ABSTRACT
AESJLUTENESS PROPERTIES IN CATEGORY THEORY
by Robert Paré - Ph.D. Thesis in Mathematics
In this thesis we study properties of diagrams in a category ,1 which are preserved by certain classes of functors; properties which we call absolute. We obtain necessary and sufficient conditions for
diagrams to possess some of the usual properties absolutely.
Chapter l investigates those properties which are preserved
by all functors (totally absolute properties).
In chapter II we show how absoluteness properties are relevant
ln mathematics by applying them to the theory of algebras over a
triple.
We study properties of diagrams fn additive categories, which
are preserved by all additive functors, in chapter III.
In the last chapter we characterize those properties of functors
which are preserved by all hyperfunctors from Cat (the hypercategory
of small categories) into itself. We see how these notions tie up
with adjointness.
The methods used are quite general and can easily be adapted
to other cases. A B SOL U T E NES S PRO P E RTl E S
l N C A T EGO R Y THE 0 R Y
by
Robert Pare
A THESIS SUBMITTED TO THE FACULTY
OF GRADUATE STUDIES AND RESEARCH IN
PARTIAL FULFILMENT OF THE REQUlREMENTS FOR
THE DEGREE OF DOCTOR OF PHILOSOPHY
Department of Mathematics
McGi11 University
Montreal
June 1969 ·e
(ê) Robert PA rp l07() 1 •
INTRODUCTION
of The a1m of this thesis is to investigate those properties class of diagrams in a category which are preserved by a given fUnctors. We call these absoluteness properties.
In the past, people were interested in classes of functors which preserve a given class of properties, e.g. monopreserving But there is a functors, continuous functors, exact functors, etc. classes of galois correspondence between classes of functors and to a given properties of diagrams, which on the one hand associates these pro class of properties the class of all functors preserving class of functors perties, and on the other hand associates to a given We propose the class of all properties preserved by these functors. to show its to initiate the study of this second association and
relevance in category theory. which are Since we will be ~tudying those properties of diagrams Ive could have invariant under certain classes of transformations,
titled this thesis "Geometry in the Category of Categories". properties Chapter Ideals with total absoluteness, the study of in terms of preserved by all functors. We obtain characterizations the usual pro equations which tell us when diagrams possess some of
perties in category theory absolutely. l to the In Chapter II we apply some of the concepts of Chapter here that theory of triples and their algebras. It becomes apparent
absoluteness properties are fundamental.
Chapter III is devoted to the study of additive absoluteness. ii.
We give necessary and sufficient conditions for certain properties of
diagrams to be preserved by all additive functors. The conditions take
the form of equations involving composition, addition, subtraction,
identity maps, and o. Certain examples of additive absoluteness are
known in homological algebra, but we shall not go into this here.
Chapter IV studies Cat-absoluteness; properties of functors
which are preserved by all hyperfunctors of Cat (the category of --- small categories) into itself. Tne characterizations are in the form
of equations involving functors, natural transformations, the four
kinds of composition, and both kinds of identities. We shall see that
these notions are closely associated with adjointness and tripleable-
ness.
This study is by no means complete. It is apparent that one
should investigate relative absoluteness, where the word relative ~s
taken in the sense of relative category theory of Eilenberg-Kelly
[51. Then the results of Chapter IV would probably fit under hyper-
absoluteness. The relationship of absoluteness with homological
algebra should also be studied.
The reader is assumed to be familiar with elementary category
theory as can be found in the first few chapters of Mitchell [191
or Freyd [71. The definitions and results which are not easily
accessible ~n the literature are given in the text when they are
needed.
l would like to thank Michael Barr for many stimulating conver-
sations. l would especially like to thank my adviser, Professor .~ Lambek, for his guidance and encouragement. CONTENTS
Introduction ~
Contents ~~~
Chapter 1. Total Abso1uteness 1
l. Abso1ute epimorphisms 2
2. Absolute weak suprema 3
3. Absolute suprema 11
4. Absolute coequalizers 13
5. Absolute pushouts 19
6. Miscellaneous results 22
7. Remarks 24
Chapter II. Triples and Absoluteness 25
1. Triples 26
2. Triples and absoluteness 28
3. Tripleab1eness 36
4. Examples 44
Chapter III. Additive Absoluteness 48
1. Absolute epimorphisms 49
2. Absolute weak suprema 50
3. Absolute suprema 57
4. Absolute coequalizers 60
5. Absolute pushouts 65
6. Absolute coproducts 69
7. Remarks 72 iv.
74 Chapter IV. Cat-absoluteness 75 l. Absolute preservation · · · · 84 2. Absolute reflection . . . . · · · · 92 3. Absolute faithfulness . . . .
4. Absolute fulness 93 96 5. Remarks . . . . · · · · 98 Conclusion . . . · 99 References · · · · 1. e CHAPTER l
TOTAL ABSOLUTENESS
if it is preserved A property of diagrams is called totally absolute is a totally by all functors. For example "a is an isomorphism" properties are absolute property. Some other trivial totally absolute , "there exists a "commutivity of a diagram", "a is an identity map"
map from A to AI ", etc. of diagrams In this chapter we consider some of the usual properties conditions for ln a category and establish necessary and sufficient equations which imply these properties to be totally absolute. We obtain are in the most that a diagram has a given property, and these equations
general form possible. we mean For the rest of chapter l, when we talk of absoluteness,
total absoluteness. obvious what they We will not state dual results unless it is not
should be. limi ts and colimi ts Following Lambek 1 s example [9] , we call we refer to them by "infimum" and "supremurr" respectively. In fact
if F: I~A is a their usual abbreviations "inf" and "sup". Thus u lS a natural trans functor, to say that sup F = (A, u) means that that if v(I):F(I)~A' formation from F to the constant functor at A such such that the is natural in l, then there exists a unique map a:A~A' ects of 1... following diagram commutes for every l E III = obj 2.
U(I)jA)A'
/v(I) F(I)
~l. ABSOLUTE EPIMORPHISMS
be (1.1) PROPOSITION. Let A be a category and let a:A~A' if and only if there exists a map ~n A. Then a ~s an absolute epi a map a': A '~A such that
a' a A'~A~A' = aa' = A'.
Proof. The sufficiency of the condition is obvious. hom functor Assume that a ~s an absolute epi. Apply the of sets. [A', - l: ~~ 12. to a, where s.. = category
(A', al: (A', Al~[A', A']
onto. Therefore there ~s therefore epi in S. But in ~ epi means Thus aa' = A' . • exists a' E [A', Al such that (A', al (a') = A'.
morphisms of DEFINITION. Let a.: A.~A' be a family of (1.2) ~ ~ = yai for all i, a category A. {ai} is called a joint epi if xai
implies that x = y. epi. In our If sup F = (A, u) then {u(I) II E.11.1} ~s a joint following result. characterization of absolute sups we will need the 3.
(1. 3) PROPOSITION. Let a. :A. __ A' be a family of morphisms 1 1 of a category A. Then lS an absolute joint epi if and only if one of the a'1 lS an absolute epi.
Proof. The sufficiency is obvious.
{a· } Assume that 1 is an absolute joint epi. Apply the functor
[A', - 1 • Then
[A', ai 1: [A', Ai l~[A', A' 1 lS a joint epl ln S. But a f~~ily of maps in ~ is a joint epi if and only if the union of the images is equal to the codomain. Thus for some index j, there exists a'e.[A' , A·l J such that [A', aj l(a')= A'.
That is to say aja' = A'. 1
§ 2 . ABSOLUTE WEAK SUPREMA
(2 - \ DEFINITION. Let F: I~A be a functor. We say that
(A, u) 1, do :eak sup of F if u(I): F(I)--+-A lS natural in l and if for any v(i): F(I)---A' which is natural in I, there exists a map a: A~A' such that
commutes for all I.
