Seminarberichte Nr. 2

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Mathematik und

Informatik

Seminarberichte aus dem Fachbereich Mathematik der FernUniversität

02 – 1976

Otto Moeschlin, Dieter Pumplün, Franz Stetter, Walter Tholen

(Hrsg.)

Seminarbericht Nr. 2

AUS DEM

Fachbereich Mathematik

DER

FERNUNIVERSITAT

NR. 2

1976

HERAUSGEGEBEN VON GÜNTHER FRANK, FRANZ LOCHER, OTTO MOESCHLIN,

DIETER PUMPLÜN, FRANZ STETTER UND WALTER THOLEN

Sie enthalten Berichte von Vorträgen, Tagungen und Seminaren, Zusammenfassungen und preprints. Entsprechend ihrem

Informationscharakter sind die Artikel in den Seminarberichten mit Schreibmaschine geschrieben und dann im Offset-Verfahren

gedruckt.

Die Herausgeber

The closed radical of a functor V C - -^• von Georg Greve

Adjoint co-triangles

von Georg Greve und Walter Tholen

Factorizations of cones along a functor von Walter Tholen

On Wyler's taut l i f t theorem

von Walter Tholen

Zum Satz von Freyd und Kelly

von Walter Tholen

Set

4 -

- 22 -

- 24 -

- 50 -

- 65 -

THE CIDSED RADICAL OF A FUNCTOR V: C ~ SET Georg Greve

In 1966 Linton gave in his paper "Autonomous Equational Categories "some criterions for an algebraic category to have a monoidal closed structure. Among other things he demanded the commutativity of the corresponding theory, which means that a certain set of eqations is fulfilled.

Now given any algebraic category one can add the missing equations to its theory and get a full reg-epi-reflective monoidal closed subcategory, which is even maximal whith these properties. The present paper glves in its third part a generalization of this construction to not necessarily

algebraic categories. In i ts second part an existence theorem for tensor products is obtained by solving the universal bilinear problem of Pumplün ( 4) andin the basic first section the relations between several types of inner hom functors are investigated.

Let us first repeat some notations we need in the following:

An inner hom functor for a functor V:C----)SET is a contra- covariant functor H:CX'C~C satisfying V⁰ H

=

C(-,-), a tensor product to His a left adjoint to Hin the usual sense ( c.p. 4). His unitary if there is an object I c ObC with C:

=

idc

=

H(I,-). Given a small category

V,

we say that V:C~SET creates identically V-limits, if for every F:V~C and every limit >..: 6 (X)--jV°F in SET there is a limi t p : 6 (A)----) F in C such that ^>..

= vo

^{p •} A morphism

m:A--,B in Cis said to be V-initial, if i t is initial in the usual topological sense: Gi ven morphisms n: C ~ 3 and x:V(C)~V(A) with V(m)x == V(n) there is amorphism f:C~A uniquely determined by V(f) == x and mf == n. Init(V) denotes the subcategory of all initial morphisms. A map y:V(A)~V(B) is called liftable, if there is a morphism g:A--,B with

V(g)

=

y.

Some remarks about inner hom functors

Throughout this section H denotes an inner hom functor to V: C ~SET. We want to discuss mainly three kinds of hom

functors H, which are characterized by the following properties:

1 Definition (a) His called BV(A) and a

ordinary, iff for all A, BE ObC the power V(A'

morphism mA B:H(A,B)~B ' exist s.th.

the following diagramm commutes:

'

V(B)V(A) - - V A ~

1

C(A,Bl_ 1 (A,B) V (mA-;B )""7V (B V (A) )

VA Bdenotes the restriction of V to C(A,B) and l(A,B) is the canonical comparison morphism commuting with the projections.

'

(b)H is initial, iff for all morphisms x:V(A)~C(B,C) the following condition holds:

There is a map y :V (B)---t C (A, C) wi th x is liftable

V (x (a) ) (b)

=

V (y (b) ) (a) •

(c) His weakly coherent, iff for all morphisms

f:A~H(B,C) there is a morphism g:B~H(A,C) such that

V(V(f) (a)) (b)

=

V(V(g) {b)) (a).

The definition of ^II weakly coherent ^II is a generalization of Linton's notion ^II coherent ^{11 (} 3 ). Every coherent inner hom functor is weakly coherent, every initial inner hom funktor is weakly coherent and every initial ordinary hom functor is coherent provided that V is faithful. The converse statements are not available.

We consider three simple

2 Examnles

(a) The usual inner hom functors of abelian groups,

R-moduls (Ra commutative ring), abelian monoids and so on are unitary, ordinary, initial and weakly coherent.

Looking only at the category of finite abelian groups, finite R-moduls (R not finite) etc., the hom functors are no longer unitary but still ordinary, initial and weakly coherent. Taking cyclic groups, His unitary, initial and weakly coherent but not ordinary.

(b) The inner horn functors to the forgetful functor

V:TOP~SET ( TOP= topological spaces) forma lattice. Three hom functors are distinguished:

(i) The greatest hom funktor (TOP(A,B) provided with the indiscrete topology) is not unitary, not ordinary, not

initial and not weakly coherent.

(ii) Providing the hom sets T0P(A,B) with the initial

topology with respect to the map vA,B :TOP(A,B)---j-V(BV(A)) one obtains a unitary,ordinary, initial and weakly coherent

hom functor.

(iii) The smallest hom functor (T0P(A,B) provided with the discrete topology is not unitary, not initial, not weakly coherent but ordinary.

(c) The usual inner hom functor to V:BAN

1~ SET(BAN

1

=

Banach- spaces with contractions, V(B)

=

unitball in B) is unitary, not ordinary, not initial but weakly coherent.

The following theorem shows the relations between the different types of inner hom functors and gives a rnotivation for the

name initial:

3 Theorem

(a) If H is unitary, C has V-powers (i.e. for all A,B E ObC BV(A) exists) and V has a left adjoint, then His ordinary.

(b) If His ordinary such that mA,B is initial for all A, B ^€ ObC (c. p. 1 (a) ) and V is representable, then H is initial .

(c) If H is initial and ordinary and V is fai thful, then mA, B is initial for all A, B € ObC •

(d) If His weakly coherent, V is faithful, has a left adjoint and reflects isomorphisms, then His unitary.

This yields the following diagramm:

C has powers

unitary ---=v--,-h-a~s'--a-~1-e-f~t-a-d~j-o_i_n_t_>

V faithful

T

V has a left adjoint 1

1

V reflects isomorphisms

1

ordinary

1 . . .

