Seminarberichte Nr. 1

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Mathematik und

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Seminarberichte aus dem Fachbereich Mathematik der FernUniversität

01 – 1976

(Hrsg.)

Seminarbericht Nr. 1

Berichte von der Sitzung des Nordwestdeutschen

Kategorienseminars in Hagen, Oktober 1975

UB Hagen

1111 1 ^{m 1}

7307777 01

H, BERNECKER, R, BöRGER, G, GREVE,

H. ^HERRLICH, R.-E. ^HOFFMANN H.-E, ^PORST, W, ^THOLEN, F. ^LiLMER, M.B. WISCHNEWSKY

BERICHTE VON DER SITZUNG DES

NoRDWESTDEUTSCHEN KATEGORIENSEMINARS IN HAGEN, ÜKTOBER 1975

HERAUSGEGEBEN VON D, PUMPLÜN UND W, THOLEN

Fachbereich Mathematik

FERNUNIVERSITAT

HAGEN 1976

Initial completions

von Horst Herrlich

Topological completion of faithful functors von Rudolf-E. Hoffmann

Gelfand monads and duality ~n

cartesian closed topological categories von H.-E. Porst

und M.B. Wischnewsky

Locally presentable categories I I

von Friedrich Ulmer

Flache und halbinjektive Moduln~

Anwendungen von Funktorkategorien auf die Ringtheorie von Harro Bernecker

Fundamentalgruppoide

von Reinhard Börger

Abschwächungen des Adjunktionsbegriffs von Reinhard Börger und Walter Tholen (G,V) - Quotienten

0

von Georg Greve

3 -

26 -

38 -

61 -

70 -

77 -

85 -

- 112 -

Initial Completions Horst Herrlich

§ 0 Introduction. Let X be a category. We investigate categories over ~ i.e. pairs (~,U) where Ais a category and U : !:_ -- X is a fai thful and

amnestic~ functor, and functors F : (!:_,U)-(~,V) over X, i.e. functors F : A -- B with V• F=U.

A pair (!:_,U) is called initially complete provided for any family (A.) of !:_-objects,indexed by

l i ^E: I

some class I,and any source (X, (f. :X-UA.) )

l l i € I

in X there exists a U-initial source (A, ( g . : A-P. .. ) )

l l i E:. I

in A wi th UA = X and Ug. = f. for each i E: I. As is

l l

well known, ~, U) is initially complete iff i t is finally complete. In case Xis complete, initial completeness of (~,U) implies completeness of !:_.

The purpose of this paper is to investigate initial completions of categories over ~, i.e. full embeddings of categories (!:_,U) into initially complete categories

(~,V) over X. In case Xis a category with precisely one object and precisely one morphism, this amounts to the investigation of completions (in the classical sense) of partially ordered sets. Especially our results extend the results of B. Banaschewski

~A faithful functor U : A - Xis called amnestic provided any !:_-isomorphism f is an A - identity iff

Uf is an ~-identity.

Herrlich

and G. Bruns [2] and P. Ringleb [13] about the Mac Neille completion and its classification as an

injective hull. In case ~=Set, the completion problem (usually in connection with the problem of embedding Top into a cartesian closed, initially complete category) has been investigated by several

authors (cf.

E.

^Spanier

[14},

^{B. Day}

[7], P.

^Antoine

[1],

M. Chartrelle

[6], G.

^Bourdaud

[4], 0.

^Wyler

[15])

^who

in general neglected set theoretical difficulties.

As we will see, the general claim that initial completions always exist is false. In addition, the completion

(f,V) - (~,V) of Antoine, even if i t exists, need not have the universal property claimed in

[1].

§ 1 Fundamental Constructions Let (~,U) be a small category over X. We will construct 7 initial completions of (~,U).

( 1) The initial comleti•n E 1

: (A, U)-. (A; U 1 ) .

Objects of ~¹ are all pairs (X,() where Xis an

~-object and ( is a set of pairs (a,A) consisting of an ~-object A and an ~-morphism a:X- UA.

Morphisms f: (X,()-(X' ,(') of A¹ are all ~-morphisms f:X-X' such that (a,A)E(' implies (a•f,A)€(. The functor

u

¹ :A¹- Xis defined by

u

¹ (X,()=X and

u

¹f=f.

(Aj

,u

¹) is initially complete. A source

((X,(),(f.:(X,()-(X.,s.)) ) is

u

¹-initial iff

l l l iEI

(={(a•f.,A)

l (a,A)€(. for some i€I}. A sink

l

Herrlich

( ( f . : ( X . , ( . ) .... (X,()). I ' (X,())

l l l l t

is

u

¹-final iff (= { (a,A) (a•f. ,A)e( for each iEI}.

l

1 1 1

The functor E : ([2, U) - • ([2 , U ) , def ined by E 1 A=(UA, {(Ua,B) ¹ a:A - Bis an ~-morphism}) and E1

f=Uf is a full embedding.

1.1 Proposition: The full embedding E1

: ([2,U) - ([2¹ ,u¹) is an initial completion which preserves final structures, but in general fails to preserve initial structures or tobe initially or finally dense.

The initial completion E1

is too large tobe of general interest. But, as sem in 2.8,it implies that any partially ordered set can be embedded into the power set of some set, ordered by inclusion, such that the embedding preserves arbitrary joins. Hence E1

is of some interest in special cases.

(2) The initial completion E2

:(~,u)-(~~u²).

([22

,u

²) is the full subcategory of ([2¹

,u

¹), . . f h 1 b"

consisting o tose A -o Jects ^(X,

U

which satisfy the following condition:

If (a,A) e ( and b:A...,.B is an [2-morphism then ( Ub • a , B) ^{E. ( •}

(~2

,u

²) is initially and finally closed in (A1

,u

¹), hence initially complete.

Herrlich

1.2.1 Proposition: The full embedding E2

:(~,U) - (~2

,u

²), induced by E1 ,

is an initial completion which preserves final structures and is initially dense, but in general fails to preserve initial structures or tobe finally dense.

Moreover, E² is characterized as the largest initially dense full extension of (~,U):

1.2.2 Theorem: (1) If E: (A,U) - (~,V) is a full extension of (~,U) then the following conditions are equivalent:

(a) Eis initially dense

(b) there exists a full (initially dense)

2 2 2

embedding F: (~,V) -(~ ,U) with E =F•E.

(2) If E : (~2 ,U2

) - extension of (A2

,u

²)

(~,V) is a full such that E•E 2 is initially dense, then Eis an isomorphism.

Proof: To show that (a) implies (b) define F by:

FB = (VB, {(Va,A) a:B EA is a ~-morphism}) and Ff =Vf. F i s full since Eis initially dense, i.e.

(B, {a:B-EA ^! a is a ~-morphism}) is a V-initial

f h b . 2 . . f 11

source or eac ~-o Ject B. E =F•E since Eis u . For (2) observe that, for any ~-object B, the pair

(VB, { (Vb,A) \ b:B - EE2

A is a ~-morphisrri}) is an

2 b. .

