morphisms S: (U, F, 7, e) , (U', F', 7', e') those functors S: A , A ' with U'S = U. Forming the monad Mort(U, F, 7, e): = (UF, 7, UeF) then defines a functor Mono: A d 0 X , M o n X while construc- tion of the Ei lenberg-Moore adjunction EiIT = (U ~r, F ~, 7 T, e ~) gives a functor Eil: M o n X --+ Ad0X. Mono is left adjoint to E, il; the comparison functor is the (pointwise) unit of this adjunction. Moreover, Ell is a full embedding.

There are various ways of enlarging the category A d 0 X by allowing more gen- eral morphisms. What fits best in our setting is the following category A d X , where a morphism (U, F, 7, e) , (U' , U , 7', e') is just a pair (S, or) with S: A ~ A" a functor and o-: U'S ~ U a natural isomorphism; composition is defined by (T, r ) o (S, or): = (TS, ~ orS) . Clearly, A d 0 X is a (non-full) subcategory of A d X . The functor Mono described above can be extended, however, to a functor Mort: A d X ~ M o n X as we are going to show now.

L E M M A 4.1. For everymorphism (S,o-): (U,F, 7, e) , (U',F',7' ,e') i n A d X there exists a unique natural transformation c~(s,~): F' ~ SF such that the


diagram

commutes.

71 I 1 ~ UfF I

~l I I U' a(s,~)

UF ~ UISF a-IF

LEMMA 4.2 ([9]). Given (S, a): (U, F, 7, e) ~ (U', F', % e) in AdX, there is the following identity."

J S = Se .a (s ,~ )U. Ua .

LEMMA 4.3. Formorphisms(U,F,~,e) (s,a! (U',F',~],e) (T,~) (U" ,F" ,r / ' , e" ) in AdX one has

a(T,T)o(S,a ) = Ta(s,a ) o a(T,~_ ).

Proof Consider the commutative diagram

1 ~" U " F " U I!

~--1Ft UIF I ~ U " T F I

] U' a(s,G ) U " T a ( s ~ ) ~ . . . ~

UF ~ UISF a-IF T-1SF

UI' OL(T,'c)o(S,a)

U " T S F

PROPOSITION 4.4. The assignments

(U, F, 7, c), , (UF, 7, UeF)

(S ,a) , ~ a F o Ua(s,~)

define a functor Mort: A d X ~ M o n X .

Proof Mort(S, a) fulfills equation (1) by definition; (2) holds by commutativity of the following diagram, where (1) and (3) commute by naturality of a(s,~ ) and a respectively, while (2) commutes by the previous lemma.

68 HANS-E. PORST

T'T'

U' e' F'

T'U' c~ T' a F U' c~T a F T ' T 'U 'S F " T 'T " U 'SFT "

(1) T ' U ' S F (3) UeF

U'e'SF ~ U'SeF

T L

T ' ' U'SF " T U' c~ aF

ToprovefunctoralityofMon, observefirstthat, for(1,1):(UF,~,e) , (UF, rhe), a(1,1) = 1 and hence Mon(1, 1) = 1. It remains to prove that, with notation as in Lemma 4.3, Mon( S, a) o Mon(T, ~-) = Mon( (T, 7-) o (S, a)), i.e.

(aF o g ' c~(s,~l) o (~F' o U" C~(T,~)) = [a o ~S]F o U" ~(vS,~o~S).

Since ~- is natural, one first has

(aF o Ut ct(s,cO) o (TF t o gU ct(T,~-)) = aF o (TSF o gttroz(s,cr)) o Ut' Ct(T,.c);

now apply Lemma 4.3 3.

REMARK 4.5. The extension of Mono: Ad0X , MonX to Mon: AdX AdX can also be described as follows: Every (S, a): (U, F, .% e) ,'- (U', F ' , ~/, e') in AdX decomposes as

(u, F, 7, ~) ( "~ ( u ' s , F, a - i F o ~,~ o Fa) (S,l! (U', F', ~', ~').

Then Mon(S,a) = Mon(S, 1) o Mon(1,a) = Mono(S, 1) o Mon(1,a) with Mort(l, a) = aF and Mon( S, 1) = U' ~(s,~).