¹

mA B initia

' 1

V representable weakly coherent ~ - - - i n i t i a l

1

Proof:

(a) If F i s left adjoint to V with unit n and co-unit s , then one has a natural isomorphism i: C

~

H (F ( 1) , -) ( 1 :

= { ~ }) •

For ae V(A) define ja:1--1V(A), ja(~):= a, and take k(A,B) tobe the unique morphism making the diagramm

H(F0 V(A),B)---k_(A_,B_) _ _ H(F(1) ,B)V(A)

~ /

H(F(j ),B) TI

a ~

✓

a

H (F ( 1 ) , B)

commutati ve ( TI the canonical proj ec tions) . a

.-1 V(A)

Define mA B:= l (B) k(A,B)H( dA) ,B).

'

(b) We show at first the direction "~" of 1 (b):

As V is representable, one has an obj ect I e ObC and a natural isomorphism j:V~C(I,-). Given a morphism x:V(A)~C(B,C)

x = V(f), f: A~H(B,C), define g:V(B)----fC{A,C) by

. V ( I)

g(b) := 71,mI,CH(J {B) (b) ,C)f, where n

11

:c

^'--7

c

denotes the canonical proj ection wi th 11 : = j -1

( I) ( I) •

Assume on the other hand x:V{A)~ C (B,C) and y:V(B)~ C (A,C) tobe morphisms with V(x(a)) {b) = V(y(b)) (a), then there is a unique h: A-1CV(B) such that nbh = y(b) (nb:CV(B4 C projections), x = V(h) and x is liftable because mB,C is initial.

(c)Given morphisms h:A~CV(B)and x:V(A)~C(B,C) with V(mB,C)x = V(h) we define y:V(B)~C(A,C) by y(b):= nbh.

Then for all b E V (B) , a e. V (A) the equation

V(y(b)) (a) = V(x{a)) (b) holds and x is liftable, that means mB C is initial.

,

(d) Let F be left adjoint to V with unit n and co-unit E

and define xA: 1 - , C (A,A) by xA (\2)) :=A. This yields a natural isomorphism i (A) :A -4 H (F ( 1) ,A) with

V(V{i(A) (a)) (z) = V(V( dH(A,A)F(xA)) (z)) (a) for all a E-V{A), z E V°F(1).

4 Corollary

If V is faithful and His unitary and initial, then C i s struc- tured in the sense of Pumplün ( 4) and closed in the sense of Eilenberg Kelly.

Proof:

We have to show that the map KB,C: C (B,C) A ---t C (H (A, B) ,H (A,C)) ,

KB,C(f) A = H(A,f), is liftable. One defines

C C

KA B:C(A,B)----raH(B,C) ,H(A,C)) by

,

KA B(g):= H(g,C) and ,

A C A

because of V(KB,C(f)) (g) = V( KA,B(g)) (f),KB,C is liftable.

Applying the preceding results to algebraic categories, we obtain a generalization of some corresponding results o f Iin ton ( 3 ) .

5 Corollary

If V is premonadic (i.e. V has a left adjoint such that the comparison functor into the Eilenberg Moore Category is full and faithful)and C has powers, or if V is monadic, then for an inner hom functor H the following statements are equivalent:

(a) His unitary.

(b) His ordinary.

(c) His initial.

(d) His weakly coherent.

Construction of a tensor product

Westart this section with a construction, which is on the one hand very simple and on the other hand very useful in various categorical si tuations (c. p. ( 2 ) , ( 5 ) ) :

6 Definition

Given a commutative triangle of functors

G C

/vo

0

(q0,q

1) (or sj_mply q

1) is said to be ^et (G,v

0)-quotient of f:V0 (A

0)---tv 1 (A

1), if the following conditions are satisfied:

(Q2) For all C1E:ObC1, x1EC1(A1,C1), Xot.Co(Ao,G(C1)) with v1 (x

1)f

=

v

0 (x

0) there is a unique y

1e

c

₁ ^(B₁

,c

1) so that y1q1

=

x1 and G(y1)qo

=

xo.

This yields the following diagram:

The relations between (G,V

0)-quotients, adjointness, completeness, image factorizations, final and initial topologies are described in ( 2). For our investigations on tensor products we need the following

7 Lemma

c

₁ ^{has (G,V}₀)-quotients (i.e. every f:V 0 (A

0)--;}V 1 (A

1) has a (G,V0)-quotient) if

(a)

c

₁ has products which are preserved by G and

(b) there are. subclasses E

1 c Epi (

c

1) and M 1 c C

1, G ( M

1) c. Init (V 0) , such that

c

₁ is E

1- co-wellpowered and has ( E

1, M1) -factorizations.

Taking a trivial triangle

the notion of a (C,V)-quotient is defined by 6. We use these quotients to get a universal construction of a tensor product which is left adjoint to a given inner hom functor H:

8 Theorem

Suggest that His an inner hom functor to V:c--;SET and assume that His initial and V is faithful or that His ordi- nary such that mA,B is initial and natural in A and B.

Then H has a tensor product if the following conditions are satisfied:

(a) V has a left adjoint and reflects initial objects (i.e. V(A)

=

~ ~ A initial object) and

(b) C has (C,V)-quotients, finite products and (infinite) pushouts of (C,V)-quotients.

Proof:

Take V tobe faithful and H initial.

first case: V(A)XV(B) f

0

Take F tobe left adjoint to V with unit n and let

i(A,B) :V(A)X V(B)~V(AnB) denote the canonical comparison isomorphism. Define xb:V(A)----:tV(A) X V(B) and

X :V(B)~ V(A) X V(B) by xb(a) := X (b) := (a,b). Assume (t ,q )

a a n n

to be the ( C, V) -quotient of n (V (A11B)) i (A, B) x , n e. V (A) XV (B) n

and consider the pushout (pn)ne: V(A) ... , V(B) of (q ) "" n n f'. V (A) l:J V (B)

Define yA(B) :V(A)--C(A,B®A) by yA{B) (b) := pbtb and y B{A) :V(A)----)C(B,B®A) by YB(A) := pata. Because of

V(yB(A)(b))(a) = V(yA{B)(a))(b) one getsamorphism nA {B) :B ~ H (A,B®A) with V(nA (B))

=

YA {B).

second case: ^V(A)X V(B) = ~

Setting B®A:= F(~) one obtains in a canonical way morphisms nA {B) :B~H (A,B@A)

If His ordinary and mA,B is initial and natural a similar argument yields a morohismn {B):B----fH(A,B®A) (c.o_.3).

- l\

Now a conventional calculation shows, that for all AEObC -®Ais left adjoint to H(A,-) with unit TJA' that means

-®- is a tensor product for H.

