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Seminarberichte aus dem Fachbereich Mathematik der FernUniversität

03 – 1977

Die Dozentinnen und Dozenten der Mathematik (Hrsg.)

Seminarbericht Nr. 3

SEMINARBERICHTE

AUS DEM

Fachbereich Mathematik

DER

FERNUNIVERSITAT

NR. 3 1977

HERAUSGEGEBEN VON DEN DOZENTEN DER MATHEMATIK

''A .-

i..t-r . --

' ' ,;,.

Die Seminarberichte erscheinen in unregelmäßigen Abständen.

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

SCHRIFT: I'

MÜLLER

Inhaltsverzeichnis

A note on a paper of Hofmann and Mislove:

Amalgamations in categories

von Walter Tholen

Semi-topological functors I

von Walter Tholen

Semi-topological functors II: External characterizations

von Walter Tholen

und Manfred B. Wischnewsky

Semi-topological functors III: Adjoint liftings von Walter Tholen

und Manfred B. Wischnewsky

An exact sequence in pair homotopy von Keith A. Hardie

Cantors Diagonalprinzip für Kategorien von Reinhard Börger und Walter Tholen

Factor categories and totalizers von Reinhard Börger

Epimorphismen in der Kategorie der Ringe von Norbert Schämann

Radikale in der Kategorie der Ringe von Norbert Schämann

Seite

- 5 -

- 13 -

- 48 -

- 79 -

- 109 -

- 124 -

- 131 -

A note on a paper of Hofmann and Mislove:

Amalgamations in categories WALTER THOLEN

In [ 3] Hofmann and Mislove prove two main results on amal- gamations in categories:

THEOREM 1. Let A be a category ·satisfying. the fol Z_owing hypotheses:

(HO) A has epic-monic factorizations.

(H1) Ais complete and has push-outs.

(H2) All objects in A are/small (i.e~, Ais @ell-powered).

Then the following statements are equivdlent:

(1) Every pair of monics whi~h can be ~eakly amalg;mated can be amalgamated ..

(2) Every ·identical· pair of monids can be amalgamated.

(3) Every monic is stric_t. {i'._e .. , is an equalize.r).

Recall that. a weak amalgamatiOn of two monics m 1, m

2 with common do~aip,is a push-out

m1

.\

(P)

-1.

^J1'

' J 2.

w.i.th monics

J

₁^, j·. - (P.) is·an·amalgamatidn-,· iff moreover (P)

2 . is a _pu11-back.

THEOREM 2. Let A be

a

c·ategor,y sa1<_i;s'fying (~O). - (H2) and suppose that at leas,t one o:[ the ·'fol iowing hypothesf:!S is sat'i sfie,cl:

(Q) . •j._op·

,d~;ries

the s-tf.uctJ.:~•J,

6f a

'very co~crete category with surjecti;e ep_ics (i. e.·, there.-. .:fi~{sts a faithfut · functor

1 : · A ⁰

P----_s-e..t

preservi,ng :Z.imi ts and .epics) • ·

·- 5 -

,- ' ! ' . .

:·,:,, -·--- ,;:,-.·:.,-1·., :· ' . ' • .'!,.,: .. · : : , .

,. ' '

, t f .·'

:,."/,

- 2 -

(I) A has enough injectives (i.e., for every object A there is a monic A - I into an injective object).

Then A has amalgamations, i f and only i f the equivalent statements of theorem 1 are valid~

Theorem 2 follows immediately from theorern 1, because i t is clear that each of the conditions (D) and (I) irnply the following condition:

(W) For every push-out (P) j

2 is monic, i f m

1 is monic.

In particular this means that A has weak amalgamations, if push-outs exist.

The aim of this paper is to show,-that theorem 1 is valid without assuming the hyptheses (HO) - (H2), one only has to assume tha_t co-kernel pairs of monics (¹i. e. , push-outs

of identical pairs of monics) exist. Moreover we show that (1) - (3J are also equivalerit to the (self-dual) condition that A is balanced, if the following weaker forms of (H1) and (W}

are fulfilled:

(H1') A has push-outs af monics .and equalizers of co~kernel pairs.

(W') For every push-out (P) j

2 is monic, i f m

1 i~ stric~ly monic and m2 is monic.

Consequently i t is possibie to orove theorem 2 in astronger form unde-r IJ1ilde.r . hypotheses.

THEOREM 1'. Let A have co~kernel pairs of monics.

Then for the statements (0) - (7) below we have the imp lications.

(0) <=> (1) <=> (2) <=> (3) => (4) <=> (5) <=> (6) <=> (7).

Moreover, under the conditions _{, , '} (H1') and (W') all statements are equival,e'nt.

/

-.6 / '·:,

1

. ,,3 /2~":~\,:j

- 3 -

(0) Weak amal,gamations are amal,gamations.

(1) Every pair of monias which aan be weakl,y amal,gamated aan be amal,gamated.

(2) EvBry identical, pair of monias ean be amal,gamated.

(3) Every monic is striat.

(4) Every monia is extremal,.

(5) Ais bel,anaed.

(6) Every epia is extremaZ.

(7) For every push-out ^(P) w~th m

1

=

m

2 monia and m

1 not an isomorphism one has j

1 :f= j 2•

1

Proof. (0) => (1) is trivial. (1) => (2) is easy to check (c~. (3], 2.3). (2) => (3): ~f we have (P) with m = m

1 ^,= mi monic ·such that , (P) ,is a pull-back, then m is an equ.alizer

'"' , . ' \

of j 1 and .j 2. (3) => (0): We prove this_ very similarly to 2.3 (1) => (2)· i_n [3] but ;,ithout using epic-monic factori- ,

• - • • .. • • 1

zations like Hofmann and Mislove. Let (P) _'

be

a weak amal--

' . .

' .

gamatio~. In .order to show that (P) is_a pull-ba~k we take x1, x

2

^{with j 1}

^x

¹

⁼

^{j 2}

^'f.