A -o Ject C Wlth EC=B.

Herrlich

(3) The initial completion E3• (A,U) - _(A3,u³ (A3

,u

³) is the full subcategory of (~2

,u

²),

')

consisting of those ~--objects (X,s) which satisfy the following condition:

If (A, ( f . : A-A. ) ) i s a

l l iE:I

U-initial source in A and

a:X - UA is an ~-morphism such that {(Uf.•a,A.) _{l l} ¹ iE I} c.

C

then (a,A) E.

s-

(~3

,u

³) is finally closed in (~2

,u

²), hence initially complete.

1. 3. 1 Proposition: The full embedding E3

: (~,U)-(~3

,u

^3),

induced by E2

, is an initial completion which preserves initial and final structures and is initially dense, but in general fails tobe finally dense.

Moreover, E3 is characterized as the largest initiality preserving, initially dense full extension of (~,U):

1 . 3. 2 Theorem: (1) If E: (~,U) - (~,V) is a full extension of (~,U) then the following conditions are equivalent:

(a) Eis initiality preserving and initially dens, (b) there exists a full embedding F: (~,V) -(~3

,u

³)

with E3

=F•E.

(2) If E: (~3

,u

³) - (~,V) is a full extension of (~3

,u

³) such that E•E3 is initiality preserving and initially dense, then Eis an isomorphism.

Herrlich

E3

is also characterized by the following universal property.

1. 3. 3

initiality preserving initial completion of (~,U) such that for any initially complete category (~,V) over ~ and any initiality preserving functor F: (~,U)-(~,V) over X there exists a unique initiality preserving functor F: (A³

,u

³) - (~,V) over X such that F=F•E³•

Proof: For each A³-object C=(X,(), let (Be, (fa:Bc-FA) (a,A)€~) be the unique V-initial source in~ with VBc=X and Vfa = a

- 3 3

for each (a,A)E:(. Let F: (~ ,U )-(~,V) be the unique functor with FC=B

C for each f-object C. To show that

f

has the desired properties, observe that a source

( (X, () , ( f . : (X, () ^l

-

^{(X.,(.)) l l iE:I) in A3 is}

^u

^{3 · · -1n1 1a t · 1}

iff ⁽ is the smallest set of pairs ( a, A) , with A

~-object and a:X-UA an ~-morphism, satisfying the following conditions:

(1) if ^ü:I and (a,A)E:C, then (a•f.,A)€(

l l

(2) if (a,A)€( and b:A - Bis an ~-morphism, then (Ub• a, B) ^{€ (}

an

(3) if (A, (gk:A - Ak) ) is a U-initial source in~, kE:K

a:X-UA is an ~-morphism, and (Ugk•a,Ak)E:( for each kE:K, then (a,A)E:(.

Herrlich

(4) The initial completion E4

: (A,U) - (A4

,u

⁴).

4 4 . 1 1

(~ ,U) 1s the full subcategory of (~ ,U), consisting of those A¹-objects (X,s) which satisfy the following condition:

If ns is the set of all pairs (A,a) consisting of an A-object A and an

~-morphism a:U - X such that for any (b,B)es there exists an ~-morphism f:A - B with Uf=b•a, then s is the set of all pairs (b,B) consisting of an

~-object Band an ~-morphism b:X - UB such that for any (A,a)ensthere exists an ~-morphism g:A - B with Ug=b•a.

(~4

,u

⁴⁾ is finally closed in (A¹

,u

^1), hence initially complete.

1.4.1 Proposition: The full embedding E4

: (~,U) - (~4

,u

⁴), induced by E1

, is an initial completion which preserves initial and final structures and is initially and finally dense.

These properties characterize E : 4

1.4.2 Theorem: If E: (~1U)- (~,V) is an initially and finally dense initial completion of (~1U) then there exists an isomorphism H: (B,V)- (~~u⁴) with V=U⁴•H.

Since, as we will see in the next section, the

~1ac Neille completion of a partially ordered

~et is a special case, we introduce the following terminology:

Herrlich

1.4.3 Definition: Any initially and finally dense initial completion of (~,U) is called a Mac Neille completion of (~~U).

Obviously the Mac Neille completion (A4

,u

⁴) is a subcategory of (~3

,u

³). This is not an accident, since the Mac Neille completion of (~,U) is the smallest completion of (~,U).

1.4.4 Theorem: Any initial completion of (~,U) contains, as a full subcategory, a Mac Neille completion of (~,U).

Proof: Let E: (A,U) - (~,V) be an initial completion of (~,U), let (~', V') be the initial hull of EA in (~,V), and let (Bx,Vx) be the final hull of EA

in (~,V'). Then the induced embedding Ex: (~,U) - (~x,Vx) is a Mac Neille completion of (~1U).

R.-E. Hoffmann [10] has pointed out that the above results 1 .4.2 and 1.4.4 immediately imply the

following generalization of an extremely useful result of H. Müller [11]:

1.4.5 Theorem: If (~,U) is a full subcategory of an initially complete category (~,V)

then the following conditions are equivalent:

( a) (~, U) is initially complete

(b) (~, U) is finally closed in its initial hull

( C) (~, U) is initially closed in its final hull in (~, V) •

(~, V)

Herrlich

The Mac Neille completion of (~,U) is

characterized not only as the smallest initial completion of (~,U) but also as the largest initially and finally dense full extension of

(~,U). These properties will be analyzed in detail in§ 2.

(5) The initial completions E-n: (~,U) - (~-n,U-n) for n=1,2,3,4.

The constructions of the initial completions

n n n

E : (~,u)-(~ ,u) can be dualized. The dual

constructions yield final (=initial) completions

-n -n n

E : (~,U) - (~ ,u). Especially the objects of A-7 are the pa1.·rs (C s, X) w ere h X · 1.s an _-o Jec X b" t and

s

is a set of pairs (A,a) consisting of an ~-object A and an ~-morphism a:UA - X, etc.

E- 3 : (~,U)-(~- 3

,u-

³) is characterized as a finality preserving final completion

of (~₁U) such that for any finally complete category (~,V) over ~ and any finality

preserving functor F: (~,U) - (~,V) over X there exists a unique finality preserving

- -3 -3

functor F: (~

,u )

(~,V) over X with

F=F

E-3. E-4: (~,U) - (~-4

,u-

⁴) is a Mac Neille

Herrlich

completion of (~,U), hence isomorphic to E . 4 l d . h t t· E4 and E-4 t th Ben 1ng t e cons ruc 1ons oge er one obtains the following perhaps most natural

description of a Mac Neille completion E~: (~,U)-(~~,U~) of (~,U). Objects of A~ are all triples (n,X,s) where

( 1 ) X is an ~-object

(2) 17 is a set of pairs (A, a) consisting and an ~-morphism a:UA-X

( 3) s is a set of pairs (b,B) consisting and an ~-morphism b:X-UB

subject to the following conditions:

of an ~-object

of an ~-object

(4) if Ais an ~-object and a:UA-X is an ~-morphism then (A,a)€n iff for each (b,B)€s there exists an

~-morphism f:A-B with Uf=b•a

(5) if Bis an ~-object and b:X-UB is an ~-morphism then (b,B)ss iff for each (A,a)sn there exists an

A-morphism f:A-B with Uf=b•a.