9 Corollary

Suggest that His an inner hom functor to V:C--fSET and assume that His initial and V is faithful or that His ordinary

such that mA,B is initial and natural in Aand B. Then H has a tensor product if the following conditions are satisfied:

(a) V has a left adjoint and reflects initial objects and

(b) C has products and there is a subcJ.ass E c Epi ( C) such that

c

is E-co-wellpowered and has (f,Init(V) )-factorizations.

1 O Remarks

(a) The construction of a tensor product in 8 can be performed without assurning the existence of an inner horn functor. Also

in this case -®- is a commutative and associative bifuntor and p ~V(AnB))i(A,B) is a universal bilinear rnap (c.p. (4),

g:V(A)x V(B)~V(C) bilinear {=! gx n liftable for all n '=-V (A) XV (B)) . (b) Take V:= V(A)u V(B) tobe a discrete category and consider the followig diagramm:

C [V, C]

6\.i /

[v,c] [v,v]

[v:v] /

~LV'

SET]

( [V

,c],

[V ,SET] functor categories)

r

^A,ncV(A)

Define F:V--+ C by F(n) := , a :VoF~ V⁰ !'!.(FoV(A"1TB)) by LB,neV(B)

a(n) := n (V(A11B) )i(A,B)x and s:F----; l{B®A) by S (n) := p t . Then

n n n

( f?, p) is nothing else than a ( !'!., [V, v] ) -quotient of a . That rneans, we can replace the assumption "C has (C,V)-quotients and push- outs of ( C , V) -quotients" by "C has ( C,

[V, v] )

-quotients for all srnall discrete categories V'! Especially every coproduct in C is a ( 4[D,~)-quotient, that rneans: the tensor product in 8 can be considered as a generalized colimit.

The closed radical

In the following section we shall construct a subcategory RAD (V)< C which is in a ca nonical way provided wi th a

bifunctor H: RAD (V)>< RAD (V)--+ C . For this purpose at first a simple existence criterion for inner hom functors is given:

11 Lemma

Suggest that V:C~SET creates identicallyV-powers(c.p. 3(a))

then V has an ordinary inner hom functor, if for all A, B ^E Ob C vA,B: C(A,B)--:> V(B) V(A) (c.p.1 (a)) is

means, there is a V-initial morphism

Proof:

initially liftable, V(A) rnA,B:HA,~ B

Define H(A,B) :== HA,B" For a morphism f:B• C one has

that such

V(A) V(f)

V(f mA,B) == V(mA,C) C(A,f) and V(A mC,A)= V(mB,A) C(f,A) and because of mA,C' m8 ,Ac Init(V), C(A,f) and C(f,A) are

liftable in a canonical way.

Now i t is obvious to introduce the closed radical as follows:

1 2 Definition

Given a functor V: C-tSET the full subcategory RAV(V)

<

C with the class of objects

ObRAD(V)

=

Ae. ObC,for all B~Ob exists AV(B)with

projections nb so that V(nb)~,11):V(A)V(~V(A) projections,and vA,B is initially liftable.

is called the closed radical of V.

1 3 Remark

In general RAD(V) is not provided with an inner hom functor, but there is a contra-covariant functor H making the following

diagramm commut.ative: _H

RAv(V)x R A V ( V ) - - - -

"'

RAV(V) (-,-)

~

SET

/

C

For the rest of the paper the case that His an inner hom functor of RAV(V) is investigated.

14 Theorem

If V: C--;SET creates identically limits, then RAV(V) is not empty and closed und er products and initial morphisms ( i. e. m: D ~ B initial, B € RAV(V),

9

De RAD(V)).

Proof:

RAV(V) is not empty because every final object of Cis inRAD(V).

Now take m: D• B to be ini +:ial, B € RAD (V) and consider the following diagram:

C(A,D)----VA,D ) V(D)V(A)=V(DV(A))

1 1

C(A,m) V(m)V(A)=V(mV(A))

l t

C (A,B)--vA,B=V(mA,B~ V(B) V(A) =V(BV(A))

P:= {(f,y)j feC (A,B) ,y:V(A)

•

V(D) ,V(f)=V(m)y} is a pullback in SET of (V(mV(A\vA,B). The map k : P ~ C(A,D) defined by k((f,y) := g, where g is uniquely determined by V(g) = y and mg = f, turns out tobe an isomorphism and therefore the

diagram (.) is a pullback. As V creates identically limits, there V (A)

is a pullback (HA,m'mA,D) of (mA,B'm ) , so that

V(HA,m) = C (A,m)' V(mA,D) = vA,D and mA,D E Init (V)' that means De. RAV (V) •

Take C to be a product of C. E: ObRAV (V) , i E: I, in C wi th

l

proj ections TT. : C

•

C .. As V creates identically products

l l

and C (A,-) preserves products, there is an object HA,CeobC

and morphisms HA, TI.

l

This yields a morphism

with V(HA,C) = C(A,C) 'i (A)

mA,C:HA,C----t- C

and V (HA, ^TI.

l

with

=C(A,TI.).

l

V (A)

TI i mA,C = mA C HA ' V(mA,C) = vA,C and mA,C E. Init (V)' ' i ,Tii

that means C E' RAD (V) .

15 Corollary

If V creates identically limits, then RAV(V) has an ordinary inner hom functor.

Proof: Lemma 11, Theorem 14.

Now applying the adjoint functor theorem i t is easy to give a criterion for RAV(V) tobe a full reflective subcategory .

..l.

Given a subclass McC, let M denote the class of epimorphisms which are characterized by Isbell's diagonalization property:

.J.

M :

=

e

eeEpiC,given a cornmutative square - e - - 4

gl ;1...-; 1 f ,m , there is a unique 1

~1.::- m

J,

with le=g and ml=f.

With this notation one obtains the following

16 Corollary

Assume that V:C~SET creates identically limits, suggest that C has (M J. ,M)-factorizations for some M~Init(V) and take C to be M -co-wellpowered.

.J,.

Then RAV(V) is a full M -reflective subcategory of C.

The next theorem gives sufficient conditions for RAV(V) to be in a sense "maximal". We say that V:C---+SET creates

identically Init(V)-factorizations, if for every factorizati- on V(f) =mein SET, e epic and m monic, there is a factoriza- tion f = ip in C with i E Init(V), V(i) = m.

17 Theorem

Suppose that V:C---tSET creates identically limits and Init(V)- factorizations and assume that V has a left adjoint with a pointwise V-final co-unit E (V-final= dual of V-initial).