^{2• Then p: = j 1m1 is monic}

'

'··---~--

"

·u 2

. 1',/2

',,

7

.. ,._: -~• ' '• •.~ l, •~- ' __ , • '••

' ; '

,\._-,',

..

\

\ 1

- 4 -

. and so an equalizer of two morphisrns f1, f2· For u. : =f . j . ,

1. 1. 1.

i

=

^{1 , 2,} we have u1m1

=

u2m2 and hence a unique ^f with _fji

=

^u;

i

=

^{1 , 2.}

For y: = j 1x

1 we get f

1y = fy = f

2y such that a unique x exists with px

are rnonic.

=

y. From this we have m.x=x , since j . , i

=

1,2,

1. 1.

(3) => (4) is trivial. (4) <=> (5) <=> (6) holds in every category as is well-known and im.mediate to check. (6) => (7):

Let (P) be as in (7). Assuming jl

=

j

2 one has m epic and hence /an isomorphism because of (6). (7)

=>

(6): Let e be epic .and let e = mg be a factorization with m monic. With the

co-kernel pair (j 1,j

2). of m one has j

0e

=

j ₁e and hence j0 = j

1 such that m must be an isomorphism because of (7).

Now let the hypötheses (H1 ') and {W'} be satisfie d. For the purpose of proving (4) => (3) we have to show that extremal monics are.strict. An extrE;:!mal rnonic has a·facto:tization

m

=

de where d is the equaiizer of,the co-kernel pair (j1,j2) of m : Now i t suffices to '.show that e is epic. If (p

1^,p2) is a co-kernel pair of e we get a morphism d' with j.d = d'p.

. ' 1. 1.,

.'.i

=

1,2. If d' is rrionic then one gets im.mediately p

1 ⁼ p

2 such that e is epic.

in

. --- ^Ji

But i t is easy to check that (*) forms a push~out of pairs '(cp. Kel~y [5]) and can be cons,truc:t~d by three push-outs:

P1. P2

d1• (I) .. .

/i""

^(II)

^1•d

. . ~ I I I ) ; d ' · ~

-<_i1---~!/-~---;;-/__.

... ,... ' , . . , , . , 1

'li.:' . -

,:

i

\' ..

r \ • .":·r,('.~-·

: .. '1

·• .• : ' (

::'f~t}ii ·:;//}\.:;:

' . - ·a.

.,,,_,,··

- 5 -

From this we see that d' is a composition of monics because of condition (W') (remember that (I) and (III) together form a push-out, too) and hence a monic.

From the equivalence of statements (0), (3) and (5) of theorem 1' one gets:

C0R0LLARY. Let A be a categoy,y satisfyinr_J lzyroihern:3 (H 1 ').

Th~H A has amabqamations i f anJ only if Ais balan~cd an.rJ Dati3fiec c:ond-i tlon (W').

REMARKS. ·1. The factorization constructed above is just the factorization of m over its dominion (cp. Isbell [4]}.

If A has all co-kernel_. pairs this facto,rization can be con.structed. for ever.y morphis:rn. Hence (H1) and (W' )' imply that A has epic-s tri<i:::t. monic · factorizatic:ms., ,It is well-known -

' .

that under the hypothesElS (H1) and (H~!) we can construct

epic-extremal monic factori,zations (cp., Herrl,ich and .Strecker

1 , ' ' 1 •

[ 2 l). So ·tn pa,rticular -(ijO).,.follpws from (H1J arid .. (H2).

But these factorizatioris ¹do not help to· ,prove ( 3). => ( 4) . Hofmann and Mislove need their epic-monic,factorizations essentially only in connection with their condition, that

, ,· 1 . • '·:" ) · i - . ~ ·~ , , ·. ,' .. •, ', • . · .· ;,

"the s_llböbjects .of· ev.ery obJect are · separated.'" This ccmdition c~m immediately be ,'Seeh'':to' :Jj~·-'eqtiivalent'. :with (3} of theorem 1 ~"

if A has: k~Lc~~oni·d fci'ctortz_at,ions and co-kernef

p.id„r-S

of monics ..

•'C,gfs 1 • "

2. Anot.her prqof of : f4J ,.=> "JO') :is po$sible us.ng: Ringels resul t

~ ,·

on amalgamations (.cp~\ [:,J}·l·., .·•But t-11.en 'on~ has

·t6

·replac:e. (Hl') apd (W') by the ,strOn'ge:it, condi tions .. (H1 "). ancf (W"):

, \., . / ·• •·· • .,.oo \. · . - ' · • ' ,

i

(H .1'' ) A. has push-outs,

jf

mpnir>s qnd pu'Zl-bar>ks of monics.

(W".) ~OY', PVP.Y'Y push"-\;ut~Jpf

"s;:·-i;,

~oniC i f ni1 and m2 aY'e Under these hypotheses_the equivalence,of the ~tatements JO) and (3). - (7) of th~orem 1' has been·_shown• in [9] (Thm. 9.17).

\ l"J".,\, " _( • ' •

Statement ("lcJ appea:ts ... · to 'the author' s kwnowledge - at first in the paper [

1]

of G:r;egoire and Schreiden.

,· ~ ,.,.;,

1 ~

/ . I

, , '

-^.' ·, '

'-~ ' :

.. ,

,, ', .j

'. _J

,, !:J-

- 6 -

THEOREM 2'. Let A be a category satisfying (H 1'}

and at Zeast one of the foZZowing conditions:

· · hf

z

^{f t} 1 1 •• A⁰P -

se~

(D'} There ex~sts a fa~t u unc or ^~ preserving puZZ-backs and strict epics.

(I'} For ever>y obj"ect A of A ther>e is a monic A - I wher>e I is injective in respect to strict monics.

Then A has amaZgamations i f and onZy i f Ais baZanced.

Proo.f. Each of the two condition~ (D') and (I') implies (W'}

1

and so, in the presence of (H 1'), theorem 2' follows from the corollary above.