Morphisms g: (17,X,s) - (n' ,X' ,s') of A~ are all

~-morphisms g:X-X' such that for each (A,a)en and each (b,B)€n' there exists an ~-morphism

f:A-B with Uf=b•g•a.

In general the initial completions Ei: (~,U) - (~i,Ui) are different for i = 1,2,3,4,-1,-2,-3.

A

B

Herrlich

§ 2 The category ~~tx Let~ be a small category.

g~tx denotes the category, whose objects are all small categories over ~, whose morphism are all functors between small categories over ~-

2.1 Proposition: If (~,U) is a g~tx -object then n n

so are (~ ,U ) for n=±1, ±2, ±3, ±4.

2.2 Theorem:

(1) If Cat init. denotes the object-full ---X

subcategory of ~~tx, whose morphism are the initially preserving functors, then the full

b t f C t init. . t· f 11 su ca egory o -~-X , consis ing o a initially complete small categories over

. . fl t. . C t ini t. Th . . t. 1

~, is epire ec ive in -~-X . e ini ia completion E3

: (~,u)-(~3 ,u3

) is the reflection.

(2) If g~txfin. denotes the object-full subcategory of g~tx, whose morphism are the finality preserving functors, then the full subcategory of g~txfin., consisting of all finally complete small categories

. . f 1 . . C t f in . Th is epire ective in -~-X . e

-3 - -3 -3

completion E : (~,U) - (~

,u )

reflection.

over ~, final is the

( 3) in-fi .

If g~tx denotes the obJect-full subcategory of g~tx, whose morphisms are the initially and finality preserving functors, then the full subcategory of Cat in-fi,

---X

2.3

2.4

Proof:

Herrlich

consisting of all initially (= finally) complete small categories over ~, is not reflective in Cat in-fi_

---X

Proof: (1) and (2) just rephrase results of

§ 1 . ( 3) follows, in view of Example 1 (below) immediately from the fact that there exist arbitrarily large complete lattices generated

(as complete lattices) by a 3-element set (A.W.Hales

[s] ).

Definition: A morphism Ein ~§tx is called essential provided i t satisfies

the following conditions:

(1) E i s a full embedding~

(2) if a ~§tx-morphism Gis such that G•E is a full embedding

then Gis a full embedding itself.

Proposition: For full embeddings E: (~,u)- (B,V) in

~§tx the following properties are equivalent:

(1) Eis essential

(2) Eis initially and finally dense (3) there exists a full embedding

F with F•E

=

E~.

(1)

=>

(2) Let E be essential. There . t . f t G (B V) - ("!::_2

,u

²) such that exis s a unique unc or : _,

~ A ~§tx-morphism is full iff i t is a full embedding.

Herrlich

for each ~-object B,

GB= (VB, {(Vf,A)

l

f:B EA is a ~-rnorphisrn}).

The functor G•E = E2

is a full ernbedding. Hence Gis full. This irnplies that for each ~-object B, the source _(B' (a.: B

l ~ EA.) ) , consisting of

l iE:I

all ~-rnorphisrns a. :B - EA. with dornain Band

l l

range in E~, is initial. Hence Eis initially dense.

Final denseness follows by a dual argurnent.

(2)

=>

(3) Let E be initially and finally dense. The unique functor F: (~,V) - (~~,U~), sending each ~-object B into

FB= ( { (A,Va) ^j a:EA-BE:Mor ~}, VB, { (Va,A) ^j a:B-EAe:Mor B}), is a full ernbedding.

( 3)

=> (

1 ) Let F be a full ernbedding with F•E

=

E~

'

and let G: (~,V) - (f,W) be a functor such that G•E is a full ernbedding. To show that Gis full, consider a pair (B,B') of ~-objects and a

f-rnorphisrn f:GB - GB'. Consider further

FB= (n,VB,~) and FB'=(n' ,VB',~'). Since F i s full, for each (A,a)e:n there exists a ~-rnorphisrn

a:EA - B with Fa=a, and for each (a' ,A')~~• there exists a ~-rnorphisrn

a'

:B-EA' with Fa'=a'.

Consequently Ga'•f•Ga:GEA - GEA' is a

~-rnorphisrn. Since G•E is full, this irnplies that there exists an ~-rnorphisrn g:A-A' with

Herrlich G•Eg=Ga' •f•Ga, equivalently with

Ug = a'•Wf•a. By definition of ~ , this ~

' ~ h'

irnplies that Wf:FB-FB is an~ -rnorp isrn.

Since F i s full, there exists

a ~-rnorphisrn f:B-B' with Ff=Wf, equivalently with Gf=f.

2.5

2.6

Proof:

Definition: A ~~Ex-object (~,U) is called injective provided for any full ernbedding E: (~,V)-(f,W) in ~~Ex and any functor F: (B,V) - (~,U) there exists a functor

~: (C,W) - (~~U) with ~•E=F.

Theorem:

For any ~~Ex-object (~,U) the following conditions are equivalent:

(1) (~,U) is initially cornplete (2) (~,U) is injective

(3) any full ernbedding E: (~,U)

~~Ex is a section

(~,V) in

(4) (~,U) has no proper essential extension (5) the Mac Neille cornpletion Ex: (~,U) - (~x,U~)

is an isornorphisrn.

(1) => ⁽²⁾ Let E: (~,V)- (f,W) be a full ernbedding, and let F: (~,V) - (~,U) be a functor. For each f-object C consider the farnily (f. :C -

l EB.) l iEI

Herrlich

consisting of all f-morphisms with domain C and range in E~. There exists a U-initial

(A (a. :A -FB.) ) such that UA =WC and

C, l C l . I C

lE

Ua.=Wf. for each ieI. The unique functor

l l

F:

^{(f,W) -} (~,U), which sends each f-object C into the corresponding ~-object A, is

C

the required extension of F.

(2) => (3) obvious.

(3) => (4) Let E: (A,U) - (~,V) be essential.

There exists a left inverse G: (~,V) - (~,U)

of E. Since G•E = 1 is full and Eis essential, G has tobe a full embedding.

Hence G and E are isomorphisms.

(4) => (5) => (1) Obvious.

2.7 Corollary: For any ~~~X-morphism

E: (~,u)- (~,V) the following conditions are equivalent:

2.8

(1) E i s a Mac Neille completion of (~,U) (2) Eis essential and (~,V) is injective (3) Eis the smallest injective extension

of (~,U)

(4) Eis the largest essential extension of (~,U).

Example 1: Let~ be a category with precisely one object and precisely one morphism.