Thus we have dropped the uniqueness from the definition of sup. 4.
(2.2) DEFINITION. Two objects A and A' are said to be connected in a category A if there exist finitely many objects
Ao' Al'····' An such that A = Ao, A' = An' and
[Ai_l' Ai] U [Ai' Ai-l] ~ (Il for every 1. = l, 2, ... , n.
Obviously connectedness is an absolute property.
We will find it convenient to use the following special case of
Lawvere's [13] comma category. Let G: A~B be a functor and let
B E I~I, th en (B, G) is the category whose objects are maps b: B--..G(A) and ,.,hose maps are commutative triangles
yG(A} A B~ !G(a} la
G(A ' ) A'
(2.3) FUNDAMENTAL LEMMA. Let F: I~A be a functor, let weI): F(I)~F(Io) be natural 1.n l (Io is fixed), and assume that for every l E ILl, w( I) and F( I) are connected in the comma category
(F(I), F). Then if v(I): F(I)~A is natural 1.n l
Proof. That weI) and F(I) are connected in (F(I), F) means that there exist a finite number of objects of l, Il' I 2 , ... , In' as many maps 1.n ~,di: F(I)--.-F(Ii ), and maps in L, b b l =4( l n+l'- I such that the bottom half of the o following diagram commutes 5.
A
F(I)
But the top half commutes by naturality of v. proceeding from left to right on the diagram we see that v(Io)w(I) = v(I) for every 1 € Ill. •
This lemma tells us that a natural transformation from F into a constant functor is entirely determined by its value at 10.
Furthermore we have a constructive (and as we shall see, absolute) way of getting it.
(2.4) THEOREM. Let 1 be a small category and let F: I~A be a functor. (A, u) 1S an absolute weak sup of F if and only if u(I): F(I)~A is natural in 1, and there exist 1 .a II 1 and o do: A~F(Io) such that for every 1. ill, dou(I) and F(I) are
connected in the comma category (F(I), F). 6.
Proof. Assume that (A, u) lS an absolute weak sup of F. By definition u(I): F(I)~A is natural in 1.
Now we shall construct a category A' containing A as a full subcategory. Let I~' 1 = 1~ + { X} where X lS an arbi trary symbol.
Let [ X, B) = 1/;, [ X, X) = {X}, and [B, X) = (B, F) the A' A' A' comma category. Compositior. lS the obvious one. This gives us our category A' .
We now define a congruence relation on A'. Two maps of
[B, X) are congruent if and only if they are connected in (B, F). A'
Make this relation reflexiveby adding the condition that any map of A' is congruent to itself. This defines a congruence relation on
A' and we can form the quotient category A" (see Mitchell [19), p.4).
The maps of A" are congruence classes of maps of A' and the objects of A" and A' are the same. A is fully embedded in A".
Let w(I): F(I)~X be the map in A" determined by
F(I) e (F(I), F). If b: I---..J ln !., then
F(b) F(I) >F(J) F(~fb)
F(I)
commutes and so F(b) and F(I) are connected ln (F(I), F). This
shows that
F(I)~ F(b)1 X F(J)~ 7.
commutes for every map b of 1. Thus w is natural. A and thus (A, u) Now (A, u) ~s an absolute weak sup of F ~n exists a map do: A~X ~s a weak sup of F in An. Therefore there that do: A~F(Io) such that douCI) = weI) for ail 1. This means to F(I) for every for sorne Ioe III and that douCI) is congruent connected ~n (F( 1), F) for l € I!.I. Therefore do u( 1) and F( 1) are every le 1Il.
a canonically Conversely, if G: A~B ~s any functor, we have being an induced functor G: (F(I), F)~(GF(I), GF). Connectedness (GF(I), GF). property, G"(d u(I)) and G"(F(I)) are connected ~n absolute o But G(dou(I)) = G(do)Gu(I) and G(F(I)) = GF(I). Consider Now assume that v(I): GF(I)~B is natural in 1.
Gdo v(lo) GA > GF(l ) ~ B. (2.3) now says that o v(l )G(d )Gu(l) = v(I). o 0 completes the Therefore (GA, Gu) is a weak sup of GF, and this
proof. •
of connectedness lt is convenient to state these results in terms see the absoluteness by stating ~n comma categories, but it is easier to though it seems more these conditions in the form of equations. Even it is sometimes simpler complicated in the general case, in special cases
and more meaningful. Thus we restate (2.4).
be a functor. Let l be a small category and let F: --l~A if there exist (A, u) is an absolute weak sup of F if and only 8.
10 t !.I and do: A--F( 10 ) such that for all lé 11.1 there exist b l ICi l and d.: F(I)--1-F(I.) such that o ~ ~ dou(I) = F(b1)dl
F(b )d = d = F(b )d 2 1 2 3 3
F(b4 )d3 = d4 = F(b5 )d5
F(b _ )d _ = d _ = F(bn_1)d _ n 2 n 3 n 2 n 1
F(b)dn n- 1= F(I) l. e. such that
F(I)
~ F(I) l commutes.
( 2 • 5 ) REMARK. If d u(I) and F(I) are connected in o (F(I), F) and if b: K~I then dou(K) and F(K) are connected
in (F(K), F). Indeed, b induces a functor 9.
to dou(I)F(b) (F(b), F): (F(I), F)~(F(K), F) which sends dou(I) l )F(b) = dou(K). and F( I) to F( l )F(b) = F(b). By naturali ty dou( F). But Therefore dou(K) and F(b) are connected ~n (F(K),
(F(K), F). commutes, showing that F(b) and F(K) are connected ~n
This proves our statement. cases. (2.5) is actually very useful when working with special
that if F (2.6) REMARK. We see immediately from theorem (2.4)
dou(I)) ~s one. has an absolute weak sup, then (F(Io )'
The next result is some sort of a converse to this.
some weak sup (2.7) THEOREM. Let F: I~A be a functor. If
of F is absolute then they ail are. if (B, v) ~s Proof. This follows immediately fr~m the fact that a weak sup of G a weak sup of some functor G, then (B', v') is B'--...B such that if and only if there exist b: B~B' and b': bv =v' and b'v' = v. •
has absolute (2.8) DEFINITION. In this case we say that F
weak sups. sup, then If F has absolute weak sups and if it has a strong not follow from this sup will be absolute as a Teak sup but it does 1 10.
(2.7) that it will be an absolute sup.
sups. Let (2.9) THEOREM. Let F: l~~ have absol~te weak exist natural transformations G: I~A be a functor such that there and G(I) G ~F~G. Assume that for every 1 E:. Ill, s(I)t(I) weak sups also. are connected in (G(I), G). Then G has absolute
F. Then (2.4) Proof. Let (A, u) be an absolute weak sup of
exist 1 €. III and d: A~F(I) such that for says that there 0-0 0 in (F(I), F). every 1 E.lli ,dou(I) and F(I) are connected induce a For every l, t(I): G(I)-F(I) and s: F-:IO-G
functor
(t(I), s): (F(I), F)~(G(I), G). F), Since douCI) and F(I) are connected ln (F(I), ln (G(I), G). (t(I), s)(dou(I» and (t(I), s)(F(I» are connected = s(Io)dou(I)t(I) But by definition of (t(I), s), (t(I), s)(dou(I»
and ( t ( 1), s)( F ( 1 » = s ( 1 ) F ( 1)t ( 1) = s ( 1 )t ( 1 ) . G) by Since s(I)t(I) and G(I) are connected ln (G(I), are connected ln hypothesis, We see that s(Io)dou(I)t(I) and G(I) and (G(I), G). If we take u' (1) = u(I)t(I): G(I)~A
) we conclude that (A, ut) lS an absolute d'o = s(Io)do : A---G(Io weak sup of G. •
with absolute (2.10) COROLLARY. Let F: I~A be a functor functor. If there exist weak sups and let G: -I~A- be another natural transformations such that 11.
then G has abso1ute weak sups a1so. •
§ 3. ABSOLUTE SUPREMA
if (A, u) ~s Obvious1y (A, u) is an absolute sup if and only joint epi. an abso1ute weak sup and {u( I) 1 le Ill} is an absolute sups. The Thus (1.3) and (2.4) give a characterization of abso1ute f0110wing resu1ts make this more precise.