Not_e that every balanced algebra.ic category (in the sense of Herrlich and Strecker (2] } with an injective cogenerator has amalgama:tions because (I) and therefore (I') holds. Porst [6]

has shöwn that such a category has an-algebraic dual with surj ecti ve epics. In ·particular conq.i tions {D) and (D') are satisfied, too.

E~M,PLES. 1. In the category .G of g+oups condition (W'} is

1 ,.: ) -

fulfillec;L (c;3-nd hence amalgamat.ions ßxist) , but not condition (W) and hence neither (D) .nor (I) . To see this, we refer to the following example of 'A~ Leu tbecher: ! Let G be a g.roup, U a subgroup of G and Na normal subgroup 0:f U. Let

N

be the normal hull of N in- G. Then one has a push-out

u G

/

l

- - - - ' - - - • GI-_{N ..}

where all morphisrns are the canonical ones.

.: {

- lO

, 1 ',

·i"

1 .-.· ·.r,

-.,_.,' ., .. ' t :~·:·-. ··,,,-••. '

lii:

- 7 -

But if Gis the symmetric group

s

₄, U the abelian subgroup_

generated by the transpositions (1,2), (3,4) and N the sub- group containing (1,2) and the identity, then Nmust be the whole group

s

₄ and U/N- G/N is not a monomorphism.

2. The categories TG of topological groups and TAG of topo- logical Abelian groupssatisfy condition (W'), because push-outs in TG and TAG are the algebraic ones provided with a suitable topology. But they do not have amalgärriations, because they are not balanced.

', v

3. The category C of compact (Hausdorff) -spaces has amalgamati,011s·,,

. . '

because C is balanced and has enough injectives (i.e. the unit interval is an injective cogenerator):

'.\

4. The category CAG of compact Abelian g.roups has amalgamations, - '_

because CAG is balanced and sat{sfies condition (W') · A push-out in CAG is the Bohr-comp_actification of the push-out in TAG and,, the cano'nical · map into th~ Bohr-coinpa~tification is one-to-ohe (cp. Semadeni [8],.p. 253} .. (Anötl}er reason for this is that

CAG

is'clually equivalent to :thecategory

A

' ' 1 , 1

of Abelian groups and therefore condi tio'n, _{• ·\ • , • <} (D') ₁ is fulfilled) • · , · . ,

5. It is

an

open problem .wether .or,not the balanced category

'

. ~ . ' - ' . • ! . . . '

. ·- ',_ ' . ^~ . ^{. .} . ^{. '}

CG of compact_ groups. sa,tisfies. condj.tion, .(~') . Usin<;,

the example in 1 (proyided w:Lth di__:5crete topologies) one sees t,hat (W) arid hence (D~· .and- (T). are. not s,ati!3fied ~ Hence i t is c:).ear' that this. probl.em_ oarinot: be solved by the, methods of [ 3] .

REMARK. All· proof~ in this paper werk .if on~ replaces "mc:mic"

byM-monic for a slliitab:Le class cf morphisms

M.

However, in view' of 'the ·exämpl~s this generali_zätion does _not seem

. .

to be justified._

... ^·

. 1

• A ."

,!. 1

',

J:

/

: ,, ' . . ,

·~

^.

- 8 - REFERENCES

[1] P. Gregoire and A. Schreiden, Generalisation de Za notion de produit amalgame et application a des categories

conretes, Bull. de la Soc. Royale des Sei.de Liege, 11 --11_ {1971), 543 - 554.

[2] H. Herrlich and G.E. Strecker, Category Theory,, Allyn and Bacon, Boston 1973.

[3] K.H. Hofmann and M. Mislove, Amalgamation in categories with concrete duals, Alg. Univ.

i

{1976), 327 - 347.

[ 4] J. R. Isbell, Epimorphisms and dominions., Proc. of the Conf. on Cat. Alg., La Jolla 1965, Springer-Verlag, Berlin 1966.

[ .5

l

G .. M. Kelly, Mono7:1orphisms, epimorphisms and pu Z Zbacks, J. Austr.al. Math. Soc-

.2.

(1969), 124-142.

[ 6] H.-E. Porst, Dua.litie,· of concrete categories, Cah. Topo. Geo. Diff.

.l2

(1976), 95 - 107.

171 C.M. Ringel, Tije intersection property of ·amalgamations, J. Pure Appl. Alg. ~ {1972), 341- 342.

181 Z. Sernadeni, Ranach Spaces of'contlnuous functions, Vol. I, ~6iish Scientific_Publisher~. Warszawa _{' ,} 1971.

1 91 W. Tholen, Re Zative Bi Zd:::er>Ze9ungen und algebPaische Kategori~n, Dissertation, Universität Mlinst~i 1974.

'(Juli 1977)

12 '.

Walter Tholen

Fachbereich Mathematik Fernuniversität

Postfach 940 D- 5 8G>O Hagen , Fed. Rep. Germany

1 /

:.·'·

'. ;'1_: '.~;,:j:4}

1

Semi-topological functors I

Walter Tholen

1. Introduction

The aim of this paper is to give an appropriate description of "nice concrete functors" such that "algebraic" as well as

"topological" functors and their compositions äre included.

Moreover, one should be able to prove all common properties of these f unctors. The means to get such a description i,s the concept of relative factorizations. In [ 11] Herrlich proved the existence of two kinds of factorizations of morphisms X -+ PA where ,p : A , .:.. X · is

a:

g'iven functor.

This concept was generalized in [2~] and extended from morphistns_

to cones over a, f ixed diagram category · in [ 31 ] . In the case ' ,, of an identical functor thi~ is nothing but the,image_

' . '

factorization of morphisms resp.· cone_s in

a'.