COROLLARY 4.6. Let (U, F, 7, e) and (U', F', 7', e') be isomorphic in AdX. (a) The EiIenberg-Moore categories ( X T, U v) and ( X T' , U v' ) are strict concrete-

ly isomorphic. (b) In particular, concretely isomorphic Eilenberg-Moore categories are strict

concretely isomorphic. Proof (a)follows from the fact that Mon and Eil are functors; starting with

Eilenberg-Moore adjunctions, one gets (b). O

REMARK 4.7. Statement (b) above does not mean that any given concrete isomorphism between Eilenberg-Moore categories is already strict. The concrete isomorphism (F, cp): (Set, 1) > (Set, 1) defined below is a simple counterexample:


Choose two sets M1, Me of the same cardinality and a bijection if: M1 ~ -~2. Let F act as the identity, except for M1, M2, which are interchanged, with appro-

priate modification of maps with e . g . X f~ M ~ > X f -~ M e , N. Let qo: t5" ~ l s e t be the natural transformation with ~m ~-1 = , ~ N = (I), ~ X = l x for X ¢ {M, N}. (Observe that (F, p) and its concrete isomorphism inverse are both fibre respecting in the sense of Remark 2.2.)

This shows in addition that EiI, considered as a functor Mon , AdX, is not a full embedding.

Though statement (a) above does not apply to concretely equivalent adjunctions (2.5 example 2 provides a counterexample), it is worth considering a concrete equivalence of the forra

(S, (7): (U, F, ')

for some monad ~'. By Lemma 3.2 and Corollary 3.5 the strict modification S ' of (S, (7) is a concrete equivalence, too (recall that U ~' is uniquely transportable!). Since the comparison functor C: A > X ? (T = Mon(U, F, 77, c)) is the unit of

the adjunction of Ell and Mono at (U, F, ~, e), S ~ factors as S ' A c X~ r L X Y' with U T' o L = U~:

A X ~ '

We claim that C is a (strictly concrete) equivalence as well. Clearly C is faithful, as is L. Given f : C A ~ CB, there exists f : A ~ /3 with S ' f = L f: S 'A = L C A ~ L C B = S 'B , since S' is full. Hence L C f = s ' f = L f , and therefore C f = f since C is faithful. This proves that C is full. Finally, for any X~'-object (X, x) there exists A in A and an isomorphism ~: L(X , x) ~ S'A, since S' is an equivalence. Now U~r'~p: U it' (X, x) = U~r'L(X, x) > U~r'S'A = UA is an isomorphism in X, such that (U ~r creates isomorphisms) there exists a unique X ~r- object (UA, ~), and U~r'(p lifts to an isomorphism (X, x) ~ (UA, ~). It follows (UA, ~) = C A since U~rC = U, hence (X, x) and CA are isomorphic.

Hence we have shown the non-trivial part of the following proposition, which clearly also follows from the version of Beck's Theorem as stated in [3].

PROPOSITION 4.8. The comparison functor of an adjunction (U, F, rl, c) is a concrete equivalence if and only if U: A --~ X is faithful and the concrete category (A, U) is concretely equivalent to (X 7' , U ~r' ) for some monad ~2'.

70 HANS-E. PORST

Notes

1. These are the concrete functors in the sense of [2]; omitting the identity transformation we might occasionally speak of the strictly concrete functor F.

2. See [2, 6.23], i.e., the 2-cells in the strict 2-category CAT/X. 3. This argument rectifies a slightly erroneous proof of the functoriality of Mono in [8].

References

1. J. Adfimek, H. Herrlich, and J. Rosick3): Essentially Equational Categories, Cahiers Topologie Gdom. Diff~rentielle (Cat@oriques) 29 (1988), 175-192.

2. J. Adtimek, H. Herrlich, and G.E. Strecker: Abstract and Concrete Categories, Wiley Interscience, New York, 1990.

3. M. Barr and C. Wells: Toposes, Triples and Theories, Springer, New York, 1985. 4. ET. Johnstone: Topos Theory, Academic Press, London, 1977. 5. S. MacLane: Categories for the Working Mathematician, GTM 5, Springer, New York, 1971. 6. H.-E. Porst: The Algebraic Theory of Order, J. Appl. Categorical Structures i (1993), 423-440. 7. D. PumplUn: Eine Bemerkung Uber Monaden und adjungierte Funktoren, Math. Ann. 185 (1970),

329-337. 8. D. PumplUn: Eilenberg-Moore Algebras revisited, Seminarberichte 29, Fernuniversit~it Hagen

1988, 97-143. 9. W. Tholen: Relative Bildzerlegungen und aIgebraische Kategorien, Ph.D. thesis, MUnster, 1974.

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