(A, u) ~s an (3.1) THEOREM. If F: I~A ~s a functor then abso1ute weak sup abso1ute sup of F if and on1y if (A, u) ~s an
of F and {u(I)1 leI II} ~s a joint epi.
Proof. The necessity of the condition is obvious. and {u(I)1 I~ 1 II} Now assume that (A, u) is an absolute weak sup weak sup there exist ~s a joint epi. Since (A, u) ~s an abso1ute l, doue I) and F( I) Io E. 1 1 1 and do: A~F(Io) such that for every
are connected ~n (F(I), F). Then by (2.3)
u(Io)dou(I) = u(I) u(Io)d = A. for all I. Since {u(I)} ~s a joint epi we get o thus proving our Consequent1y, {u( I)} ~s an abso1ute joint epi,
theorem. • I)I lE:I II} an Theorem (3.1) shows that the map which makes {u( theorem (2.4). We abso1ute joint epi can be chosen to be the do of 12.
can now state theorem (3.1) ~n its final forme
(3.2) THEOREM. Let l be a small category and let
F: I~A be a functor. (A, u) ~s an absolute sup of F if and only if u(I): F(I)--.-A is natural ~n 1, and there exist l E: 1 l 1 o - and do: A~F(Io) such that
u(I )d = A o 0 and dou(I) and F(I) are connected in (F(I), F) for every 1_1 Il .•
(3.3) COROLLARY. If F: I~A has absolute weak sups and has a (strong) sup then it is an absolute (strong) sup. • A map a: A __ A is said to be an idempotent if aa = a. It is said to be a split idempotent if it factors
a A >- A \1 A' with a'aii = A'.
(3.4) THEOREM. Let F: I~A have absolute weak sups. If
(A, u) ~s one of the absolute weak sups of F, th en using the same
notation as ~n (2.4), d u(I ) is an idempotent and the following are o 0 equivalent:
(i) dou(Io ) ~s a split idempotent, (ii) F has an absolute sup,
(iii)F has a sup. 13.
Proof. By (2.4), douCI) and F(I) are connected 1n (F(I), F)
for al1 1. By lemma (2.3), (dou(Io))(dou(I)) = douCI) thus
(dou(Io))(dou(Io )) = dou(Io). Thus dou(Io) is an idempotent.
(i)~ (ii). Let dou(Io) = d'oa' such that
d' a' A' -_o-:>-~F(Io) -~~~A' - A'. Define u' (1) = a' doue 1): F( 1 )---'lIPA' .
d'ou'(I) = d'oa'dou(I) = dou(Io)dou(I)
= dou(I)
Thus d' u'(I) 1S connected to F(I) 1n (F(I), F) for every 1. By o theorem (3.2), (A', ut) is an absolute sup of F.
(ii)~ (iii) Obvious.
natural in l, thus there exists a unique d'o: A'~F(Io) such that
commutes. By lemma (2.3), u'(Io)dou(I) = u'(I), consequently
u'(Io)d'ou'(I) = u'(Io)dou(I) = u'(I). Since {u'(I)} is a joint
epi, we get u' (1 ) d '0 = A' . 0 •
§4. ABSOLUTE COEQUALIZERS
Because absolute coequalizers are important in the theory of 14.
we shall investigate triples and their algebras (see [20] , and chapter II) us a chance to these coequalizers in greater detail. This also gives see how the conditions break down in special cases.
a a o (4.1) THEOREM. Al ____~:_Ao~A ~s an absolute weak coequalizer al and a finite number of ~n A if and only if there exist b: A~Ao
aa aao = l
ba = av(l)bl
av ( 2) bl = ay(3)b2 ay(4)b2 = ay( 5 )b3
ay(2n)bn = Ao
where v( i) = 0 or l, n ;. O.
a a o if ~s an absolute coequalizer ~n A if and only Al ___~>_A , 0 -----A al bi: Ao-.-"..A there exist b: A ----,- Ao and a fini te number of maps l
satisfying the same equations as above as well as ab = A.
about Ao' Proof. Remark (2.5) says that we only have to worry an easy exercise The first equation expresses naturality of u. It is above equations. to verify that the conditions of theorem (2.4) give the 15.
1 The second part follows from (3.2).
A o:>-A coequalizer. (4.2) EXAMPLE. o ~s an absolute
(4.3) EXAMPLE. If a: A~A' ~s an absolute epi then there to see that exists b: A' --:;"..A such that ab = A'. It is easy
ba a A :::;:A~A' is an absolute coequalizer. A a a is an If a: A--"-A is an idempotent, then A. :::::A~A A a weak coequa absolute weak coequalizer. Thus A ~ A has absolute A a has a coequalizer. lizers and a splits if and only if A ---;,:~ A
a ao Beck defines [2] A _~:_A ~A to be a (4.4) EXAMPLE. l 0 al
contractible coequalizer if there exist b: A----.-Ao
such that
ab = A
alb = A . o 0
Contractible coequalizers are absolute.
are absolute The natural question arises as to whether there is yeso Consider coequalizers which are not contractible. The answer A' and only if ;. A' which is absolute but ~s contractible if 16.
a is an absolute epi.
is (4.5) DEFINITION. We shall say that an absolute coequalizer which (4.1) is valid. n-contractible if n is the smallest integer for
a A' A ~ A' ~A' ~s O-contractible. ~ a a a o coequalizer and if a ~ If Al > A ~A ~s a contractible o --.,>... 0 al
then it is l-contractible.
in ~ We shall show that there exist n-contractible coequalizers we shall establish for each positive integer n. Before we show this, coequalizers in ~. a graph-theoretical characterization of absolute
f graph by letting Given a pair of set maps X ~Y we define a f 2 be the elements of the vertices be the elements of Y and the arrows of f and f is the X such that f (x) ~f2(x). The coequalizer l 2 l projection set of components of this graph with the natural
f f X~Y ___ Z ---f 2 A path from y to y' ~s a finite number of vertices i there is an such that y = Yo' y' = Yn' and such that for each is said to be n, and the arrow Yi~Yi ... l or Yi ... l-Yi. The length -l, one for each arrow in the ~ is defined to be a sequence of +1 or 17.
f (4.6) THEOREM. If X --1.-Y ~s a pair of set maps, then the r-2 and on1y if there exists a coe~ualizer of f_ and f is abso1ute if .1 2 induced by (f , f ) path type t;,. such that each component of the graph 1 2 every vertex of the on Y has a vertex, ca11ed the centre, such that component. graph has a path of type 6 to the centre of its
f f coe~ualizer, Proof. Assume that X --==- Y ~Z is an abso1ute f 2 g: Z~Y and g.: Y~X such that then by (4.1) there exist ~
gf = f v (l)gl f -f V(2)gl- v(3)g2
f v (4)g2= f V(5)g3
f g = Y. v(2n) n we can shorten the We can assume that v( 2i - 1) .;.. v( 2i ) for if not
chain. For any y ~ Y, If z is a component, define its centre to be g(z). gf(y). The the component containing y ~s f(y) and its centre
above e~uations give
gf (y) = f v ( 1) gl ( Y )
f v ( 2 )gl (y) = f v ( 3) g2 ( Y ) 18.