~ategory (cp. [? l, [24] and references there). 1. • . , .• , 1, , , . ,

A special kind of th.ese iela:ti ve factorizaticns o·f cones was impl.ic1tly 'used

to

introciuce the 'diffet-ent notioris of

topological functors. ·,wh.ich are based on· a, categorical treatment

. \

of ini tial.i ty and linality:

Cp.

Brümrner [ 3] , Hoffmann [ 15] ,

' - ' '

Wischnewsky·

J

37

L.

[3ä].;, Wyler (40], · [

4 i ]'

(earlier ·or related

• l \ i ; i ß , 1 ' • • - •

investigations can be t°öunc;i ':ii-r ['1'] ; [ 9] ,· [20], · [ 22 }, [ 25], [28] ;.

mostly based on~yl:e~'s'.p'ap:er:~·:are [4], [6],' (23],· [ .. 27},

[29]}.

The first explicit ·use of' b,on:e f.actorizations to introduc~

a: -

generalize.d type of topo,logical fun'ctors·appeared in

Herr,lich's paper [12]. Qn'.'the¹bther· hand, Herrlich [13·]--has-··

' \. .

also us'ed cone 'factorizations to describe an •ialgebraic" 'type ' of ftmotor~'- 'In hpth pap·ers ,afl con~s a;e all'owed

tö

be large· so:

\ .

' \

13

t

. ' --~ ; ,. '. '

' 1

\

,. ' j

'

, I

;, • I~ __ '. • L

. ;_· : . \';

- 2 -

that any smallness conditions are superfluous. Extending his ,relative cone factorization to arbitrary diagram categories the

author introduced the notion of an (orthogonal) M-functor

'

-

([32], [331) which includes both "a~gebraic" and "topological"

functors. A related notion was introduced by Hong [19].

But there are sorne reasons to generalize this notion once more.

Topological functors are self-dual ([15], [25]), i.e. they c~n b~ equivalently described by initiality or finality.

HofTmann [16] proved a duality theorem even for Herrlich's

generalized topological functors using so called semi-identifying lifts (cp. [5], [10], [30], [42] for related notions). Semi-

topological functors as introduced in this paper generalize

!:!-functors just so far such that they allow both an initial ap~ a final characterization.Thi~ is the anly ju$tification _foi;- the name "semi-topolögical".

' ' I

Se~i-topological functors arise everywhere: All types of topolögicai functors treated in the papers ment.ioned before

1 ' '

are semi-topological; monadic functors'. over Ens and regular '._, ',:j~_upctpts [ 13] are semt-topologica;l.; wellbounded categories . [ 39]

· .•_cmd . locally' presentable categor;iei:; _ [ 8] admi t a semi-topological

••· ,, ßµn.ctor into some power . .pf Ens. Semi.;.topplogical functors have

- , , ! , ' -,· . . - - - · f / ' . . '

. a1J :the · nice.- lifting properties c:me c~n ~xpect ~ ( In this paper

, ' ,

,':o

r: ,

"we; oply prove "internal" liftiI:1.g properties; for external

• 1

>.'

^0-·., qne~ . .see [_36]) ._ Furthermore they pan be externa_lly characterized

; ., _in the language of 2-categories;, this will_ be outlined in [35].

Tl:le main ·r~sults

ol

this. pape,r are. the generalized "Duality Theorem.II 3. 1 . (already aI?,nqunced in [ 34 l) , the charact!erization

.

.

^'

, 1 .

. ~theo:i;:-em 6. 3 (which· generali.Ze$ corresponding resul ts of

• • - • ' • , • • 1 •. J •

. He,rrlich [ 12] and Hoffm~I?,p [ 1 7]) and .the representation theorem ,~ .• 3 ~ ;which was proved a1:, first "dire_c;tlyl' by. ' M. B. Wischnewsky

\

_and the . author during the meeting on · "Kategorien" in Oberwolf ach

•~~ 1977. The essentially new proof 3.2 of faithfulness of semi-

14

•1 .}.::

' ! \~ ~--

_:_(.~?/J

- 3 -

topological functors was inspired by dLscussions with.R. B6rger.

The method used there leads also to the useful corollary 6.4 and will be worked out in [2]. Besides M.B. Wischnewsky and R. B6rger

I am indebted to D. Pumplün and G. Greve for some useful dis- cussions on the subject of this paper.

- 15 -

,,

' I

1, ... < .,

f • ~· \ ' ••

,. f.

i.. '/

;,-

. . 1

., ,:-- .

"'.' 4 -

2. Semi-initial and semi-final liftings

Let P A .... X be a functor. A P-cone is a triple (X, E,, D) , where X is an object in ~. D is a diagrarn in A (i.e., a functor D : _Q „ !2,, D any category) and E, : AX .... P ⁰D is a natural transforrnation (A

=

^{A always} denotes the canonical

D

functor into a fun~tor category). Often, for brevity, we call

E; a P-cone. Cone (P) denotes the "class" of all P-cones.Rever- sing the dire·ction of E, one gets the notion of a P-co-cone and the "class" Co-Cone (P). In case D = 1 (one point category), P-cones can be presented as pairs (x,A), where Ais an object of A and x : X .... PA is a rnorphisrn in X They form the class Mor (P) of so called P-morphisms. Dually one gets the class Co - Mor(P) of P-co-morphisms. An object of ~ can be regarded as a P-cone or a P-co-cone over the empty category

_!.

2.1 Def~nition. Let E, ,: AX „ P⁰ D be a P-cone. A P-semi-initial lifting of E, consists of a P-morphism e : X .... PA and an A-cone

a : AA .... D with

(Poa) (Ae) ·= E,,

such that condition (SI) holds:

For all P-co-;morphisms y

(SI) ß AB -+D witti E, (4·y)

=

.mo:i;phisrn b : B -+A with

: PB .... X and A-cones Poß there is a unique _\ ey

=

^{Pb and a (Ab)}

=

^ß.

Often we call (*) a P-semi-initial factorizatJon of E,. If e can be chosen as an isomorphism (identi;cy morphism), a is called a

· (proper) P-initiaL liftiftg of E,.