Then every full subcategory c

1

<

C which has an ordinary inner hom functor and which is closed under products is already a subcategory of RAV(V).

Proof:

Let F be left adjoint to V with unit n and co-unit ^E and take H

1 tobe an ordinary inner hom functor of c

1 exactlv of V/c

1) -T.ve show B

1 ^€. RAD (V) for all B

1 ^E ObC 1 :

[ ) . V(B )

Define jb 1

:1 :=

0

~ V(B

1) by Jb

1 ⁽⁰⁾ := b₁, fb 1

:F(1)---;),Bl 1 by V ( f b

1 ) n ( 1 ) = j b , h: F ( 1 )

~

B

~

( B 1 ) by ^TT b h = f b

V(B ) 1 1 1

( b

1

:B.i 1 --tB

1 the canonical projections) and consider the following diagram:

C (F(1) ,B V

1) - - F(1) ,B 1 V is monic, C (h, B

1) is epic, V V (B ) is liftable

F(1),B1 1

B1 ,B1

and as V creates identically Init(V)-factorizations there is

· ·t· l h' VoF(1)

an ini ia morp lSffi mF(1) B :H (1) _, ~ B1 with 1 F ,B

1 V(mF(1) ,B

1)

=

^{VF(1) ,B}

1.

This yields annother commutative square:

C(F(ja) ,B 1)

i

C(F(1),B 1) (j :1--;V(A) as usual)

a

Because C(FoV(A) ,B

1) is a product of C(F(1) ,B

1) with projections C(F(ja) ,B1) there is a product HFoV(A) ,B

1

with projections HFoV(ja) ,B

1 :HFoV(A) , B ~ HF( 1 ) ,B

1 in Cso that V(HF(ja),B

1

)

=

C(F(ja),B 1 ). We set sx:= (V⁰ s)(A)(x)6.V(A), x E; VoFoV(A), and considering the projections

·BVoFoV(A) B d th . t· Vc.F(1)

TTx• ₁ ? 1 an eproJecionTTn(l)(0^{):B.1 B1 one}

. . V<>F⁰V (A)

obtains a morphisrn mFoV(A) ,B

1 :HFoV(A) ,B

1--7B 1 which is uniquely determined by TTxmFoV(A) ,B

1

=

TTn(1) (y'.\}mF(1) ,B

1HF(j Ex) ,B 1 · This last mentioned morphism is initial and satisfies the

equation V(mFoV(A) ,B1) .= VFoV(A) ,B1.

Finally consider the following diagram:

(u)

C(A,B V

1) - - - A,B 1

1

Q:= {<f,y) 1 f:FoV(A)~ B1 ,y:V(A)--tV(B 1 ), V(f)

=

y(V•s )(A)}

SET ( ) ^{V( Vo} s(A)) and

is a pullback in of (V mFoV(A) ,B

1 , B1

defining l:Q--+(\A,B

1) by l((f,y)) :=g, g uniquely determined by V (g) = y and f = g⁰E(A) ,we obtain an isomorphism 1 which proves that (••) is a pullback._Now the argument in the proof of 14 shows that there is an initial morphism

V(A) .

mA B :HA B ~ B 1 with V(m B ) = VA B .

' 1 ' 1 A, 1 ' 1

Therefore V A,B 1 is ini tially liftable for all A E ObC, that means B

1 ^~ RAV(V), q.e.d ..

Now i t is easy to derive various corollaries from the preceding results. We confine ourselves to the following statement:

1 8 Corollary

(a) If v:C ---;SET is premonadic (c.p. 5) and creates identically limi ts and Ini t (V) -factorisations ( c. p. 1 7) or if V is mona- dic, then RAD(V) is a maximal monoidal closed subcategory of C •

(b) If v:C-4SET is topologic (c.p.(5 )) then RAV(V) = Cis a monoidal closed category.

Finally let us consider two

1 9 Examples

(a) The closed radical of groups and ringsare the abelian groups, the radical of the categories of associative -, Lie -, Jordan- algebras ( over a commutative ring R) is the category of

R-moduls.

(b) The radical of the category of metric spaces (with con- tractions) are the metric spaces with finite diameter.

( 7 )

( 2)

( 3)

( 4)

( 5)

( 6)

(July 1976)

Greve

Greve

Linton

Algebraische Konstruktionen von Linksadjungier- ten, Diplorrarbeit, Universität Münster, 1975

(G,V0)-Quotienten, Preprint in Kategorienseminar Nr. 1, Fachbereich Matherratik, Fernuniversität Hagen,1976, 112 - 125

Autonorrous Equational Categories, J. of Math.

and Mech. Vol. 15, no 4, 637 - 642

Pumplün Das Tensorprodukt als universelles Problem

Tholen

Tholen

Math. Ann. 171, 1967, 247 - 262

Adjungierte Dreiecke, Colimites und Kan-Erwei- terungen, Math. Ann. 217, 1975, 121 - 129

Relative Bildzerlegungen und algebraische Katego- rien, thesis, Universität Münster 1975

Georg Greve Fernuniversität

Fachbereich Matherratik D - 58 Hagen

Postfach 940

Adjoint cotriangles

Georg Greve and Walter Tholen

Abstract of a lecture given at the Category Theory Meeting Sussex 1976

The following problem is investigated: given a commutative

"co-triangle" of functors

K1 G K

~ ^{/ /1}

⁰

p1 "'

^'·

'

', /Po

'-.

K

such that ^p

0 and have left adjoints, we ask for a left adjoint of G. By techniques of [1] and [3] a generalisation of a result of Hardie [2] is proved:

Theorem: (Existence theorem) Let M

1 be a class of mono- morphisms of K1 containing all equalisers and closed under

composition, pullbacks and intersections. If

i) K

1 has equalisers, pullbacks and intersections, ii) G preserves these limits,

iii) K1 is M

1-wellpowered and the unit of P

1 is a pointwise M

1-morphism, then G has a left adjoint.

Two corollaries follow immediately from the theorem:

Corollary: (Pushing colimits) Let P 1 : K functor having a left adjoint and let M

1 be as above.

If i) and iii) are fulfilled and if K is D-cocomplete, then K₁ is D-cocomplete (D diagram category).

Corollary: (Special adjoint functor theorem) Let K

1 be complete and G : K

1 K

0 continuous. Let M

1 be as above and K₁ be M₁-wellpowered. If there exists a set of M

1-cogenerators, then G has a left adjoint. (Uc Ob K 1 is a set of M

1 -cogenerators, iff for every A E Ob K₁ the canonical morphism

A

TT ^TT

^u

uEU K

1 {A,u) is in M

1.)