= y f V(2n)gn(Y) (y) we get a path of 1ength Thus taking the vertices to be f v (2i)gi by v as fo11ows: n from y to gr(y). The path type is determined , n). ~ = (v(2n-2i+1) -v(2n-2i+2) 1 i = 1,2, .•.
for which each Converse1y, assume that there exists a path type g(z) = centre component has a centre. Define g: Z~Y by putting of y is gf(y). of z. Gi ven y ~ y , the centre of the component
By hypothesis we have a path
gi (y) = xi for i = 1,2, .. ,no (~stands for Il ---,>,or ~ "). Define for the f's, Using the path type to choose the appropriate subscripts • we see that we have an abso1ute coequa1izer.
the n ~n the Notice that the 1ength of the path is the srume as
~n S if and characterization, thus a coequa1izer is n-contractib1e
only if the minimum path 1ength is n.
fo X, Let X = Y = {a, 1, 2, ... , n} and Z = {a}. = The graph induced on f (i) = min (i+1, n), f the on1y possible map. 1 y is
a 1 2 3 n-1 n .~.~.~. '~0 1 19.
We take n as the centre and the length of the path is n. Thus
f f X 0: y --:-Z is an n-contracti ble coequali zer. This example f l was suggested by Michael Barr to replace a more complicated one.
We now give an example in S where both f l and f 2 are onto (absolute epi) but where the coequalizer is not absolute. Let
X = y = {O, l}, Z = {Ole f l = X, f 2 (i) = 1 -~. The induced graph is 0....---....1 .~.
Clearly there can be no centre.
We also g~ve the following example to show that there exist n-contractible coequalizers in the categorJ Ab of abelian groups.
Let C2 be the cyclic group of order two, and consider the following diagram:
C n 0 C n 2~C n~{O} 2 f;;'-- 2 where f: (~, a2 , a3' ..• ' an)~(a2' a 3 , •.• , an' 0). This is an n-contractible coequalizer in Ab. The details are left to the reader.
§5 • ABSOLUTE PUSHOUTS 20.
In view of remark (2.5), the conditions of theorem (2.4) applied to
pushouts specialize to:
is an absolute weak pushout if and only if there exist do: A~Ai
(say Al: for simplicity) and a finite number of maps d., d'. such ~ ~ that
a Plal = P2 2
doPl = aldl doP2 = ald\ a d d d 2 l = a2 2 a2d'1 = a2 '2 a d a d a d'2 l 2 = l 3 l = al d'3
a d' = A 2 m 2
This diagram is an absolute pushout if do satisfies the further
relation
The first column of equations may reduce to doPl = Al' but the
second column never reduces this much. We conclude from a2d'm = ~ 21.
e We also see that either is an ~s an absolute epi. that a 2 ~ mono. absolute epi or Pl is an absolute an absolute In the case of absolute pushouts '·le see that a 2 is
ep~ or is an isomorphism. epi and either al ~s an absolute Pl
( 5.1) EXAMPLE •
~s an absolute epi. ~s an absolute pushout if and only if a
( 5 •2 ) EXAMPLE •
is an absolute pushout f'or· all a.
~s just the pushout where The cokernel pa~r
a Pl ~s an absolute As we saw above, if ~O_Al--,.:'_A P2 We conclude that cokernel pair, then a must be an absolute ep~. 22.
pairs are the Pl = P2 = isomorphism. Thus the only absolute kernel trivial ones.
of We finish this section with the following characterization the absalute pullbacks in S.
and only (5.3) PROPOSITION. A pullback in S ~s absolute if
if it is of type (5.2) or a non-empty intersection.
The proof is straight forward and will be left out.
§6. MISCELLANEOUS RESULTS
absolute is e~uiva In ~, the fact that all epimorphisms are all mono lent to the axiom of choice. It is a trivial fact that
morphisms with non-empty domains are absolute. = sup of the Obviously there are no absolute initial abjects
empty functor).
= + ~2 where If l ~s disconnected, in the sense that ~ 1i have an absolute sup. Il 1: C/J 1: 12, then no functor I~A can of 1. Indeed, 10 1n (2.4) must be connected to every object are the It follows from this that the only absolute coproducts
l-fold coproducts, i.e. when ~ has one object.
[9]. Now we give an alternate proof of a result of Lambek 23.
and let l be (6.1) PROPOSITION. Let F: I~A be a functor u(I) : F(I)~A, a connected small category. Assume that there exists l, then natural in l such that u( I) is an isomorphism for every ~ (A, u) = sup F and this sup is absolute. u( I). v(I) is Proof. Let v(I): A__ F(I) be the ~nverse of I. Let l and natural in I. Let weI): F(I)~AI be natural 1n
Il be any two objects of I. There exist
b l II( n+l Il since l 1S connected. n w: The following diagram commutes by naturality of v and
AI A
i F( Il)
Thus w(I)v(I) = w(II)v(II) for any l, Il E.III. But making the w(I)v(I)u(I) = weI), thus w(I)v(I) 1S the unique map
following diagram commute: . IA~(I) u(I) AI
F(I) ~ 24.
Therefore (A, u) = sup F are But that u is an isomorphism and that l ~s connected of F. 1 absolute properties, thus (A, u) is an absolute sup
one object (6.2) COROLLARY. If A ~s a group considered as a are absolute.1 category, all sups of functors from a connected category
one element Lambek proved that if A ~s a group with more than
then a functor from a disconnected category has no sup.
§7. REMARKS
enough In all of these absoluteness theorems it would have been embeddings. to require that the given property be preserved by full
Thus these are really results of embedding absoluteness. sups it For the characterization of absolute epis and absolute be preserved would have been enough to demand that these properties epis and sups by representable functors. For this one must know what be used to characterize are ~n S. This process, however, cannot by repre the dual concepts of mono and inf which are always preserved and inf are defined. See sentable ~ŒOctors. In fact this is how mono
(20) for an example of this method applied to coequalizers. 25.
CHAPTER II
TRIPLES AND ABSOLUTENESS
and In this chapter we examine the relationship between triples absoluteness properties. of §l, The reader is assumed to be familiar with the material the leader is which is included for completeness. For more details referred to Manes' thesis [18]. properties in con- ~2 studies the occurrence of absoluteness nection with triples and their algebras. replacing In §3 we reformulate Beck's tripleableness theorem, absolute his original contractible ccequalizers by the more natural a more coequalizers. Although absolute coequalizers constitute are just general class coequalizers, in practice the conditions in as easy to verifY. The idea 1S not to use the characterization from these terms of equations, given in chapter 1, but to get away by all equations entirely, using only absoluteness (preservation than computa- functors). This way the proofs are conceptual rather
tional, and thus clearer. advantage, In the VTT, absolute coequalizers do give us a slight making it making the conditions weaker. New conditions are given
eas1er to prove VTT in some cases. how these §4 is devoted to some known examples, just to show
conditions can be used. 26.
§1. TRIPLES
A (1.1) DEFINITION. Let A be a category. A triple on is a functor and ~s a three-triple (T, n, \.1) where T: A~A 2 such that n: A-+- T and \.1: T --;;-T are natural transformations Tn T\.I 2 2 T3 ,.. T T :> T
nTl \;l "Tl 2 l" 2 +T T »T T p )J the multiplication of commute. n and )J are called the unit and and the triple. The above diagrams express that )J is associative that n ~s unitary.