--. _: __ / _ _. >PA

AI;_b,.. - ,..

7

_{Po a}

Figure l

-- 16 _{'. "- .}

/

- 5 -

A factoriziation (*) of Eis called rigid,if condition (R) holds:

(R) For all endomorphisms t : A .... A wi th (Pt) e = e and a(At)

=

^{a one has t}

=

^A.

2. 2 Remarks. 1. Condition (SI) is fulfilled, if in (*)

a : 6A • D can be chosen as an ·P-ini tial con.e, i. e. a ·is an P-initial lifting of Poa : APA .... PoD.

2. Condi tion (R) is fulfilled, if in ( *) e: X .... PA can be chosen as an P-epimorphism, i.e. for all f ,g : A .... c the · equation (Pf)e

=

^{(Pg)e iinplies}

f =

^g.

3. For every functor; all P-mqrphisms . x : X .... PA have a

' • ·

trivial rigidP-semi-initial factorization: _' x

=

^{(PA) x.}

. ,,

To illustrate the notions · in 2. 1 ·• we restrict 9ur.selves here , to.

,,,_-,.

two characteristic examples,.,.

, 1

1 •'!;

2. 3 Examples. . .1 • Let P : Top .... Ens be the f orgetful functor of the .,

:/

category •of topologica.l' spaces and let E AX .... P•D be a ::·:

' • , , ' . 1 '

P-cone. Then there is a c9arsest topblogy on the set X

' , ' ' ,' '

making all mappings X-+ Dd, d E Ob ,Q_, continuous. In this way one gets a space · · A and a P-±n'i tial cone

1

with P⁰a

= ~.

.In particU:la_r, , (Poa) / a 11.,. -(AX) •

=

^E

is a r1gid,P-semL·,·1nitial _{. '} f a;.ctor±zat:ion.

\ ·, -. ' .

2. Let p : Grp · .... Ens.'

the

"'forge'l;f~l;, .fun9tor' of ~t.he category of groups and let . ,x •. : X -+ A be·.· a mapping

f

rom a set to a group. Then its factoriz.ati:~m over the subgroup <im X> of A '

generated by im ~ • yields a rigid P-~emi-initial factorizati.on

of x •. In 'case of. a ·sma;ll

p;,.~'.öne ~· :

^{AX -+ P00D (D :} .Q. ~ Gr]?, D

smäl.1) ,.

i t suffices to factorize the .i,.'nduc,~_d "'mapping · x : X ....

T1

^{Dd. '' ·}

17

i .

!! i{

j,;

1

fi-

1\

lt l! . ~ ~

\i.-

1'· ' .

1 ' , :

_.,-· !

l

\1 ,,

l!'-.

- ··~ . r ..

1

r.

I-': ,·, 1,.

.<,, t,,

,_

- 6 -

If l'; is an arbitrary P-cone one first has to factorize all inappings sd : X-+Dd and then to choose a representative set { l';' i : X .... <im (sd.)>

J_

!

iE!_}, which reduces s to a small P-cone s'• In general, this procedure is treated in 7.5.

The usefulness of semi-initial liftings,is demonstrated already by the following lemma.

2.4 Limit Lemma. Let {*) be a rigid P-semi-initial lifting of a limit-cone 1;, then <1 is a li:mit-cone and e · an isomorphism of X.

Proof. The limit property oft; gives us a unique morphism

y : PA .... X with s {l\y)

=

Poa. Necessarily we have ye = X and, by {SI) , there is a morphism t : A .... A with Pt = ey and

a. {l\t),

=

a.. Because of ( R ) , t must be an identity morphfsm, and therefore e is an isomorphism. Hence a. is a P-initial

lifting of a limi t-cone and so- a limit-cone i tsself {cp. [12], [ 31 l), 2. 5 Definition. Let s : P⁰ D .... l\X be a P-co-cone.

A

P-semi:-final lifting of E. consists of a P-morphism e :X .... PA and an A-co-cone a : D .... 6A wi th

. (6 e ) ~

=

P ^a a. , such that condition (SF) ho1ds:

For all P-morphisms y X .... PB and A-co-cones (SF) ~ :, D .... l\B _ with {l\y)· I;

=

P⁰ ß there is a unique

morphism t : Al .... B with {Pt) e

=

^y and (!lt) a

_= ·

ß.

~PA

?i

P o D - - - l\X I l\Pt

---

^.·

~ - -

_~ ~

_tiPB

18.- ^/

\ ,"

1 '

- 7 -

Often we.call (**) a P-serni-final prolongation of ~ . If e can be chosen as an isornorphisrn (identity rnorphism), a is called a (proper) P-final lifting of ~ .

2.6 Rernarks. 1. "Final" is the dual notion of "initial", buf

"serni-final" is not dual to "semi-initial". The dual notion~

are "co-semi-initial" and "co-semi-final~.

2. Obviously, a P-semi-final prolongation of an P-co-cone is uniquely determined up to a canonical isornorphisrn, whereas

a P-cone can have many different rigid P-semi-initial · ₁^, factorizations.

3. A P-morphism e : X .... PA is called a P-quotient, if i t appears in some P-semi-final prolöngation, i.e. there is a P-co-cone E. : P⁰ D ... tiX and a cone a : D ⁺ tiA wi t};l ( **) and (SF). Let us consider the following subclasses of M?r(P):

Iso (P)

=

^{{ (e,A)} e is an isomorphisrn~of ~},

- -

Quot (P) :

=

^{{ (e,A)} (e ,A) is a P-quotient},

, .

Epi (P) :

=

^{{ (e,.A)} (e ,A) is a P-epimorphism} ( cp . 2 . 2 , 2 : ).

For ^p faithful ·we., then have the inclusions Iso (P) ^C Quot (P). ^C Epi (P) •..