If (~,U) is a ~~!x-object, then the faithfulness

Her~lich

of u implies that (A,U) can be considered in the usual way, as a set with a reflexive and transitive binary relation, and the amnesticity of U implies that this relation is antisymmetric. Hence ~~tx-objects are just partially ordered sets and ~~tx-morphisms are just monotone maps, i.e. ~~tx is the category POS of partially ordered sets and monotone maps. Initial source means meet, esp.

initially dense means meet dense, initially preserving means meet preserving, initially complete means complete in the order theoretic sense. Final sink means join, etc. Essential means essential, Mac Neille completion means Mac Neille completion, and our results of this

section obviously generalize the corresponding results of B. Banaschewski and G. Bruns [2] and P. Ringleb [13] concerning the categorical role of the Mac Neille completion. The completions

n n

E :P-P n€ {1,2,3,4,-1,-2,-3} are usually

4 -4

different. P and P are the Boolean algebra of subsets of the underlying set of P, ordered by inclusion, hence isomorphic, but the embeddings E4

and E-4

in general fail tobe equivalent.

If P has the diagram

p 0

0

Herrlich

n n {

then the four completions E :P-P n € 1,2,3,4}

of P are different.

The diagrams of P1 P2

P3

, , ,

and P ⁴ are as follows:

0

/i\

G 0

\ XX

• 1/

The difference of P3

and P4

demonstrates that the Mac Neille completion in general fails to have the universal property of E3

:P-P3

. This error of N. Bourbaki [3, P.104] has been pointed out before by P. Ringleb [13]. If Q is the rationals with their usual order, then E3

and E4

are the usual embed- dings of Q into the reals R, but E2

is the embedding of Q into the linearly ordered set obtained from the reals by doubling each rational point.

HerrJich

§ 3 Initial completions of large categories

Let~ be large. If (~,U) is a small category over ~' then the fundamental constructions

n n n

E : (~,U) - (~ ,U) of § 1 can still be performed, but each of these initial completions and, in fact, any initial completion of (~,U) is large. Hence these constructions cannot be performed in the category of small categories over X. Nevertheless most of the results of § 2 remain valid, if properly reformulated. In case (~,U) is large, several of our results still remain valid, - especially (~,U) is initially complete iff it is injective with respect to full embeddings over ~ (this surprising result has been obtained independently andin a rather different looking form by G.C.L. Brümmer and R.-E. Hoffmann

[s] -,

but our fundamental constructions cannot be performed any more.

Consequently several unpleasant things may happen, e.g.

(1) there may not exist any initial completion of (~,U) at all; equivalently, there may not exist a Mac Neille completion of (~,U) (2) there may exist a Mac Neille completion

of (~,U), but no initial completion of (~,U) with the universal property of E . 3

(3) there may exist a Mac Neille completion of (~,U) with large fibres, even though

(~,U) has small fibres etc.

3. 1

Herrlich

Example 2 Let X be the c~tegory Set of sets and functions.

(a) Let (A,U) be the concrete category, whose objects are pairs (X,T) consisting of a set X and a collection of subsets of X closed under finite unions and finite intersections, and whose morphisms f: (X,T)-(Y,o) are

functions f:X-Y such that f-1

[s]€T for each SEo.

Let (B,V) be the full subcategory of (~,U) consisting of those objects (X,T) for which T is closed under arbitrary unions and arbitrary intersections, i.e. (f,V) is the category of finitely generated topological spaces. Let (f,W) be the full subcategory of (f,V) consisting of those objects (X,T) for which Xis finite, i.e.

(f,W) is the category of finite topological spaces.

The embedding (f,W)--- (f,V) is a Mac Neille

completion of (f,W), and the embedding (f,W)-(~,U) is equivalent to E3

: (f,W)- (f3

,w

³). This

dernonstrates that the construction v: (f,W)

(f, W)

of P. Antoine [1], which is a Mac Neille cornpletion of (C,W), does not have the universal property

of E3

: (C,W)

(f

³

,w

³), as clatrned in [1].

Herrlich

(b) Let Q be a proper class. Consider the following subcategory ~ of Set:

objects are all one element sets of the form { (a,i)} with aEQ and

iE:{1,2}, and a function f:{(a,i)}-{(ß,j)}

is an ~-morphism iff either (a,i)=(ß,j) or i<j and atß. Let U:A

embedding.

Set be the

(~,U) has no initial completion. Otherwise i t would have a Mac Neille completion (Ax,Ux), which in turn would have the property that, for any non empty set X, the Ux-fibre of X would be in a one-to-one correspondence with the "conglomerate" of all subclasses of Q. But this, for any decent set theory, would give rise to a Russell type paradox.

Let B be the subcategory of Set which has the same objects as ~, but such that a function f:{ (a,i)}-{ (ß,j)} is a B-morphism iff (a,i)

=

(ß,j) or i<j. Let V:B-Set be the embedding.

(~,V) has a Mac Neille completion with large fibres, but no initial completion with the universal property of E • 3

Herrlich

(c) Let~ be the subcategory of Set whose objects are all sets whose cardinal nurnber is either a lirnit cardinal or a direct successor of a lirnit

cardinal, and whose rnorphisrns are all functions f:X-Y between ~-objects satisfying at least one of the following conditions.

(1) f is bijective or constant (2) Card Xis a limit cardinal and

Card Y is not a lirnit cardinal and not a direct successor of Card X.

Then (~,U) has no initial cornpletion, even though i t is transportable and any constant rnap between ~-objects is an A-rnorphisrn.

Sufficient conditions for the existence of initial cornpletions of (~,U) can be found in forthcorning papers by H. Herrlich and L.D. Nel [9], R.-E. Hoffmann [10] and 0. Wyler [15].

Herrlich

References

[1]

[2]

[3]

[4]

[ 5

J

[6

J

[7

J

P. Antoine

B. Banaschewski and G. Bruns

N. Bourbaki

G. Bourdaud

G.C.L. Brümmer and R.-E. Hoffmann

M. Chartrelle

B. Day

ttude elementaire des categories d'ensembles structures.

Bull.Soc.Math. Belge 18 (1966), 142 - 164

Categorical characterization of the Mac Neille completion.

Archiv Math. 18 (1967), 369 - 377.

Theorie des ensembles. Ch. 3 Ensembles ordonnes.

Hermann, Paris 1963.

Some cartesian closed topological categories of convergence spaces.

An external characterization of topological functors.

Proc. Int. Conf. Categ. Topology, Mannheim 1975.

Construction de categories auto-dominees.

C.R.Acad.Sci.Paris Ser.A-B 274 (1972), A 388- A 391.

A reflection theorem for closed categories.

J. Pure Appl. Algebra 2 (1972), 1 - 11 .

[ 8

J

[9]

[1 o)

[11]

[1 2]

[1 3]

[ 1 4]

[1 5]

A.W. Hales

H. Herrlich and L.D. Nel

Herrlich

R.-E. Hoffmann

H. Müller

G. Osius

P. Ringleb

E. Spanier

0. Wyler

On the non-existence of free complete Boolean algebras.

Fund. Math. 54 (1964), 45 - 66.

Cartesian closed topological hulls.

Preprint.