U pair of functors, ~.e. we have Let _--E--_B~A be an adjoint F such that natural transformations ~: FU~ and n: A~UF
U nu>o UFU~U = U
F Fn.FUF~F = F.
n, U~F). This adjoint pair induces (see Huber [8]) a triple (UF,
form Eilenberg-Moore [6] showed that every triple was of this togethe.r by constructing the category of algebras over a triple,
with a canonical underlying - free adjoint pair.
Let 'li' = (T, n, )J) be a triple on A.
(1.2) DEFINITION. A 1r-algebra ~s a pair (A, a) where 27.
a: TA~A l.S an ! morphism such that
Ta 2 nA T A ------....- TA A ------?l>_ TA "1 la la TA ------~~~ A A a commute. a l.S called the structure of (A, a) and the above dia- grams assert that a l.S associative and unitary.
(1.3) DEFINITION. A ~-homomorphism f: (A, â)~(AI, al) l.S an A - morphism f: A~A 1 such that
a TA ------~>__ A T'1 l, TA'------~~__ A' al commutes, i.e. f preserves structure.
We denote the category of 1r-algebras and 1r-homomorphisms by
We have the canonical underlying functor UT: ~ ~A defined by U1r(A, a) = A and ~(f) = f. This functor has a left adjoint
FT: A~A1r which is defined by FT(A) = (TA, ~A) and P(f) = Tf.
The adjunctions are 2f': FTUI' _AT where ~(A, a) = a and nT: A~ U'li'F'1r where nT = n. The adjoint pair r--; UT induces the triple 1r = (T, n, ~); the one with which we started. 28.
U B On the other hand, if we start wi th an adjoint pair =!, F
form F ~ U, form the triple T = (UF, n, UEF) and then UT original category B and -AT --+-FT -A, we do not in general recover the "comparison" functors F and U. We have, however, a canonical
functor ~
A
~(g) Ug. ~ is the unique ~ is defined by ~(B) = (UB, UEB) and =
functor satisfying the fOllowing relations:
U'1rg'"Jr ~ = UE
FT = ~F.
(1.4) DEFINITION. We say that a functor U: B~A ~s defined com- tripleable if U has a left adjoint and if the above
parison functor ~ is an equivalence of categories.
§2. TRIPLES AND ABSOLUTENESS
be a Let '1' = fT, n, jJ) be a triple on A and let (A, a) 29.
11' -algebra. By associativity of a, Tn+1a Tn+2A ~ Tn+1A
Tn"Al Tna (2.1)
Tn+1A ~ TnAI Tna
o be commutes for al1 n~O (T J.s defined to ~) . diagram commutes The ~-associative 1aw shows that the fo11owing
for a11 n ;. 1: Tn~A Tn +2 A ------..,>_ Tn+1A Tn-l"TA~ lTO-1"A (2.2)
Tn+1A------~--__~.-TnA Tn-1~A
By naturality of ~ we get that
Tn~A n 1 Tn +2 A ------~:.- T + A Tn-i"TiAl ITO-i"Ti-1A
Tn+1A ___~ __~~_ TnA n - 1 T ~A
commutes for n ~ i ;;!l: 2.
Tia and (2.4) LEMMA. Al1 maps TnA~A built up from
Tj ~~A are equal.
Proof. For n = l, there J.S on1y TA~A. 30.
For n = 2, the only possibilities are T2A Ta~ TA~A and T2A \JA.. TA~A which are equal by (2.1).
Assume that we have proved the result for n + l where n ~ 1.
We shall prove it for n + 2. Call the unique map TiA~A, a·l. for (ao = A). Consider
There are four possibilities for x:
(i) Then since
(ii) x n = T \.lA. Then an+l.x = a n+ 1.Tn\.lA. n-l (iii )x = Tn-1\.lTA. Then since ~+l = ~.T \.lA, n l n l an+l·x = ~. T - \.1.A T - \.1 TA which, by (2.2) is equal to n-l n n A ~.T \.IA.T \.lA = ~+l.T \.1 • Then
which by (2.3) is equal to an.Tn-i\.lTi-l.Tn\.lA = ~+l.Tn\.lA.
This proves the lemma. •
De fine ~ T and
\.1 T: T~T. 0 =
n+l ~n-l. LEMMA. Let c. Ta· .\.1. l'r A for a rixed n l. = n-l. l.-
and 1 ~ i ~ n. Then the following diagram commutes: 31.
c. ~ T2nA >- Tn+1A
Oi+11 1"~-lA
Tn+1A ~ TnA Tna ---Proof. Consider the diagram: 2n i Tn+l~ . ~. T - A -~ ~-l T2n-iT1A Tn+1A T2nA )0 ..
(i) commutes by naturality of ~;
(ii) commutes by definition of ~i; diagram is (iii)commutes by lemma (2.4). Commutativity of the outer • what we had to prove.
n Tn Ti-ln . We are now in Define n. = A~T ---"T2 ~ ••• ---'-T~. ~ - a position to prove the main theorem of this section. 32.
(2.6) THEOREM. If '"Ir = (T, n, lJ) J.S a triple on A and
(A, a) is a 1r-algebra, then
J.s an absolute coequalizer in A for any n ~l.
Proof. Lemma (2.4) shows that a coequalizes Tna and n n l n lJT - A. Let do = nn (A), do: A -+oT A. Us ing the fact that 2 n-l ~ = a.Ta.T a ••• T a we show that
nnA A ------>-~ TnA lan
A commutes. Now
>A
c01llIilutes by naturality of nn. Thus n n = T ~.nnT A n Tn+l TnA = T a. ~-l.nn 33.
(by (2.5))
n-l n n n A.c .n T A = T a.c .n T A ~T n-l n n n ~Tn-lA.Cn.nnTnA = ~Tn-lA.Tn~lao.~n_1TnA.nnTnA n-l n n lT A.n T A = ~T A.~ n- n = ~ Tn-lA.n TnA n n = (~ .n T)(Tn-lA) n n = TilA
This completes the proof of the theorem. •
reference Of course this 1S not an elegant proof~ but no further with it. will be made to the proof nor to the definitions connected has been stored The necessary information that the equations contain theorem in the fact that we have absolute coequalizers (the next (including testifies to this). From now on we forget these equations and work solely the ones used in defining contractible coequalizers) proved using only with absoluteness. We see how Beck's theorem can be
b~ applied. absoluteness and in the last section we see how it can
n-contrac- Note that generally~
~or some t1ble, but in some cases it may reduce to i-contractible
i.< n. In any case it is a good candiJate for an n-contractible
coequEllizer. 34.
and let (2.7) THEOREM. Let 'lI' = (T, n, lJ) be a triple on A 2 i-l A-morphism. Let a. = a.Ta.T a .•• T a. If for a: TA----A be an 1. Tna l absolute coequalizer, some n ~ l, r-+ A ==~~:: is an lJTn-IA then (A, a) 1.S a 1r-algebra.
n a .lJTn-1A but an = a.T~_l and Proof. We have an.T a = n
T~-l TnA ~ TA
"Tn-1AI i"A Tn+1A ~ T2A 2 T a n-l
commutes by naturality of lJ, therefore n n~ ~l an.T a = an.lJT A = a.Tan_l.lJT A 2 = a.~A.T a n- 1 But 2 = a.Ta.T a n-1
thus
so 2 n+l 2 M·l a 1.T a = a.Ta.T ~_l.T a a.lJA.T n- 2 a.lJA.T2~ = a.Ta.T an.
But ~ is an absolute epi, therefore (The a-associative law).
Now a.nA.a = a.Ta.nTA = a.lJA.nTA = a. 35.
a is epi Thus a.nA.~ = an and since n a.nA = A ( The a-uni tary law).