Again we considet. our two ·bp.s.~9. e~amples 2. 3:

2. 7 Examples. l. .. Fqr

P:

To_p , ... ', Ens let. ~ :. P⁰D .... tiX be a, P-co-cone •. Then there .:is a :e-ines:t topoJogy on X rnaking. all

. ' ^, '

mappings· .Dd ^•

x,

^c;1 E'oh'..g~,. c'ont:i,n~oµs. So we' have a proper

P-final. li:fti~g .of ·F>.

whicl:t fn

pa-:r;ti?u1ar, yi~lds a. P-semi-final · ..

prolo!ilga tion of , 0 ; ·

.· -1 . t ;: . ,..,

2. In ca-se Grp ^•-

En.:s: ,

¹ förm the fr~e grcmp· FX on

·x

and cons-ide'r the smalle~t c~ri~'rue~ce re:tatiort'

.;,_.qri'"

>FX · such that all rnappirlg~

Dd .... FX/ ~ become honk?rnorphisms. •Then the. canonical map X .... FX/~ gives q; P-Semi---final piolongation of ' .

_, 19 -

''

• ✓ : ..

..

j'

i 1

! '[

!F 11

1 11 ' t)i

lij

i' )_

! 1 -

',, i

'1

, ^__

"

1 J ',

'1

, i ,

·. ', ·_

,

^.

. ,-:...;.

\_ . ^' '

, , l

- 8 -

2.8 Co-Limit Lemma. Let (**) be a P-semi-final lifting

,'

of a co-lirni t-co-cone E. then so is ^{a •} In general, e is not an isornoprhisrn in X.

Proof. Straightforward~ using property (SF). Co-limits are not necessarily preserved by P, hence e is not an

isomorphism in general.

The fundamental connection between semi-initial and semi-final liftingsis given by the following lernma, which we shall use later.

2.9 Diagonal-Lemma. In the commutative diagram of figure 3

ö X - - - - = ' - - - - -

Pol;

ö P B - - - ~ Figure ~

p ^o a

let (p,A) be a P-quotient {cp. 2.6,3.) and let (P.,µ) (öq)

= :

~

be a P-semi-initial factorization. Then there exists a unique morphism t : · A - B with {Pt)p

·=

qx and µ {ot)

=

a~

Proof. ''By def ini tion, there exists a P-co-cone , i:;, :

PoC-+öX /

,, an A-co-cone y : C - öA wi th C : C :...: A such that

{öp) i:;. = P•y is a P-semi~final prolongation. Because of (SI), for, all c E Ob C there is a unique Be : Ce ... B with

qX ( l; C)

=

ß : C - t.B

Pßc and µ(ößC) = o(öyC). So we getan A-co-cone with (öq) ^{öX) r:

=

Poß and therefore, because of ( SF) , a tmique morphism t ^{: A •} B with ,(Pt)p

=

^qx

and (L\t)-y

=

ß

.

Now the_equation ^\J (t,t) = Cl

- and the uniqueness of t is proved canonically.

'

- 20 -

and

- 9 -

3. The generalized "Duality Theorem" .

Fundamental for the definition of semi-topological functors is the· following theorem generalizing the well know~

"Duality Theorem" for topological functors {cp. [15), [25);

see 4.3 below).

3. 1 Theorem. Let P : A ^• X be a functor and let' Q c Mor (P) be a subclass. Theh the following assertions are equivalent: ~

(i) Every P-cone

t

has a rigid P-sem{-initi~l lifting (*) with {e',A) E Q.

(ii) Every P-co-cone f has a.P-semi-final lifting (*i) with (e,A) E Q.

In order to prove this theorem we need first the following lenima •.

3.2 Lemma. Each 0f the c0nditions (i) and (ii.) implies that P i s faithful.

Proof. ( i) . Let f, g : A „ B be ~:-morphisms wi th Pf

=

Pg a:nd f

+

g. Consider the I-indexed discrete P-cone (PA, xi, Bi; iE!_) ··.

with

! :

⁼ Mor(~)and

.· ,I

B. : _l = _· B, x. ··: _l =,Pf : PA,....PB. _{. l} and.form a P-semi-initial factorization

(P a.}

l

e '

=

^{x; ,} _{·,:1 . , l} ^{a. :} _{· •·} ^{C-+- B.} _l

for all iEI ..

for a+l iEI.

= {

^h·: A-C, ^·I a.h € { f,g,} for all i E'I}

' 1 ·. "·. .

'

The class K

is not empty In f±,gure 4, take.cili bi tobe .f and apply (SI}.

7

^{- .}

PA

/ Figure- 4

' , .. ^{, ·'}

y «

'.-~' ,

PC

Pa. l

1 \

. -,.,

\

li

1

1. ·.

i'i _.

ii . 1l.

1 .'\_, \

i ;I

! ,'

- 10 -

Hence there exists a surj ection o : I - K ( take o /!S_ to be idK and

o/!,~

tobe constan~) and we can define

b. l

=

r , f in case .

1

^{g in case a. o(i) a. o(i) l} l

^{= =}

^{g f}

Application of (SI) yields an h for all i EI.

A .... C wi th a . h

=

^b.

l l

Hence we have a j E _! wi th h

=

^{o ( j)} and the contradiction aj o ( j )

=

^{f < = > a j o ( j )}

=

^{g .}

(ii) Form the _I-indexed discrete P-co-cone (PB, x., A.,;iEI)

l l -

with A.

=

A

l

analogous

and I , x.

- l

way as above.