Topologische Vervollständigung treuer Funktoren.

Nordwestdeutsches Kategorien- seminar, Hagen 1975

Über die Vertauschbarkeit von Reflexionen und Coreflexionen.

Quasi-Top-Kategorie über Meng.

Preprint, Bielefeld 1974 Eine axiomatische Struktur- theorie.

Thesis. Free University Berlin 1969

Untersuchungen über die

Kategorie der geordneten Mengen.

Thesis, Free University Berlin 1969.

Quasi-topologies

Duke Math.J. 30 (1963), 1 - 1 4.

Are there topoi in topology?

Proc.Int. Conf. Catego Topology, Mannheim 1975.

"Topolo,11;ical Completion of Faith:ful Functors"

Rudolf-E. Hoffmann

This note is the introduction (§0) of a (handwritten) manuscript

1975/76

entitled "Topologica] Completion of Fai thful Func tors ^{11 [} 9] . I t descri bes the con ten ts of that paper. The biblio~raphy given at the end is

different from the bibliography of the whole paper.

Hoffmann

The main purpose of the present paper is to investigate possibilities to find for a given faithful functor

V:C---.D a topological functor W:X-D and a full embedding C--X such that V is the corresponding restriction of W.

This program is carried out in several steps.

In section 2 we associate to an arbitrary functor V:C-.6 a .6-identifying functor ,6V: L'.1~-D, i.e. a functor 6 V admi tting identifying lifts for 6 V-data of U-small discrete type (U denotes the universe we are wor- king in) , and a full embedding 6 E: 6

c -

D such tha t

V= /:::,. V " b.. E. This construction fulfils an obvious "univer- sal property". As a byproduct we prove that V i s a b..-semi- identifying functor, if ßE is a right adjoint, i.e. V is a "reflective restriction" of a ~-idt. (= ß-identifying) functor. In this connection, i t is proved that every

semi-idt. (=semi-identifying) functor V whose codornain is co-complete can be obtained as a "reflective restric- tion" of an idt. functor. This, however, is done by means of a different construction, which is well known as

"comma-category" (D,V). (We briefly comment the basic role a suitable ¹¹faithfulization¹¹ of this construction plays in the theory of ¹¹topological functors admitting genera- lized Cauchy-completions" developed in an earlier paper

[ ß] • )

Another (universal) ¹¹faithfulization" applied to the above

11 ti.-construction¹¹ gives us a construction, called "A-con- struction" which associates to a faithful functor a faith- ful idt. functor (which, however, may fail tobe a topo- logical functor) satisfying a universal property. The A- construction was introduced by G. Osius [12] •

In section J we study a similar construction for a faith- ful functor U:C- D, called B-construction, which is strongly related to construction(s) introduced by Ph.

Antoine [1] in case D = Ensi in particular, i t coincides

Hoffmann

with O. Wyler's construction for base category D = Ens ( [15] , who has also corrected some mistakes in [ 1

J ) •

Let us summarize our main results on the B-construction:

The B-construction associates to a faithful functor

V:C-D a category B,which is legitimate only with respect to the universe

u+

next to U, a functor T:B___..D, which is topological in the universe

u+,

and a full and faithful functor F:C---..B with V=TF. The B-construction has a "uni- versal property". More important, however, is the follow-

ing "maximality property":

Let

J

F

C - - - B

K - - - D G

T

commute, let J, G be faithful, and let J be full, then according to a theorem from G.C.L. Brümmer - R.-E. Hoffmann [J] we have a "T-finest" diagonal K :K--- B. Moreover, K is full

( and, obviously, fai thful), provided for every XE: ObK there is a G-identifying co-cone with codomain X, whose

"diagram part" fac tors over J: c ~ K. Such a pair ( J, G) with GJ=V is called a join-extension of V; i f Gis a topo-

logical functor, (J,G) is called a topological join-

completion. (F:C---B, T:B-D) is the "maximal" topologi- cal join-completion.

Given any join-extension (J,G) of V, then the above

"T-finest diagonal" K:K~B takes G-co-identifying cones into T-co- identifying cones, in particular, so does

J: C ^---+ B *). If G is a topological functor, then K is

- up to equivalence - a T-co-identifying subcategory of B.

*) (idC' V:C~D) is a join-extension of V:C-D.

Hoffmann

The B-construction "categorizes" some ideas implicit in Spanier's concept of quasi-topology. In order to illustrate this, we give an ad hoc-definition of V-quasi-topology for any fai thful functor V:C ~ D. In section

4

we are in the heart of the paper. We study the B*-construction, which associates to a faithful functor V:C--+D the "minimal"

topological j oin-completion (F: C - B*, T*: B* - D); accor- ding to the above, B* is a full T-co-idt. subcategory of B, i t is the smallest T-co-idt. subcategory of B contai- ning (the image of) C (under F), T* is the corresponding restriction of T. Note that B* need not be U-legitimate, topologicity of T* is tobe understood with respect to the universe

u+.

Let us summarize those results on the B*-construction, which in our opinion are most important:

(i) If V:C-D is topological, then (and only then) F:C--.B* is an equivalence

(4.7).

(ii) The B*-construction is self-dual:

Let V:C-D be a faithful functor. If we apply the B*-construction to

v

⁰

P:c

⁰

P - o

⁰P, we arrive at

(F*⁰

P:c

⁰P~B*op , T*⁰P:B*⁰

P - o

⁰

P).

In connection with (i)

(4.7),

this duality implies the Antoine- Roberts-duali ty ( [ 1

J , [

13] ) • Cf.

4.

5.

(iii)Let (J:C-Y, U:Y-D) be any topological completion of V:C-D, i.e. J is full and faithful and U:Y-D is a topological functor with UJ=V.

F*

Let K* denote the U-idt. hull of the image of C under J, and let M* denote the (U/K*)-co-idt. hull of C in K*. Then there is an equivalence E* :M*- B* rendering commutative

C ---+M* J*

- /..

B*

I

M*

U/M*

E B*

/"V

T*

n

_Cf.

_4.8.

Hoffmann

(iv) = dual of ( i i i ) . Note that M*$ M*. Cf. ¹±.9.

(v) If both (E:~--Y, U:Y--D) and (E⁰P,

u

⁰P) are topological join-completions of V:~--, D and

v

⁰P, respectively, then (E, U) is up to an equivalence the B*-construction applied to V. Cf.

4.14.

(vi) Let U:Y--D be a topological functor, and let C be a full isomorphism-closed subcategory of Y, then the following conditions (a), (b), (c), (d) are equiva- lent:

(a) U/C is a topological functor.

(b) There exists a functor G:Y- C with

(U/C ) ^o G=U and

G/ Cis the inclusion of C into Y.

(c) Let K denote the U-co-idt. hull of C in Y:

Cis a (U/K)-idt. subcategory of K.

(d) Let K' denote the U-idt. hull of C in Y:

Cis a (U/K' )-co-idt. subcategory of K'.