This completes the proof. 1
category of If A and B are categories we let ~ denote the as morphisms. If functors A~B with natural transformations G~: It.-.~ G: B~C is a functor we have an induced functor transformation defined by composition. If t: G~H is a natural also then we have an induced natural transformation tA: ~ ~H~ functors defined by composition. ()A sends categories to categories, and to functors, natural transformations to natural transformations,
respects all kinds of composition and identities.
Let 1l' = (T, n, ).l) be a triple on A. We have an induced A triple TA ::: (~, on A-. Since (T,).l) i3 a ~-algebra
we get the following theorem. on ~, then (2.8) THEOREM. If 1r= (T, n,).l) ~s a triple
Tn).l ).ln n 2 n l in ~ for all T + ==::;:-: T + ~T ~s an absolute coequalizer ).lT rt-
n ~ O. •
From (2.8) we see that Tn+i).lTj . " Ti Tj .. Tn-t2+i ~j =:;.~=::::~: Tn+l+~+ J __).l_n_~.,..~ T~+J +1 T~).lTn+J
all possibilities is an absolute coequalizer in AA. This takes care of powers for pairs of maps of the type TP).lTq between two consecutive
of T. If we apply the substitution functor that
us that is an absolute coequalizer in A. Theorem (2.6) tells Tn+ia Tia . Tn+l+i (A) _'"""':"'""_--::-_~~_ Tn-ti (A) _--,n=!,.~ Tl. (A) TijJTn-l(A)
a 1r-algebra. l.S an absolute coequalizer in A, where (A, a) is
This takes care of all possibilities of pairs of maps
§3. TRIPLEABLENESS
Let U: B~A be a functor. (3.1) DEFINITION. We say that B has U-absolute sUEs if sup in A, every functor G: I---B such that UG has an absolute
has a sup (not necessarily absolute) l.n B. (3.2) DEFINITION. We say that U Ereserves U-absolute sUEs UG has an if whenever a functor G: !.~B has a sup in Band sup G absolute sup in A, then U.sup G ;!! sup UG, i.e. if should exist an iso- c (B, v) and sup UG = (A, u) then there
morphism a: UB~A such that a.Uv = u.
sUEs (3.3) DEFINITION. We say that U reflects U-absolute that if for every v(I): G(I)~B, natural in 1, such v) - sup G (not (UB, Uv) ~sup UG which is absolute, then (B, 37.
necessarily absolute).
on A. Then (3.4) THEOREM. Let T= (T, n, \.1) be a triple sups, and Arr has U'lt'-absolute sups, UT preserves U'll'-absolute
ur creates UT-absolute sups.
= H and Proof. Let G: l ~ AT be a functor. Let UTG that H has an ~ETG = h, then G(I). (H(I), h(I)). Assume we see that absolute sup in A, (A, u) = sup H. Applying T A is natural sup TH = (TA, Tu). S~nce TH(I) h(I)~ H(I) u(I)~ A such that ~n l, there exists a unique a: TA __
a TA >= A Tu(,)l fu(,)
TH (1) ------...- H(1) h(I)
commutes. the We shall show that (A, a) ~s a 1r-algebra. Consider
following diagram: a .. TA ~ A ~ 'Tu(I) rU(I) Th(I) > TH( 1) ~ H(I) >- \.IH(I) h(I) a, the upper The square on the right commutes by definition of square on the square on the left for the same reason, and the lower 38.
left commutes by naturality of ~. Therefore 2 a.Ta.T u( r) sup, {uer) Ir E.lr I} for every r E Ir 1. Since (A, u) ~s an absolute
a.~A. ~s an absolute joint epi, therefore a.Ta =
Next consider the following diagram: TA
nA/ ~a A/ Tu(r) ~A
u( r) uer) 1 TH(r) 7·~ H(r) » H(r) H(r)
left by naturality The right square commutes by definition of a, the axiom. Therefore of n, the bottom triangle by the h(r) - unitary {uer)} ~s a we have a.nA.u(r) = u(r) for every rf:III. Since proof that (A, a) joint epi, we see that a.nA = A. This finishes the
is a 1['-algebra. follows That uer): (H(r), h(r))~(A, a) is a T-homomorphism in r follows from from the definition of a. That u(r) is natural
the fact that it is natural ~n A. in r. Then Now let ver): (H(r), h(r))~(A', a') be natural Since applying uT, ver): H(r)~A' is natural in 1. f: A~A' such (A, u) = sup H, there exists a unique A-morphism
that f A-----~A'
commutes. following dia- To see that f 1S a 1r-homomorphism consider the gram:
f A ~ A' ~ A:; H(l)
a a' h(r)
Tf TA ... TA' ~ Âl) TH(l)
f.a.Tu(l) = f.u(r).h(r) = v(r)h(l) = a'.Tv(r)
= a'. Tf. Tu( r ) Thus f But since {uer)} is an absolute joint epi, f.a = a'.Tf. (A, a), u) = sup G. 1S a 1r-homomorphism and we conclude that
Thus AT has UT-absolute sups. 1r rt is obvious that U preserves these sups. that Since the a was uniquely determined by the requirement 1r reflects these uer) be a homomorphism it is also evident that U • sups.
We now state Beck's tripleableness theorem. 40.
only (3.5) THEOREM. A functor U: B~A is tripleable if and and U if U has a left adjoint, B has U-absolute coequalizers, preserves and reflects U-absolute coequalizers.
B A~ and Proof. If U is tripleable, we can assume that = rest follows from that U = U~. Then U has a left adjoint and the (3.4).
Now assume that U has a left adjoint F with adjunctions has U-absolute coequalizers E:: FU ---.B and n: A~UF, and the.t B the inverse of and U preserves ~~d reflects them. We obtain i,
~ as the following coequalizer
n - FT A~~ (A, a)
is any'1r-algebra. where n is a fixed integer (n ~O) and (A, a) pair of maps, we This coequalizer exists since, applying U to the
get
which is an absolute coequalizer by (2.6). equivalences The details which show that ~ and ~ are inverse are left out. • are standard (see Beck [1], [2] or Manes [18]) and
In Beck's original statement of the theorem, contractible absolute coequalizers. coequalizers (see (l, 4.4)) were used instead of
v U U and Consider the following situation C~B --..A where 41.
tripleable (e.g. V are tripleable. It does not follow that UV ~s
Torsion free abelian groups~Ab~§).
U, ensure Beck has conditions (VTT) which, when inr,i)osed on replacing that UV is tripleable. We now give these conditions
contractible coequalizers by absolute ones. that the (3.6) VTT. Let U: B--A be tripleable and assume and reflects U-absolute coequalizers that B has and U preserves V: f. ---B, UV are themselves absolute. Then for any tripleable 1 is tripleable.
for it is This theorem ~s not equivalent to Beck's VTT, pairs be quite conceivable that coequalizers of U-contractible
absolute but not contractible.
to use in We now prove a sort of VTT which seems to be eas~er
practice.
theorem.) (3.7) THEOREM. (RVTT = relatively vulgar tripleableness B has Let C ~B ~A be such that U has a left adjoint, and there exist split idempotents, there exists a functor G: A~B
natural transformations sand t such that V~GUV.....t--V = V through V, then for any tripleable ~~ctor W: D~B factoring 42.
W D ------,,..-B \) c uw ~s tripleable.
VT = VT, Proof. since V~ GUV ~ V V, VT sT'>a GUVT -..-t1'- thus vi sT... GUW tT~ W = W. in A. Then Let H: 1~D be such that UWH has an absolute = WH by GUWH has an absolute sup in B. Since WH~Gm1H~WH split idempotents (1, 2.10) WH has absolute weak sups. (1, 3.14) and
~s tripleable, imply that WH has an absolute sup in B. Since W sup (since it is W-absolute) H has a sup ~n D. W preserves this
and then UW obviously preserves it. (D, u) To see that UW reflects UW-absolute sups let sup H = in 1, be UW-absolute. Assume that u'(1): H(1)~D' is natural
th en there exists a unique map d: D~D' such that
and let its commutes. Assume that UWd is an isomorphism ~n A
inverse be a: UWD'~UWD. 43.