[1] G. Greve: Algebraische Konstruktionen von Linksadjungierten.

Universität Münster {1975)

[2] K.A. Hardie: Special adjoint functor interpolation.

Algebra Symposium, Pretoria {1975)

[3]

w.

Tholen: Relative Bildzerlegungen und algebraische Kategorien. Thesis, Universität Münster {1974)

[4] G. Greve, W. Tholen: Adjoint co-triangles. Math. Colloq.

Univ. Cape Town X (1975-76), pp. 81 - 92.

(July 1976)

Factorizations of cones along a functor

Walter Tholen

Abstract.

First a general Galois-correspondence is established, which generalizes at the same time the correspondence between classes of monomorphisms and injective objects and the

correspondence between classes of epimorphisms and monomorphisms in a category. This correspondence arises naturally if one

tries to generalize some concepts of "topological" or also of ''algebraic" functors. Both kinds of functors admit certain factorizations of cones, and just this fact implies some of their cornmon nice properties, lifting limits, continuity and faithfulness, for instance. These properties can be shown without having a left adjoint. Therefore the theory yields also applications to functors which are neither ''topological"

nor "algebraic".

1. Introduction

In category theory several Galois correspondences have been established. First there is a correspondence between classes of generalized injective objects and monomorphisms

(cp. Maranda [16], Pumplün [19]):

A morphism e : X---+ A (think more of "embedding" than of "epimorphism"), and an object B in a category are called weakly orthogonal, written e ~ B ,

x: X - B has a factorization d: A--+ B.

X e

A X

j

^{/ / d /}

'

B

iff every

Defining M.L ^:= {e I e ^.i. B for all BE M} for each class of objects M and

~.L : = { B I e .L B for all eEE} for each class of morphisms E one gets the so called "Maranda correspondence".

In connection with the "orthogonal subcategory problem"

Freyd and Kelly ^{[ 5 ]} define a similar correspondence:

They call e and ^B to be orthogonal, writing e b B

'

iff every ^X has a unique factorization d. Then for

..L and E in

classes E and M one gets classes M-

- b the same way as above.

Factorizing m~rphisms over certain kinds of (epi)morphisms and (mono)morphisms many authors (cp. Dyckhoff [3], Freyd and Kelly [ 5] , Kennison [ 1 5 ] , Marny [ 1 7 ] , Nel [ 1 8 ] ,

Pumplün [19], Ringel [20]) look for pairs of classes of morphisms E and M which fulfil Isbells diagonalization property:

Two morphisms e and m are called weakly orthogonal (orthogonal), iff for every pair of morphisms x , a

with mx

=

ae , there is a (unique) diagonal d rendering commutative the diagram

X e

A

/

j j

^{/ /}

X ^/ a

,,,,,, d

k ,,

B C

m

In this case we write again e ~ m (e ~ m) and ~ ~ M

(~ :b

!'i) ,

iff for all e E E and m E M ^{e ..1. m ( e de m)} holds. Given a class M there is a maximal class

M J.. (M J: ) such that _M ^{.1. .l} _M (M.L _- ¾ M) • D f. . e ining th e class ^{~ .i..} (EJ:) analogously one gets the so called

"Isbell correspondence".

By Freyd and Kelly [5] and Herrlich [10] this correspondence has been generalized to a correspondence between classes of

(epi)sinks and (mono)morphisms and dually between classes of (epi)morphisms and (mono)sources, where a sink is a discrete co-cone and a source a discrete cone.

The characterizing diagrams are

e. e

X. _l

- - -

^{l •} A _X --- --- ---• A

1

1 1

1 X. ^l

j

^d

_l

^{a X}

_{1 / /}

^{/ d,., /}

_l

^{a. l}

/_. Jt

B • C B C.

m m. ^l

l

In case of general co-cones or cones over a diagram category D these pictures get the form

f\e

D f\A f\X f\A

1 Ax

j ^{/ßd '}

,

j

~ 6d, ^/ 6a a

1

;:_ ~

flB ^firn

--

^{f\C f\B µ --+ D}

where D is a diagram of the given category K and 6 the embedding of K in its functor category [Q,~].

In [21] this concept has been generalized again replacing

f\ by an abstract functor G : A - + X. The so def ined

correspondence describes the appearing classes of morphisms if one factorizes "G-morphisms" X---+ C::A or GA---+ X in a natural way (cp. Herrlich [8]). In the first case we have the following diagonalization property:

G B - - - - - Grn

GC Ga

In the sarne way as above one has a Galois correspondence between classes E of "G-(epi)rnorphisrns" e : X - - - and classes M of (rnono)rnorphisrns of A. We write e ^.L. m , e :! rn etc. again. For instance, if e is G-universal e b rn holds for every rn in A.

GA

However, looking to the concept of topological functors (cp. Brürnrner [2], Herrlich [10],Hoffrnann [11],Wischnewsky [22], Wyler [24]), these factorizations seern tobe not enough general. There one looks at "G-initial liftings"

JJ : 6A - D of cones y : 6X - G ^O D , D : D - A

a diagrarn:

6GA

/ ~ • p

.J

6 X - - - -

y

If G is not transportable the identity on the left rnust be replaced by an isornorphisrn e: X--+ GA:

6X - ^{G, D}

y

So we have a factorization of ^y along G, and for G · · t · 1 ne " ·. 'B - E the obvious

every -ini ia eo ^v o

diagonalization property is fulfilled

6X 6e 6GA

l

^{/ / / G ,, a}

6x /6Gd

/

+

v

6GB G "E G ^{,_ \)}

This diagram defines a Galois correspondence which includes all correspondences mentioned above as

special cases (§ 2) and leads us to a concept of functors which generalizes at the same time concepts of topological and algebraic functors ^(§ 3).

2. A general Galois correspondence

From now on let G : A - - " 7 X be a functor and D a category which may be assumed as small.

We call a pair (e,A) with e:

x -

^{GA in X and}

A E ObA a G-morphism. Often we write e : X - - GA instead of (e,A).

Mor (G) :

= {

(e,A) 1 (e,A) G-morphism }

is the class of obj ects of the comma-category <X, G

>;

if G is the identity of A

= ~,

Mor(G) is up to a canonical isomorphism the class of morphisms of A.

We call a pair (B,µ) with a natural transformation µ: .6B D , D : D - A , and B E ObA a D-cone . Often we write µ: .6B - D instead of (B,µ).

Cone(D)

= {

(B,µ) 1 (B,µ) D-cone}

is the class of objects of the comma-category

<

.6, [Q,~]

>;

if D

=

^~ is empty, Cone(.!2_) is the class of objects of

~ , and if D

=

1 is a category with exactly one morphism, Cone(D) is the class of morphisms of A.