(c) and (d) were proved by Dr. H. Müller (Bielefeld) in an informal manuscript (

[11]

he gave a talk on his results in Münster

1974)

in a special case:

D=Ens, U = Horn (t,-) where t denotes a terminal object of Y. Müller's proof seems tobe quite different from our proof in section

4.

H. Herrlich has pointed out that (vii)B* is not necessarily U-legitimate.

(viii) B* fails to have the universal property Ph. Antoine [1] claims. Cf.

4.

16.

(ix) We give sufficient criteria for B* tobe U-legitimate

(4.19).

These criteria are based upon our theory of

"topological functors admitting generalized Cauchy- completions" developed in an earlier paper [

8] .

A generalization of these criteria is desired.

I presume the importance of the B-construction and the B*-construction is understood best from the special case

Hoffmann

D =

1.

In this situation the faithful functor

V:c~o

11becomes" a preordered class C:

ß*is the "lower end completion" of C.

ß* is the "Dedekind-Mac Neill-completion" of C.

Part of our results are tobe considered as generalization~

of results of B. Banaschewski and

G.

Bruns

[2].

In section 1 we give a detailed introduction into the general theory of identifying l i f t s , i.e. of identifying and, resp., topological functors, which was developed in our doctoral thesis

[6] •

In section

5

we give some supple- mentary material: A comparison between

1) (E,M)-universally topological completion [

7]

2) "canonical extension" in the sense of [ 8

J

and

3)

B*-construction;

some examples;

a complete description of reflective restrictions of topo- logical functors in terms of semi-identifying l i f t s .

Finally, we should like to say something on the origin of this paper. In§

5.5

of our doctoral thesis

[6]

we have considered the following series of commutative squares

Top-Ca t ~ Co-id t. < - - + Fi b '---+ [ 3_, Ca t]

I II III

Cat = Cat = Cat = Cat and, resp.

Top-cat ~ fib

~[_g,

cat1

II* III*

cat = cat = cat

Hoffmann

Cat dsnotes the category of U-legitimate categories and functors (Cat is, of course, not a U-category). Objects of Top-Cat are all topological functors in U, which l i f t

isomorphism uniquely; morphisms V-W in Top-Cat are commu- tative squares of functors

G

vj ^w

- - - -

F

such that G takes V-co-idt. cones into W-co-idt. cones.

Similarly we define Co-idt.: its objects are all co-idt.

functors in U with a fixed choice ("canonical") of

"precise" co-idt. lifts and the morphisms in Co-idt. are commutative squares with the above preservation property for "canonical" co-idt. l i f t s . Fib denotes the category of fibrations (with a fixed choice of co-idt. l i f t s of morphisms) and commutative squares of functors

G

w

F

such that G preserves ¹¹canonical¹¹ co-idt. lifts for mor- phisms. ~ denotes the ca tegory O - 1; [~, Cat] denotes the functor category, i.e. the category, whose morphisms are commutative squares of functors.

cat, Top-cat, fib, [~, cat] denote the according ¹¹small"

analogues of the above categories (i.e. the domains and codomains of the functors are assumed tobe U-small, hence ~ ' Top-cat, etc. are U-legitimate). We observe

that a co-idt. functor, whose domain (and co-domain) is

Hoffmann

small automatically satisfies the smallness condition (for functors), hence is a topological functor; th~s the

"small" analogue of square I is trivial. The horizontal arrows c..__..,. denote the obvious inclusions; the vertical arrows Top-Ca t ~ Ca t, etc. denote the func tor ^II co-domain"

assigning to V:C-D its underlying category D and to the square

G

F

its bottom functor F. The forgetful functors Top-Cat - Cat, etc. behave nicely. Top-Cat-,.. Cat admits co-idt.

lifts for U-small-indexed data. The situation with the other vertical arrows seems tobe similar. The preserva- tion properties of the horizontal arrows (inclusions) seem tobe rather good. In particular, Wyler's theorem on

adjointness for "taut lifts" [16] (whose presuppositions are not satisfied here!) suggests the question whether ~

are right adjoint. This question is answered in the posi- tive for Fib - [~, Cat] and fib - [~, cat] by

J.W. Gray

[4] •

The ß-construction (="generalized lower end completion^{11 )} established in section J of the present paper gives a positive answer for the embedding Top-cat

~ [~,c~J • For Top-Cat---. [~, Cat] the problem re- mains open (the answer seems tobe negative), since the category ß constructed from V:E_--,.. D in section 3 need not be U-legitimate, even i f C and D are U-legitimate.

I t would be of interest to obtain a complete answer for all of the above commutative squares (and for some re- lated categories, e.g. faithful co-idt. functors). As far as we have carried out this program (in our doctoral

thesis we have announced a paper ''On categories of topo- logical functors, identifying triangles [ = identifying

Hoffmann

functors], and cofibrations"), the material is included into the present paper.

We note, that the present paper to some extent overlaps with work of O. Wyler [15] , and E.G. Manes [10] (we have not seen his paper!), and, in particular, H. Herrlich

[5] , who also discusses the B-construction and the B*-construction.

Hoffmann

References

( 1

J

^a ANTOINE, Ph.: Extension minimale de la categorie des espaces topologiques.

Compt. Rend. Acad. Sei. Paris, ser. A, 262, 1389-92 (1966).

I 2]

[ 3

J

[4]

[5 J

[ 6

J

b - Etude elementaire des categories d'ensembles structures.

Bull. Soc. Math. Belgique 18, 142-164 and 387-414 (1966).

BANASCHEWSKI, B. and BRUNS, G: Categorical Characterization of the Mac Neille completion.

Arch. d. Math. (Basel)

1.Q,

369-377 (1967).

BRÜMMER, G.C.L. and HOFFMANN, R.-E.: An external characterization of topological functors. International Conference on Categorical Topology, Mannheim 1975.

GRAY, J.W.: Fibred and cofibred categories.

Proc. Conf. La Jolla 1965 on Categorical Algebra, pp. 21-83.

Berlin-Heidelberg-New York: Springer 1966.

HERRLICH, H.: Initial Completions.

Nordwestdeutsches Kategorienseminar Hagen 1975.

HOFFMANN, R.-E.: Die kategorielle Auffassung der Initial- und Finaltopologie.

Dissertation Universität Bochum 1972.

[7]

[3]

L 9]

[10]

[11]

[12]

[131

[14]

[15]

Hoffmann

- (E,M)-universally topological functors.

l~bilitationsschrift Düsseldorf 1974.

- Topological functors admitting generalized Cauchy-completions. International Conference on Categorical Topology, Mannheim 1975.

- Topological completion of faithful functors.

(Handwritten) Manuscipt Düsseldorf 1975/76.

(Nordwestdeutsches Kategorienseminar Hagen 1975).

MANES, E.G.: Algebraic Theories.

Springer Graduate Texts in Math. (to appear).

MÜLLER, H.: Über die Vertauschbarkeit von Reflexionen und Coreflexionen, Quasi-Top- Kategorien über Meng.

(Informal) Manuscript Bielefeld 1974.