Wd WD .. WD'
STDj !STD' GUWd GUWD .. GUWD'
lGa GUWd GUWD
ltTD ltw.
WD ::!II WD' Wd commutes, the left half showing that
(tTD.Ga.sTD').Wd = tTD.sTD = WD and the right half showing that
Wd. (tTD.Ga.sTD') = tTD' .sTD' = WD' . that U reflected Therefore Wd ~s an isomorphism. This shows W-absolute sups) this sup. Since W is tripleable (thus it reflects
in ~, W also and since this sup which U reflected is absolute sups. Thus reflects it, thus showing that UW reflects UW-absolute 1 UW is tripleable.
U has a (3.8) COROLLARY. Let C ~B ~A be such that there left adjoint, B has split idempotents, V is tripleable, transformations exists a fUnctor G: A~B and there exist natural
s and t such that 44.
V~GUV~V = V.
Then uv ~s tripleable.
and (3.9) COROLLARY. Let U: B--..A have a left adjoint that assume that B has split idempotents. Assume, furthermore, natural trans- there exists a fUnctor G: A-+-B and there exist
formations s and t such that
Then U is VTT.
§4. EXAMPLES
of sets. The category of groups is tripleable over the category It is weIl known Let U: Gr~~ be the usual underlying functor.
f o be two groups and two U has a left adjoint. Let X -.....::"",..... X that I --~... 0 fI
group homomorphisms and assume that
f f X 0 ... X----X l ~ 0 fI a group structure is an absolute coequalizer in sets. We must define the coequalizer on X in such a way that f is a homomorphism and
of fo and fI in Gr.
Consider the diagram: 45.
(f, f) .. XxX
1 1 :m ;::r .1 f Y ~ x .. x »- 0
respec- where m and are the multiplications of x and o o because tively. The upper and lower squares on the left commute Thus f.mo·(f ' fo) = f.mo·(fl , f ). fo and f l are homomorphisms. o l is the result But the upper row is a coequalizer diagram because it diagram. of applying the squaring functor to an absolute coequalizer the square on the Therefore there exists a unique m: XXX~X making right commute.
Next we use the same reasoning on the following diagram f f c ~ X ~ X f l 1. I~ 1 i1r :{ 1 f f 'W' Cl .. X , X Xl ".. f 0 l
of X and to get a unique where ~ and are the inverses o ~l o commutes. ~: X~X such that the right hand square
The following diagram gives us the unit of the group: l , l >=1 1 1 1 lu 1 uol 1 ~I fo f ~ .. X .... X Xl ;;am 0 f l 46.
on X. One easily verifies that this defines a group structure that f This structure was entirely determined by the requirement
f f be a homomorphisme Furthermore Xl-~o~;--,..X ~X is a coequalizer f 0 l (3.4). in Gr. The details are very similar to those of theorem
This sketches the proof that U: Gr~S is tripleable. [15]). This example is typical of varietal categories (see Linton over S. Indeed varietal categories are known to be tripleable
Now let B be a full reflective subcategory of A and the left adjoint of U we U: B~A the inclusion. Then if F is idempotents, then have FU ~ B. If we assume that ~ has split (3.9) implies that U is VTT.
A refinement of (3.7) and thus of (3.8) and (3.9), specifying inclusion of which idempotents must split, gives the result that the
any full reflective subcategory is VTT.
category Let X be a dis crete category and let A be a pointed idempotents. having X indexed products. Assume that A has split pair AX~A where U is the product and F We have an adjoint -~- F of A the is const, the functor which associates to any object
constant functor vith that value.
We have natural transformations
( nA. ) is defined by s(A.).: A.--nA. vhich where s (A. ): (A.) -.- ~ J J ~ ~ ~ ~ ft ~s determined by
if ~ = J A.~A. ~ J ={; if ~ :F J •
Then t.s = A!::. thus u ~s t ~s defined by the projection maps.
VTI'.
U B has split Let B~~ be an adjoint pa~r and assume that F says idempotents. Then one of the adjunction equations F Fa,. FUF E.F ~ F = F.
~s a tripleable Thus U is RVTT through F, Le. if W: D--B
functor which factors through F, then UW is tripleable.
U B has split Let B ~A be an adjoint pair and assume that F R. Then idempotents and assume also that U has a right adjoint
U ~s RVTT through R. 48.
CHAPTER III
ADDITIVE ABSOLUTENESS
i.e. In this chapter we propose to study additive absoluteness, are preserved by properties of diagrams in additive categories which of some of the all additive functors. We obtain characterizations take usual properties of category theory. These characterizations subtraction, the forro of equations involving composition, addition,
identities, and zero. A such that By an additive category we mean a pointed category the usual the hom functor , ] : A op x A _ S factors through to sets. underlying fUnctor from the category of abelian groups
~s called an A functor G: A~A' between two additive categories
additive functcr if the induced maps
are abelian group homomorphisms. algebra, Additive absoluteness has been used often in homological exact sequences e.g. split short exact sequences are absolute shclrt acyclic. We and for chain complexes, contractible means absolutely work. leave all questions of homological algebra for a later in this It is understood that when we speak of absoluteness
chapte~ we mean additive absoluteness. 49.
§ 1. ABSOLUTE EPIMORPHISMS
and let (1.1) PROPOSITION. Let ! be an additive category is an absolute epi if and a: A~A' be an ! morphisme Then a - A'. only if there exists a map a': A'-A such that aa'
Proof. The sUfficiency is obvious.
Now assume that a is an absolute epi. Then (A', a) is (A', a): (A', A)_(A', A') is an epi l.n Ab. Thus aa' = A'. • onto and therefore there exists a' E. (A', A) su ch that
and let (1.2) PROPOSITION. Let A be an additive category an absolute A._ A' be a family of maps l.n A. Then {a.} is a·:l. l. l. maps a'.: A'_A., on1y joint epi if and only if there exist l. l. finitely many non-zero, such that a' i A'. l:ai =
l:a. a' . = A' and assume that xa. = ya. Proof. Assume that l. l. l. l. =ya. a' . and Exa.a' . = l:ya.a'. but for all i. Then xaiai l. l. l. l. l. l. x = y and {a. } l.S a joint epi. l:xaia ' i = x and l:ya·l. a'· l. = y thus l. additive However the fact that l:aia'i = A' is preserved by all
functors, thus {ai} is an absolute joint epi. an absolute joint epi. Apply the Now assume that {a.}l. is a joint epi in functor (A', -). (A', ai): (A', Ai)~(A', A') is map Ab. But this is a joint epi if and only if the induced Therefere there J.gA', ~)-(A', A') is epl. in Ab, i.e.onte. 50.
AI, i. e. there exists an e1ement of lkA l, Ai) which is sent onto such that exist al.: A'~A., on1y finite1y many non-zero, 1 1 îa. al. = AI. • 1 1
92. ABSOLUTE WEAK SUPREMA
(not Let A be an additive category, l a sma11 category necessari1y additive), and F: I~A a funcLor. J) with rows (2.1) DEFINITION. An F-matrix lS a matrix (al , by the objects of l such that a F(J)~F(I) and columns indexed l, J th row and the J th co1umn, and where al , J is the entry in the I such that on1y finite1y many rows are non-zero.