In general there arethe following important subclasses of Mor(G) and Cone(D):

Univ(G)

= {

^{(e,A) 1 e X - GA} is G-universal}

(i.e. for all x: X - - + GB there is a unique d : A - - - + B rende:cing commutative the diagram

X e ),GA

X

l

_"' ^{„ Gd}

GB

Iso (G) == { (e,A) ¹ e X---+ GA is an isomorphism of X}

Epi(G) == {(e,A) ¹ e

: x -

GA is an G-epimorphism}

(i.e. for all a,a' : A---+ B with (Ga) e == ( Ga ' ) e a == a' holds)

Lim(D) == { (B,µ) I µ: LIB--+- D is a limit-cone } (i.e. for all a: LIA-+ D there is a unique d: A - B rendering commutative the diagram

LIA

/' /

Lid / ^/

a

/

/ +

""

L\B ^• D

\J

MonoCone(D):

= {

(B,µ) 1 µ: L\B _ , . D is a mono-cone}

(i.e. for all b , b ' : A - B with µ(lib) == ;~(l'ib') b == b' holds)

Init(Q,G)

: =

^{{ lB,µ) 1 ]J : l-.B D} is G-initial ^}

( i . e. for all ^{X :} GA-r GB and a : l-.B D with G " a

=

^{(G 0 ]J) llx}

there is a unique d :A

--

^{B such}

that the identities Gd

=

^{X and}

µ ( lld)

=

^{a hold}

l-.GA l-.GA

ßx

j

/

j

/ /

/ G ^o a

,,,,,,, l-.Gd

/

~

6GB G ,. D G o µ

If G is the identity on ~

= ~,

Univ(G) and Iso(G) are both the class of isomorphisms of ~ , Epi(G) is the class of epimorphisms of A and Init(Q,G) is the class of all D-cones.

If D

= ~,

Lim(D) consists of all terminal objects of

~ , MonoCone(D) of all "pre-terminal" objects of A and Init(Q,G) of all "G-indiscrete" objects of A.

If

Q = l ,

Lim(D) is the class of isomorphisms of ~'

MonoCone(D) the class of monomorphisms of A and Init(Q,G) the class of "G-initial" morphisms of A.

(1) Definition. We call (e,A) EMor(G) and (B,µ) ECone(Q) weakly orthogonal (orthogonal), iff for all (x,B) EMor(G) and all (A,a) E Cone (Q) with (G o a) L. e

=

^(G o µ) t, x

there exists a (unique) morphism

d:A---+- B in A such that the identities (Gd)e

=

x and µ(6d)

=

a

hold; we write (e,A) .1. (B,µ) ( (e,A) 4: (B,µ)).

6X 6e

6GA

/ /

ßX 6Gq_ ^/ G ,, ^r-.

/

+

^{,,,,. /}

+

6GB G ,, D G .. µ

and M c Cone(~) we write For classes E c Mor(G)

~ J.

!i (~

±.

!i) ,

iff all (e ,A) E E and (B, µ) E

!i

fulfil the relation (e,A)1. (B,µ) ((e,A)± (B,µ)). There is a maximal class M.J.. (Ml) such that M.1. M (Mo'=. M)

- - - . . L - - ± and

a maximal class ~ ^.L ( ~ ... ) such that ~ ^.J., ~ ^{.L. (~} ± ~ "'= ) • Sometimes we write more precisely ~-'-(!::>,G), !2_-:1=(D,G), E (D,G) and ^{E (D,G)}

- .J. - -:!:, .

(2) Proposition. The assignments

M 1 - - - M..L

E +---, E

-.J.

and

M

E. - - , E

-~

define two Galois correspondences, which lead (by special choice of G or

Q)

to all correspondences mentioned in§ 1.

The proof is obvious.

From now on, for abbreviation we only consider the relation.-L . But many of our statements are still valid (in similar form) for ~ ; moreover one has the

following two lemmata:

(3) Lemma. If M C Cone (D) is a subclass, then

holds, if one of the following conditions are fulfilled:

a. M c MonoCone (.!2_), b. MJ. C Epi (G),

c. A has pullbacks, G preserves pullbacks, D i ~ and for all commutative diagrams

vr"

^{D' µ t:,A}

a'

l

^pullback

l "

t:,B ---1--D µ

(A, v) is in M whenever (B, µ) is in M. (In case D

=

¹ this last condition is fulfilled, if M is closed under pullbacks and if M contains all co- retractions whose retraction is in M).

Proof.

Cases a. and b. are obvious. c. is proved by Ringel [20] Lemma 6 in the case D

=

1 and G

=

identity;

but his proof works also in the general case.

Analogously one proves

(4) Lemma. If ^E C Mor ^(G) is a subclass, then

-E ...

holds, if one of the following conditions are fulfilled:

a. E c. Epi ( G) ,

b. ~ .... ^c MonoCone ^{(D) ,}

c. X has pushouts, G is faithful and for all commutative diagrams

X e

GA

X

l

^pushout

^j

^x'

GB e' X'

"-l

^{GB u}

(u, B) is in E whenever (e ,A) is in E.

Now we want to investigate subclasses which are closed under the closure operator given by the Galois

correspondence b .

First we dencte by ~G the class of all D-cones

l1 : llB - D , such that G ^{O l1 :} llGB - - G ^O D is a mono-cone in X. Then we have:

( 5) Lemma. Univ(G) _b Cone (Q) ,

Mor ( G) ^..L. Lim (Q)"' ~G, Iso ( G) _b Init(Q,G)

.

The proofs are trivial.

(6) Corollary.

Univ(G).L

=

Cone(D) , Mor (G)

=

(Lim (!?_) " ~ ) b

Iso (G)..L

=

Ini t (Q, G)

(7) Remark. 1. If D

=

^~ or if A has a terminal object which is preserved by G, then

Univ(G)

=

Cone(D)

±

2. If X has an initial object, then

Mor(G)b

=

Lim(Q) ⁿ ~G

3. Let D be not empty and let D be either connected or let G be faithful. If G ist Q-topological (i.e. for every cone y: 6.X-- Go D there exists e : X - GA in Iso (G) and

JJ : 6. A - D in Init(D,G) such that (Go JJ) (6.e)

=

y) ,

then

Iso(G)

=

Init(Q,G)J::

Although the assertions of (5) and (6) are very simple there is one important remark following from (6):

Generally Init(~1G) is a non-trivial subclass of Cone(D), which is closed under the closure operator given by the Galois correspondence ,.k. • That is why this class plays an important part, if one tries to factorize cones along G ( § 3) .