OSIUS, G.: Eine axiomatische Strukturtheorie.

Dissertation Freie Universität Berlin 1969.

ROBERTS, J.E.: A characterization of initial functors.

J. of Algebra

f,

181-193 (1968).

SPANIER, E.: Quasi-topologies.

Duke Math. J. ~ ' 1-14 (1963).

WYLER, O.: Are there topoi in topology?

International Conference on Categorical Topology, Mannheim 1975.

[16]

Hoffmann

- Top categories and categorical

topology. Gen. Top. Appl •

.!.

17-28 (1971).

Rudolf-E. Hoffmann Universität Düsseldorf Mathematisches Institut 4 Düsseldorf 1

Universitätsstr. 1

GELFAND MUNAD3 AND DUALITf IN CAHtESIAN CLOSED fOPOLOGICAL CA~EGOüIES

H.-E. PO~ST and M.B. WISC3~EJ3~Y Fachsektion Mathematik

Universität Bremen

The lack of natural function space structures in the category rop of topological spaces makes it an awkward category notably for homotopy theory and topological algebra. To remedy this inconvenient situation several authors advocated its replacement by other "topological categories" like the cartesian closed catec;ory k-Haus of compactly generated Hausdorff spaces (Dubuc-Porta [5],

Gabriel-Zisman [24-] , Mac Lane [ 181 ifobertson [20]) or the cartesian closed category Lim of limit spaces (Binz [3]).

Finally Franke considered abstract cartesian closed categories [8] and showed that the functor "function space" has a left adjoint called the spectral functor.

In this paper we generalize the Dubuc-Porta paper on algebras over the category of compactly generated

spaces to arbitrary cartesian closed topological categories (over 3et). Just the combination of the concept of a

cartesian closed category with the concept

of a topoloßical category simplifies many so called

Porst, Wischnewsky

classical theorems on function algebras over compactly ßenerated or limit spaces.

üur aim 1s to show that most things done as yet in this field can be reduced to some standard cate[orical con- structions using methods of enriched category theory.

Nevertheless there are still some open problems mainly in duality theory for the working mathematician in this special field which can't be solved mainly by abstract categorical techniques.

Finally we want to remark that everything done here for cartesian closed topological categories over the category of sets can easily be generalized to cartesian closed Top-categories over more general base-categories.

{~1 CAi?.'l'ESIA~': CllX:_1ED TOPOLOGICAL CA'l':2GORIES

Let us first recall sorne basic definitions and propositions.

A category C with finite products is called cartesian closed if for any object

C E C

the functor

C X - : C ---;l> C

has a

right adjoint denoted by Q(C,-). Let T: [ ~

~

be a functor and

D : 2_ ----1 ~

be a diagrarn.

A

cone ':f :

^{6.K--,. D}

frorn the vertex

K

to the base Dis called a T-initial cone if for any c one

j_ :

D

^{X ~ D}

f rom the vertex

X

to D and any _!:-rnor- phism t :

'I1X~

':rK with

("I1

f ) ß t

= T)C_

there exists exactly one K-rnorphisrn t* :

X ~ K

wi th

f'6.t* = X and ':rt* = t.

'1

'he functor

T : K ~ L

is called a topological functor if

( l , ^{. '\}

for every srnall category Q and every functor

D : D ~ K

and every cone f.._: ßL---','rD there exists a T-initial cone f: ^ßK.

^----;tD

generated by):_i.e. T :f =X.

(ii) T is fibresrnall i.e. the fibres of the functor T are sets up to isornorphisrns.

(iii)

T

creates identively isornorphisrns. (cp. [11] ,[21])

( 1 .1)

T:--IEOREVi ( l12 J ) • Let

K

be

^_.§:_

topological ca tegory over Sets. Then the following conditions are equivalent:

(i)

(ii)

(iii)

K

is cartesian closed.

lor any K-object

K,

the functor

I(~ - : K~~

preserves colirnits (or equivalently coproducts and quotient rnorphisms).

For any pair (K,Y') of K-objects the set K (K,K')

- - - - - - - - - - - ' - - - - - -0

can be supplied with the structure of a K-object

d eno l,e

^~

d b

-:; _ K(v v1·.," ¹⁾

suc

l· 1

that

Porst, Wischnewsky

(a) The evaluation map

eK K' : K:Xf(K,K') ---'7K'

'

( k , f ) )---t f ( k ) is a ~-morphism, and

(b) For any f-object K, the map

~ : ~(K'',ß(K',K))-- lS:(K'

'X

K' ,K) eK, JdK'

X f).

f

,

is surjectiveo

(1.2) EXAMPLES.

a) The following categories are cartesian closed topo- logical categories over Sets.

'1) k-Haus

2) Lim

3) Con 4) FsTop

5)

Fil

6) Preürd

7)

Born

the category of compactly generated Hausdorff spaces (~ k-spaces ⁼ Kelley spaces)o

the category of limit spaces (Bastiani[2], Cook[4-] , Fischer [7],Binz[3] ).

the category of convergence spaces (Kent

r~)

the category of pseudotopological spaces (MachadoÖ7] , Nel

[19] ) •

the category of filter-merotopic spaces.

(Katetov [13], Robertson

[29).

the category of preordered spaces.

the category of bornological spaces.

b) Let U C K be a full subca tegory of a cartesian closed topological category. Then

'l) If U is bireflective in

K

whose reflector R satisfies For any catsgory K we denote by K (K,K') the set of all [-morphisms from

K

to K' where K-;°K¹E: K.

R(TJX.l'()

U )( R(K) then Q is cartesian closed (Robertson[2O])

2)

If

U

is a non-trivial coreflective subcategory closed

under finite K-products then .1L is cartesian closed.

In particular the coreflective hull of any non-

trivial subcategory of

K

is cartesian closed (Ne1[19]).

We will need the following two Lemmata.

(1.3) LEMMA ( Leroux)

Let rr K ~ L be

~

pullback preserving functor with an adj oin t righ t-inverse J and uni t f : ^Id

^~

JT. rrhen a

K-morphism K~ K' is a cartesian morphism

(=

T-initial cone where D

=

1) generated by T if and only if the

following diagram is a pullback

- - - - - -

-

K - - - , , l\'

f

can l

JTK

_ _ _

JTf

___, J'rK'

(1.4)

LEr:1MA. Let T : K -

~

be

_§:

topological functor over

~

category

_!:.

wi th products. Then a cone j: ^ßK---t

^D

is a T-initial-cone over a diagrarn - - - - - - - - - -

- - - - D :

- D~K if and only -

^-

- - - - - if the K-morphism

- - - - - - morphism.

( f

d ) : K

--TTD(d)

d f

^!!_

is a cartesian

( 1.

5)

PROF'OSI'rION. Let Y.. be

§:_

cartesian closed Top-ca tegory over 0ets. Then the representables

Y._(X,-) : Y..~Y.. ,XE:,Y:, preserve initial-cones

(are initial continuous).

-

Proof:

a) V(X,-) preserves codiscrete objects.