3.3 Proof of the theorem.

as before and proceed in an absolutely

(i) => (ii). Let!; : P•D-tiX be a P-co-cone and consider the category

D

whose objects are all P---morphisms (x,B)., such that there is a co-cone

ß : D .... tiB wi th ( tix)· /;

=

P^O ß; öecause of 3. 2 ß is uniquely determine.d and denoted by ·. ßx,B • A morphism

r :

(x,B) .. (y,C) is given by an ~-morphism f : B ~ C with (Pf) x

=

y. We have a canonical f unctor

5 : 5 -

^A and a P-cO~e

r :

^{~X - p O}

^D

^{wi th}

i

^(x,B)

=

x · for all (x;B) E Ob5, ~hich has arigid P-semi-initial

''

factorization (ie) (Po;)

= r

^{with e : X 7 PA in}

-

^Q. Now, for every ^{' ' ' .} d E Ob D , we define ßd .(x,B) :

=

ß _x,B · d for aH. (x,B) E Ob Q,

and get an. ~-cone _

ßd :

~Dd ^••

D

:because of 3. 2. Furthermore, we have i(ö!;d)

=

P⁰jd', and applying (S

1

I) we get a morphism ad: Dd-A with,e (!;d)

=

^Pa.d.

(

Figure 5

.'

- ,11

a. : D ^• 11A 'is a cone with (/1e) /; = Poa.. It remains to show that this is a P-semi-final prolongation of /; Let y and ß be as in (SF), that is (y,B) E Ob

Q

and · ß = ß B"

y,

With t : = ;(y,B) we then have (Pt)e = y (and therefore (/1t)a. = ß). Any other s : A ^• B with (Ps)e = y leads to a morphism

s

(e,A) ^• (y,B) of

D

and hence to the equation

- -

s a. (e,A) = a. (y,B) = t. But the morphism a : = ; (e,A) : A ^• A fulfils the equations (Pa) e=e and ~(/1a) = ~ and hence must be the identity morphism because of (R) .

(ii) => (i). We proceed in an analogous way as before. Given

:...

a P-cone I; : /1X ^• P⁰ D we form the category Q_, whose objects are P-co-morphisms (x,B), such that there· is a cone ß = ß :t1B • D

x,B · wi th I; (11x) · = P^O ß. We consider a P-semi-fin~l · pro~ongation ·

( lie) I; = P⁰ ä wi th e : X • PA in Q of the canonica1 P-co-cone

·i _:

^{Po ß _•}

^'lx.

^~ For e~ery d E Ob :Q_ we have an ?±-cone ßd : D .. ~Dd wi th ( 11 l;d) [, = P• ßd and get a. morµüsm ad : A ^• Dd wi th

1 -

(Pa.d) e = l;d.- Tb prove that (P•·a.) (!1e) = I; is ·a rigid P-semi-initial-,- factorization, let (y, B) be in Ö. _{- ,} Then, wi th ,b : = ~ (y ,];3)

we have,. ey = Pb. (R) is fulfilled because .of 2.2,2.

, 3. 4 Remarks. 1. To 'prove the (i)..-version of 3 ._2. we have not u.sed condi tion . ( R l and' tl;J.e unisruenes$ par;t; of (SI) , which·

1 '

follows neci~ssarily~

2. B~cause of 3.2 -(ii•),_statement 3.1 (ii) is equivalent to·the

1 ' - ... , •• - • • . . •. _;· \ ' .< -,. ,, : • ' "".', • : • ' ••• ' ; · , • ~

corresDonq.ipg- statement aboüt di-scret'e ·:P-co,..cortes~

,,. ~ ~ ~ ' ' ' '

- '

'-,"' c;

3. Note that :the proof_ of

_3.2,

wqrks'withou,t assuining that the resocctive ca,fegorie's,· _{• ' ·. - . . . . .} '. . •. _ .. ·; • ~ . have small .. hom-c1'asses . as Hoffmann . :r, : ' '., .:: ' - -. . • . ~ •• • , ... · .. - • :_~ . - .• ' . ·; . -i ' . ·: • '~-~ , ,.,

apd ,H.err~icn .dt,9:n:,:, __ . i11 th:Er·i,~_9e_:Si9-,l. ,GqS,~. Q{ '(§_,~) ~top_o+9qica1 functors, (cp.

[·12,J,. r1iv

^{!_:.' ,, . . . . . -} ~ . - - . .

\ '

1 •

4. Lemma 3 ._2 become~ · wrong

i.f

one restricts cono.i tions ( i) , (ii)-_,resp. ·to sma-ll·c_o~es, co-.cones resp. For ·a.-.counter-example consider, · the functor A .... 1 , ·where ~ is. any complete,

, J - '

co-complete· T,e!:;p. ca;t~gory being not a lattice.

- 23

. · : '

\j/' . ,··

·'

- 12 -

4. Semi-topological functors

The generalized duality theorem leads to the following definition:

. 4. 1 Definition. 1. P : A -+X is called semi-topological, _ iff every P-cone has a rigid P-semi-ini tial lifting.

Dual notion: co-semi-topological.

2. P i s called (properly) topological, iff every P-cone has a (proper) P-initial lifting. Dual notion: (properly)

co-topologica1.

3. P i s called a (proper) fibration, iff every P-morphism has a (proper) P-initial lifting. Dual notion: (proper) co-fibration.

We have as immediate consequences of 3.1:

4.2 Corol1,ary~ The following staternents are equivalent:

(i) P is_semi-topological.

(ii) Every (discrefe) ~-co-cone has a P-semi-final lifting.

(iii) Every (d{screte) P-con~ E has a P-semi-initial lifting (*) wi th (e ;A) E Epi (P). ¹·

(iv). Every (d.i'screte) P-cone

t

has a P-semi-in{fial lifting ( *) · wi th {e ,A) E Quot (P) ~

'

4.3 Corollary. 'The following are equivalent:

(i) ^p is (properly) topological.

(ii) ^p is (properly) co..:.topological.

(iii) ^p is semi..:.topological and·a (proper) fibration.

(iv) ^p i's co~semi-topo16gical and a (proper) co-fibration.

Proof. It remains to show ( iii) => CiY : One gets a (proper) P-initial lifting of a P-cone E. taking first a ' P-semi„initial - lifting (*) and then a (proper) P-initial lifting of e.