3. Factorizations of cones along a functor

First we prove the following theorern which is wellknown in case D = 1 and G = Id .

(8) Theorem. Let E cMor(G) and !ic.Cone(Q) be subclasses wi th ~ :!: ~ and let every cone

have a factorization

y : 6X - G ^o D , D : D - A ,

y

= (

^{G o µ) ( 6e)}

where e : X - GA is in E and µ : 6A - D in M

Furtherrnore, i f i : A - - B is an isornorphism of A , let ( (Gi)e,B) be in E for all (e,A) EE and let (A,µ(!::,i)) be in M for all (B, µ) EM . Then the following implications are valid:

1. Q =f

0

^{and D connected ~ E}

=

^MJ.

2. Q

=F0

^and

E'.

cEpi (G) ^~ E

=

^M=b

3. ~cinit(D,G) ^~ M

=

-¼ E " Init (Q,G)

4. ~ cinit(D,G) and M cMonoCone (D) ^=;, M

=

^E _-;! ^{n MonoCone (D)}

5. ~ c:Init (Q,G) and G faithful ^~ M

=

_{~ J.. •}

6. ~ c:Init(Q,G) and

Q

* ⁰

^{and D connected ~ M}

⁼

^E _-::!:

7. Iso (G) CE ^::::;, M

=

_~~

Proof.

1 • , 2. : I t remains to show ~-c ~ . ^.L Let q:X

-

GA be in M= and take a factorization (G .. µ) (!::,e)

=

^{6 q,}

e : X ---- GB in E and JJ : 6B - 6A in M . Under the assumptions 1. or 2. JJ can be written as JJ = L,m

m: B ---- A. Furthermore, there is a d :A - B rendering commutative the diagram

q

X -+ GA

l

^Gd

II

e

GB ^1::- GA

Gm

So we have md = A, and looking at the diagram e

X GB

GB GA

Gm

we get also dm= B. Therefore, in the comma-category

<X,G>

itself.

(q' A) is isomorphic to (e,B) EE and hence in E

3. - 7. : Again we only show the non-trivial inclusion.

Let et : L,A D be in _§_ b and take a factorization

(G ~ µ) (L,e) = G cet with e and JJ as above. Because we have Mcinit(D,G) (alsoincase7.: see (9) below), there is anunique d :A ---- B with Gd= e and µ(L,d) = e t , and furthermore we have a unique d' : B ---- A wi th

(Gd')e = GA and a(L,d') = JJ • Looking at the diagram

L, e

L,GA ,,.,-

L,

el ~~~

tc<

L,GB

G " JJ

G ~ D

we get immediately dd¹ = B. lt remains to show d'd = A.

If a is G-initial, this equation follows from (i) a(t:id'd) = a , (ii) Gd'd = (Gd')e =GA.

If a is a mono-cone, ( i) implies already d' d = A , and if G is faithful, (ii) implies already the desired result.

If Iso(G) c E holds, we have a E~::!: c Iso(G)~ = Init(_Q,G) as above. At last, we have to show that the assumptions of (6) imply the assurnption of (7), i.e. Iso(G) c:~ .

For that, because of (9) i t suffices to show But this is already proved in (1).

(9) Remark. For any functor and for classes E and M as above one has the following implications:

Iso(G) CE ~ Iso(G) ^C Mk ~> ~clnit(_Q,G) .

If G has a left adjoint F with co-unit ^{E ,} then the condi tion ~ c Ini t (Q, G) is also equivalent to:

For all A E Ob A ^E A : FGA - A is in ~,b(Q, idA)

We see from (8) that in all interesting cases the classes E and M determine each other uniquely. Hence we define:

(10) Definition. Let ~cCone(_Q) be a subclass, which is closed under composition with isomorphism of A from the right (cp. (8)). We say that G admits (epimorphic,

orthogonal) M-factorizations [with respect to Q], if for all cones y: t:iX - G" D, D: D - ~, there is (e,A) EMor(G)

(Epi (G); ~.da) and (A,JJ) M , such that the diagram

commutes.

/j/

6GA ^{~ o µ} t:.X - - - + G o D

y

Functors admitting such factorizations arise in topology as well as in algebra.

(11) Examples. 1. Let G : A - X be an (~,~) -topological functor in the sense of Herrlich [10] and let

~,G denote the class of all cones (B,µ) E Init(_Q,G) such that

(B,Gµi) iEOb D

and orthogonal

is a source of M. Then G admi ts epimorphic

~Q,G-factorizations for all categories ^{D •} Especially, this is fulfilled if G is a top-category in the sence of Wyler [24] •

2. If G is a regular functor in the sense of Herrlich [9], then G admits epimorphic and orthogonal MonoCone(_Q) -

factorizations for all categories D • (Note that for regular functors MonoCone(_Q)cinit(_Q,G) holds.) Especially, every

monadic functor over Set admits such factorizations.

3. Let G: A - Set be monadic and Gf. _{in -}: Af. _{i n}

-

^{- -Setf. i n}

the restriction of G to the finite objects of A. Then Gfin is not regular (in general), but Gfin admits

orthogonal and epimorphic MonoCone(l)-factorizations.

4. If G is a locally adjunctable functor in the sense of Kaput [14] and if A has and G preserves pullbacks, then G admits orthogonal Cone(l)-factorizations; moreover,

if G is even strongly locally adjunctable (cp. [1 ]) ,

then G admits epimorphic and orthogonal Cone(~)-factorizations for all non-empty and connected categories D.

Although the concept of functors admitting factorizations of cones is a common generalization of topological and

algebraic functors, i t is still possible to prove some nice properties of such functors.

(12) Proposition. Let McCone(D) be a subclass.

1. If ~cinit(Q,G) holds and if G admits (epimorphic;

orthogonal) ^~ - factorizations, then A does.

2. If G has a left adjoint and if A admits (epimorphic;

orthogonal) ^~ - factorizations, then G does.

Proof.

1. For every a : A - D there is a factorization (G ^c µ) (6e) = G • a with (B,µ) E~ and (e,B) EMor(G) (Epi(G) ;~*). The desired factorization of a in A is

given by the unique d :A - B with Gd= e and µ(6d) = a . 2. For every y: fl-X -

y = (G ^c a) (fl-nX) , where

we have a factorization n X : X

-

at _{X '} and a can be written as y

= (

^{G o} JJ ) ( 6 e) w i th e :

=

(Gd) n X .