Let JS be a codiscrete object in V with TJS S.

Let t : TV--tT(Y(X,JS)) be a mapping where V is an arbitrary _:{-object. We have to show that t is actually a y-morphism. Since JS is codiscrete the mapping

X '>( V ~

JS : (

X , V) ~

t

(V) (X)

is a _:{-rnorphism. Hence we obtain that t : V~ y(X,JS) is a y-morphism. Hence y(S,JS) is a codiscrete y-object.

b)

V(X,-) preserves cartesian rnorphisms.

Let m : V

---J

V' be a cartesian morphisrn. Let X be any y-object and consider a commutative diagram

'r(y(x,

V'))

T(y(x,m)) T(_y(X,V))

where f : X'---1 V(X,V') is a y-rnorphism and g : TX'--J T(Y(X,V' )) is a mapping.

ile

have to show that

g

is a Y-morphism.

Consider now the diagram

~V'

Tm

T w ( ~

T(X'X X)

where w : y(X' ,_y(X,V')) ---_y(X'XX ,V')

^';;;

is the canonical isomorphism and

TX' 1

g

*

'l1 ( X ' )( X) - - ' >

'r

V : ( X ' , X) ^j---) g ( X ' ) (X) •

g

Since rn is a cartesian rnorphisrn g°"' is a ,Y-morphism and hence g: x•~ V(X,V) is a ,Y-morphism.

c) Y(X,-) preserves arbitrary initial cones.

Since ,Y(X,-) is a right adjoint it preserves products.

Hence By LEMMA ('1.4) and part b ,Y(X,-) preserves arbitrary initial cones.

§2 CATEGORIES OF TOPOLOGICAL ALGEBRAS

Let V be a cartesian closed Top-category over Sets.

Let R be a commutative y-ring. Denote by R-Alg(y) the category of all R-algebras in .Y and R-algebra homo- morphisrns in y (cf. [5} ). Given A,B E R-Alg(,Y) R-Alg

0

(,Y)(A,B) shall denote the set of

all

R-algebra morphisms from A to B.

( 2. '1) PROP0,3ITIOE. R-Alg(,Y) is

_.§:_

,Y-category and the underlying functor

1 1 :

R-Alg(,Y)

~

.Y is

~

,Y-functor.

Proof: Since y is a cartesian closed Top-category over Sets y( IAI, jB/) is a Y-object. Supply now R-~lg

0

(y)(A,B) with the initial structure generated by the inclusion

R-Alg

i

0

(y)(A,B)-1--->> T(,Y( IAI, IBf)

~

y

₀ ⁽

^IAI ,1 ^BI).

'I1he corresponding ,Y-object is denoted by R-Alg(y)(A,B)o The rest is completely straightforward • .l!'or example the

composition c : R-Alg(,Y)(A,B)X R-Alg(,Y)(B,C) • R-Alg(,Y)(A,C)

is defined in the diagram:

R-Alg(y)(A,B)X R-Alg(B,C)

^C

can

X

can

l ^can l

y( jA / , / B / )

X

y( i Bi ,

¹

C

j )

can'

Since can is a cartesian rnorphism and can' is a V-mor- phisrn, c is also a y-rnorphism.

Let T: K-71 and T' : K'--'>L' be topological functors.

A functor M : K

----1

K' is said to be initial continuous if f or any initial cone '::f over f M j is an initial cone over K'

(2.2) THEOREM ( c.f. [21}) Notation as above.

1) The underlying functor RT: R-Alg(y)---'7R-Alg(Sets) is a topological functor.

-

-

- - - - - - - ~ -

is initial continuous.

- - -^---- - - - · - - -

3) ^{1 / :}

R-Alg(y)---? Y.. is rno~adic. ·1.'he free R-algebra

functor F : Y---"l R-Alg(y) is a V-functor and Y-left-adjoint to

^{1 / •}

Proof: The above TH80REM is well-known and its proof is categorical routine (~~). The fact that Fis a V-functor and y-left-adjoint to

^{1 1}

follows from a well-known criterion by M.Kelly [15].

We omit the standard corollaries which follow immediately

frorn the above

rL¹

'.:~0REn ( cp .[21] )

Let

^!::._

be a y-category, A e A an object in ! and V E Y an object in V. The cotensor of V with Ais a representation for the functor y(V,!(-,A)) : !op___,,. y . The representing . object is denoted by !':!_(V,A). ! is said tobe cotensored

if all the representables !::._(-,A) : !op--'7-Y have a Y-left- adjoint ( cf. l6] ) •

(2.3) PROPOSITION. The category R-Alg(}'.:) is a cotensored y-category and the forgetful functor / /: R-Alg(}'.:)---f Y preserves cotensors.

Proof: The proof is the same as in [5] • We have to show that for all A E R-Alg(y) and VE. y there is an object R-Alg(y)(V,A)( R-Alg(}'.:) and a V-natural isomorphism

R-Alg(}'.:)(-,R-Alg(y)(V,A)) = Y(V,R-Alg(}'.:)(-,A)).

Define /R-Alg(}'.:)(V,A)/

:=

}'.:(V,/A/). '.l:hen the R-algebra structure on A induces a ~-algebra structure on y(v, /Al) in a natural way. It is clear that this is true for any

cartesian closed category y i.e. R-Alg(y) is cotensored for any cartesian closed category y.

Gi ven BE. R-Alg(y) • Consider the following diagram R-Alg(y)(B,R-Alg(y)(V,A)

^r:::::/

Y(V,R-Alg(y)(B,A))

l

¹

can

c;-J

l ^Y(V,can)

}'.:( /Bi , y(v,

/Aj) ^,...___, = ::,,.

y(

V

,y( / B / , / A

^{J ) )}

It is easy to see that the isomorphism restricted to upper level provides a bijection. This bijection is a Y-isomorphism since can and }'.:(V,can) are cartesian morphisms„

Cotensors realize the classical construction of algebras of continuous functions and play an important role in

duality theory.

(2.4) PROPOSITION. The category R-Alg(_y) j-s Y-cocornplete

0

The colirnits are forrned in rl-Alg(Sots) and supplied with the coinitial structure (induced by the topological

functorRT : R-Alg(y) --~ R-Alg(Sets).

Proof: Since R-Alg(Sets) is cocornplete and R-Alg(y) is a Top-category over R-Alg(Sets) R-Alg(y) is cocornplete and the colirnits are constructed as rnentioned above.

Sin0e R-Alg(y) is cotensored the representables R-Alg(y)(-,A) preserve colimits. Hence R-Alg(y) has V-colimits.

The above statement says that for any diagram D Q~ R-Alg(y) the colimit ColimD exists and that for any R-algebra A inV there is a Y-isomorphism

R-Alg(y)(colimD,A)

~

lim R-Alg(V)(D(d),A).

d E :Q_

( 2.

5)

PROr_osrrrroN. 'The category R-Alg(y) is Y-complete i.e. R-Alg(y) is complete~~ ~he representables

R-Alg(y)(A,-) preserve the limits.