- 24 -

- 1 3 -

4.4 Examples (of semi-topological functors).

1. All topological functors are semi-topological, for instance the underlying Ens-functor of the categories of topological space·s, uniform spaces, proxirni ty spaces, limi t. spaces,

nearness spaces, rneasure spaces, Dynkin-s~s~erns, pre-ord~red sets etc.

More general, any (~1~)-topological functor in th~ sense of Herrlich [121 is semi-topological.

1-·

/, i

2. All monadic functors over Ehs are semi-topologicat, .for -.:., instance the underlying En~-functor of the categöries of

semi-groups, monoids. groups, rings, R-rnodules, (associative) R-(Lie-; Jordan-) algebras, loops, graphs, cornpact _T

2-spaces:

etc. More general, any regular ·functo_r in the sense of

Herrlich [13]is semi-topö1ogical. .\

From these two gronps of examples _one gets :i;nany. n~~ ones

applyinq the fol lowing proposi tion .. Moreover, ;Ln sectiqns 6 and 7

'

we shall give yery gene;;ral cptegorical methods to construct further examples (cp. 7.8).

4.5 Proposition. Anyfunctor ^p A + X of ~h~ followin~

is semi~topologibal:

a. P is the embe'ddinq of an arbitrary ftlll· refler.tive subGategory.

b. P is the r.Omposi tion of two semi-topoloqical functors. _.

c ._ P is the composi tion

i

^{J (j}

A .:-... B. _._

!5:;-:-~

X

,

^..

, where- Q i s topoloqü~ 9 ~-• . ^J is_ c1 fu 11 epi-ref lecti ve arrl I a full' co-refl~ct'ive. emb.eddinq with~co-reflection tnap o such that Q or, is an isomorph i sm.

- 25 -

·-,_, -~~ .,, .:

. '

', , , \

\. :1.J

~ ., .·

••_,,,

:, ' -· ,,/ . ~- -

- '14 -

Proof. a. Every P-cone factorizes over a reflection rnap and every ~-cone is P-initial, because P i s full and faithful.

b. Let Q: K-+X and R: A-+K be semi-topological with P

=

^{Q• R and let} [, : L'IX->- P ⁰D be a P-cone. Form a Q-semi- ini tial factorization (QoK) (6e)

= [

with a Q-epimorphism e and an R-semi-initial factorization (Roa) (L'lk)

=

K with an R-epimorphism k . .

c. Because of b., a P-cone E, : L'IX-+ P⁰ D has a Q':

=

Q⁰J- serni-initial factorizatibn (Q' ⁰ ß) (L'le)

= [,

where ·e X-+ QB can be chosen as an epimorphism of X. Then a :

=

ß(L'lpB) is an A-cone and (Po a) L'I (QpB) - 1 ( L'le)

= [,

is a P-semi-ini tial

factorization with theP-epimorphism (QpB)-1 e.

4. 6 Remarks. 1. By 4. 5 ,' a., b., ,restrictions of topological functors to arbitrary full reflective subcategciries are semi- topological. In 8- we shall show that - vice versa - every semi-topological functor has a presentation as a full reflective restiiction of a topölogical functoi.

2. Ad~mek

'(43]

has cpnsfructed ~ monadic category over the . category of graphs whi,ch is not co-complete. From this example

i t follows that a monadic functor over an arbit:rary base- category fails tobe semi-topological, in geheral, as can be seen from the following theorem which coliects some of the internal lifting properties of semi-topological functors.

4.7 Theorem .. ¹ Let p : A-~ X

be

semi-topological. Then one has:

1 -. p is faithfu1.

2. ^p ·has a left adjoint.

:l • If X is (D-) complete, so is A.

4. If X is (D-) co-complete, so is A.

5. If X has a set of generators, so has A.

- 26 ^{1 '}

,

,..;;_.-,'

I

- 15 -

Proof. 1 . Cp. 3. 2.

2. Apply (SF) in case of empty co-cones.

3.4. Cp. 2.4 and 2.8.

5. This follows from 1. and 2.

Further (external) lifting properties of serni-topological functors are proved in [36].

- 27 -

• i':

' .

:; ~ ). 1' '. ':

' . '-1 -11: /

' r ,

···1'

' J

r .. ,

- 16 -

5. Locally orthogonal Q-functors

In this section we try to get a finer description of the different kinds of P-semi-initial factorizations

(Poµ)(ll.q) = !; of a P-cone !;. The duality theorem 3.1 suggests to consider only those factorizations where q

belongs to a fixed subclass gcMor (P). For some considerations i t is- also convenient to fix a subclass ~cCone (~) and to demand µ E ~- In the following we shall always assume that g (~) is closed under composition with

-from the left (right), i.e. for all p isomorphisms. i : A + B of A and all µ

(Pi)p in

g

^and µ(ll.i) in M.

~ - isomorphisms

X + PA in Q, all ll.B +D in M we have

5. 1 Defini t;i.on. A lo.cally orthogönal (Q,~) -factorization

of a P-cone !; : ll.Y +PoD consists of a P-morphism q : Y +PB in Q

and an A-cone µ : ll.B + D in M with .(Poµ) (ll.q) = !;,

' -

such that·the following "local diagonalization property" holds:

For all p : X ⁺ PA in

g, ·

all !~morphisms _ x : X ^{+ Y} (LD) · and all A-cones· ex : ll.A + D wi th (P oex) ( ll.p) = ' !; (6x)

/

there exists a unique t : A +B with (Pt) p

= qx

and µ(At) = ex (cp. figure 3).

P A + X is called a locally orthogonal (Q,~} -functor, iff

g

contains the class Iso (P) (cp. 2.6) and everr P-cone pdmits a localiy orthogonal Cg,~}-fa~torization.

Sorneti~es we replace the prefix (Q,~) by

g,

if the class M is not specified, i.e. M = Cone (~).

The following bheorem establishes the connecti'on between semi.;..

topological and locally orthogonal Q-functors.

28 -

·.

,.