GFX is the unit of G

a

=

µ (6d) . So we have

(13) Remark. The following assertions are equivalent:

i. G has a left adjoint,

ii. G admits epimorphic and orthogonal Cone(D)-factorizations for all categories D.

iii. G admits orthogonal Cone(D)-factorizations for all categories D .

iv. G admits orthogonal Cone(~)-factorizations.

(12) 2. tells us, that the concept of functors admitting factorizations of cones leads us to new aspects "only" in case of functors, which have not a left adjoint. Hence to study continuity properties of such functors becomes more interesting. First we note the following wellknown

(14) Lemma. Every G-initial cone which is mapped into a limit-cone by G is already a limit-cone itself.

From now on, let us assume ~ c Init (Q,G) .

(15) Lemma. If G admits epimorphic or orthogonal

M-factorizations, then Lirn(D) cM holds; especially, every limit-cone is G-initial.

Proof.

By (12) 1. A adrnits epimorphic or orthogonal ~-factorizations.

From this fact the inclusion follows straightforward (cp. [10]).

(16) Theorem. If G adrnits epimorphic or orthogonal M-factorizations and if X is Q-cornplete, then so is A and G preserves D-lirnits.

Proof.

Let D: D - A be a diagram. We take a limit K :l'.K - G ⁰ D

in X and a factorization (Go;,_) (l\.e) = K with (L,>-) EM and (e,L) EEpi(G) or (e,L) ~~- The unique k: GL - - - K with K(.6.k) =Go A fulfils the equation ke = K. Because of (L,>-) Einit(D,G) we have a unique 1 : L - L with

Gl

=

ek and ;,_(61)

=

>-. Then (Gl)e

=

e holds and therefore ek

=

GL if e is either G-epimorphic oder in M•

So Go>-

=

^K is a limit-cone, and we are ready by (14).

If the factorizations are epimorphic and orthogonal, the

assumption of _Q_-completeness of X is superfluous for G to be D-continuous.

(17) Theorem. If G admits epimorphic and orthogonal M - factorizations, then G preserves D - limi ts.

Proof.

Let >- : l\.L - - - D a limit-cone and let y : X - Go D be any cone. y has a factorization (Geµ) (.6.e)

=

a , and there is an unique m : A - B such that >- (6m) = µ holds. The

morphism x:= (Gm) e is uniquely determined by (G ^{O ;,_)} (6x) = y • Indeed, since ;,_ is in M by (15), we have an unique

d : A - L rendering commutative the diagram

t.e

l\.GA ßGd ...- ...- ...- - ...- - -

l

^G ' µ

-

K ...- .6.GL

G ";,_

for every y : X - GL with y

=

^(Go o) (.6.y) . But because \ is a limi t-cone, d must be equal to m and therefore y

=

(Gm) e .

(18) Corollary (cp. [1]). Every strongly locally adjunctable functor preserves all non -empty connected limits, especially all filtered limits.

The following theorem generalizes corresponding results in [9], [10] and [11]. For every subclass E cMor(G) and every

X E Ob X let <X,~> be the comma-category of ^~ - morphisms wi th domain X:

A morphism a (e,A) - (e' ,A'), where (e,A) and ( e' ,A') belang to ^~ , is given by a morphism a : A - A' of A such that the following diagram commutes

Ga GA

~

_X

(19) Theorem. If for each non-empty set I there exists a class

~1^c: Init(_!,G) (i.e. a class of I - indexed G - initial sources), such that

1. G admi ts orthogonal ~I - factorizations, 2. ~I

= ~

1 , where 1 is a singleton set, 3. <X,~

1> has a small skeleton for all X EOb X , then G is faithful.

Proof.

Using conditions 1. and 2. one can immediately prove, that all categories <X,M

1 > have products (of non-empty families of objects). Because they have also a small skeleton, i t follows from a well-known theorem of Freyd, that they must be pre-ordered classes. Now, if we have x

=

Ga

=

Gb with a,b: A - B in

!2.,

we get a factorization (Gm)e =x with m

E~, ,

(e,C) EM1 ^J. . Furtherrnore, we get rnorphisrns da,db :A - C with

Gd a

=

Gd b

=

e and mda = a , rndb = b . So we have

<GA,~

1 >-morphisrns da,db (GA,A) - (e,C) , which rnust be equal, and therefore also a and b are equal.

GA - - - ~ GC

'b/Ga~

G A - - - + GB Gb

Conversely, continuity and faithfulness of G yields existence of factorizations of cones in many cases:

(20) Theorem. Let A be cornplete and wellpowered and let G be continuous (i.e. a limit preserving functor) and faithful. Then for every small category D there exists a class ~cinit(Q,G) , such that G admits epimorphic and orthogonal M - factorizations.

Proof.

For every cone y : !'::.X - - G • D we have an induced rnorphisrn g: X - G

TT

^Di

iEObD

By the usual technique of Isbell - Kennison - Herrlich one can construct a factorization g

=

(Gm)e, where rn is a G-initial rnonornorphisrn and e a G-epimorphism

(cp. [6]; note that every equalizer is a G-initial monomorphism, because G is faithful and preserves equalizers). Going back to the cones one gets immediately a factorization y

=

^(Geµ) (!'::.e), where µ is in M:= Init(.Q_,G) ~ MonoCone(D)

[ 1 ]

[ 2 ]

[ 3]

f 4 ]

[ 5]

r

⁶ J

[ 7 ]

[ 8]

[ 9]

Börger, R. and Tholen, W.

Brümmer, G. C. L. :

Dyckhoff, R.:

Eilenberg, S. and Moore, J. C. :

Freyd, P. and Kelly, G.M.:

Greve, G. and Tholen, W.:

Herrlich, H.:

Herrlich, H. :

Herrlich, H.:

R E F E R E N C E S

Abschwächungen des Adjunktionsbegriffs.

manuscripta math. 19 (1976), 19 - 45.

A categorical study of initiality in uniform topology,.

Thesis, Cape Town 1971.

Topics in general topology.

Bicategories, projective covers, perfect mappings and resolutions of sheaves. Thesis, Oxford 1974.

Adjoint functors and triples.

Ill. J. Math. 9 (1965), 381 - 398.

Categories of continuous functors, I.

J. of Pure a. Appl. Alg. 2 (1972) , 169 - 191 •

Adjoint Co - triangles.

Math. Coll. Univ. Cape Town X (1975/6), 81 - 92 .

Topologische Reflexionen und Coreflexionen.

Lecture Notes in Math. 78 (1968) • Factorizations of morphisms

f : B - - FA .

Math. Z. 114 (1970), 180 - 186.

Regular categories and regular functors.

Can.J. Math. 26 (1974), 709 - 720