__ b: J-I be a map ln I. Define (2.2) DEFINITION. Let o 0 are zero except M(b) to be the F-matrix (al , J) where a11 al, J = is understood that if al F(b) and a -F(J ). It 0, Jo Jo, Jo 0 = F(b) F(J ). l = J then a 0 0 0 al 0' Jo Jo' Jo
matrix (2.3) DEFINITION. Define a D-co1umn to be a co1~~ th entry is an A- indexed by the objects of l such that the I being zero. morphism D~F(I), a11 except finite1y many
F-matrices Note that the domains and codomains of the maps in can compose F-matrices and D-c~lumns are so arranged that we D-co1umn. The together and we can compose an F-matrix wi th a al1 sums make finiteness conditions in (2.l) and (2.3) insure that
sense. 51.
whose (2.4) DEFINITION. Let E(I) denote the F(I)-co1umn zero. Ith entry is F(I) and whose other entries are
and (2.4) lS The termino1ogy of definitions (2.1), (2.2), (2.3), not standard.
objects of (2.5) LEMMA.. Let [v( 1)] be a row matrix indexedby
th F( 1) --->- A for a l where the I entry lS an A-morphism v( 1): on1y if [v(I)] M(b) - fixed A. Then v(I) lS natura1 in l if and °
for a11 maps b of 1.
Proof. Let b: J ~I , then o 0 v(J )' 0, ... ) which is [0] if [v(I)]M(b) = (0,0, ... , O,v(Io)F(b) - o if and on1y if the fo11owing a.nd on1y if v(Io)F(b) = v(Jo )' l.e.
diagram commutes
The 1emma is now obvious.
chapter. He can no", state and prove the maJ.n theorem of this and l a smal1 (2.6) THEOREt.1. Let A be an additive category
I-,-A lS a fu..Ylctor, (not necessari1y additive) category. If F: if and on1y if then (A, u) lS an abso1ute weak sup of F an A-co1uron (al) u(I ) : F(I)-rA is natura1 in l and there exists 52.
number of I- such that for each J €: III there exist a fini te B , B ,···, Bn morphisms b , b , •.. , b and as many F(J)-columns l 2 l 2 n such that
Proof. Assume that there exists an A-column (al)' are satisfied. Let al: A~F(I), such that the above conditions a map f: A~A' such v(I): F(I)~A' be natural ~n I. We want
that fu(I) = v(I) for all I. indexed Define f = fV(I)aI and let [v(I)] be the row matrix,
by objects of l, whose Ith entry is v(I). n (aI)u(J) = E(J) T.E M(b. )B. ~=l ~ ~
[v(I)](aI)u(J) = [v(I)]E(J) +i!l [v(I)]M(bi)Bi
that (A, u) ~s a Therefore fu(J)= v(J) for all J. This proves involving weak sup of F. But the above conditions, being e~uations zero, are preserved composition, addition, subtraction, identities, and weak sup of by all additive functors. Thus (A, u) is an absolute F. F. Assume that (A, u) ~s an absolute weak sup of contains A We shall construct an additive category A' which is an arbitrary as a full subcategory. lA' 1 = 1 AI + {X} where X C ( lAI, symbol. (X, X)A' = Z, the ring of integers. For Ab). (X, C) i:J. = {O} and (C, X)~ = ll(C, F(I))A' (the coprC'duct in - I - Composition in Thus a map C ~X can be thought of as a C-column. a scalar. As we A' is just ordinary multiplication of a matrix by A as a full have defined it A' is an additive category containing 53.
subcategor.{ .
Next we define an additive congruence relation, - on the hom f = f. sets of A'. (1) For all A'-morphisms f, we have
, b , ... , b , and as many C-columns a finite number of maps of l, b l 2 n (a - b )= f M(b. )B .. r r i =1 1 1 and This relation is reflexive, symmetric, transitive, additive, on A'. !!!1J_lt.i_plicative, i.e. it is an additive congruence relation additive and Form the quotient category A'/=. This category is maps contains A as a full subcategory. We have the following by any one of its E(r): F(r)~X (we represent a congruence class elements) .
Now E(r) lS natural ln r, for if b: J-----r, then ln A'. There- E(r)F(b) - E(J) = M(b)E(J) and so E(J)F(b) = E(r)
fore in A'/= F(II)~
F (b) X
F (J) ~
commutes.
is also a Now Slnce (A, u) is an absolute weak sup in A it ): A_Xl such weak sup in A'/=. Therefore there exists a map (ar
that 54.
A -x U(~ IJJ \ / F(J) commutes for all J. Thus (aI)u(J) = E(J) 1.n !l'l,=:, 1..e.
(aI)u(J) '=: E(J) in A' for every J. Therefore, for every
JE III there exists a finite r-umber of b , b ,···, b , ~-morphisms l 2 n and as many F(J)-columns Bl' B2 , ... , Bn' such that n (aI)u(J) = E(J) +i~l M(bi)Bi · •
(2.7) REMARK. If the condition of theorem (2.6) is satisfied for J and if b: K~J, then this condition is also satisfied for
K. n Indeed, if (aI)u(J) = E(J) +.E M(b1.. )Bi then 1.=1 n (aI)u(J)F(b) = E(J)F(b) +.E M(b1.. )B.F(b). By naturality u(J)F(b) = u(K). 1.=1 l. Also BiF(b) is an F(K)-column for all 1.. Finally, note that
E(J)F(b) = E(K) + M(b)E(K), thus n (aI)u(K) = E(K) + M(b)E(K) +.E M(b. )B.F(b). 1.= 1 l. l. This proves our claim.
(2.8) REMARK. If one weak sup of F is (additively) absolute, then all weak sups of F are. In this case we say that F has absolute weak sups.
If F: I_A is a functor from a small category l to an 55.
F has a totally additive category A then it is obvious that if absolute. We now absolute weak sup then this weak sup is additively the conditions of show how the conditions of theorem (1, 2.4) imply
theorem (2.6).
Let x and y be two maps of A such that
E(J)x E(I)y ~n commutes. Then M(b)E(J)x = E(I)y - E(J)x thus = and A' (defined in (2.6)). Thel'efore if x: F(J)->-F(I) then E(I)x = E(K)z. z: F(J)~F(K) are connected in (F(J), F)
2.4) d u(I) and F(J) are connected in Since by theorem (1, o
(F(J), F) for all J, we see that
therefore n E(I ) d u(J) == E(J) +1L M(b. )B. o 0 =1 ~ ~ additively some b and Bi. This shows that (A, u) ~s an for i absolute weak sup.
sups. Let (2.9) THEOREM. Let F: I--A have absolute weak natural transformations G: I~A be a functor such that there are exist I-morphisms G---1....F---..l3...... G. Assume that for every 1, there
G(I)-columns B'l' B'2' •.• ' B'n such that b , b 2 , ..• , b n , and 1 n s ( l )t ( l ) E ' (I ) = E ' (1) +- • L M' (bl.. ) B '.; ~=l ... ,.rhere the prlme;~ inrEcate that [;he mat~J.ces are riefinerl ~.Tith~'espect to G. Under these conditions G has abso1ute weak sups a1so.
Proof. Let (A, u) be a weak sup of F. Now assume that ver): G(r)~A' is natura1 1n r. Then we have
F(r) s(r» G(r) v(r)~ A' thus there exists f: A~A' such that
f A ------:> A' ~) uer) r G(r) F(r) ~ commutes. Then fu(r)t(r) = v(r)s(r)t(r). But n v(r)s(r)t(r)E'(r) = v(r)E'(r) + l: v(r)M'(b. )B'. i=l 1 1 = v(r)E'(r).
Therefore v(r)s(r)t(r) = ver) for a11 r, thus fu(r)t(r) = ver).
We see that (A, u.t) is a weak sup of G. Since F has abso1ute weak sups and the hypotheses are preserved by additive functors, then
(A, u.t) is an absolute weak sup of G. 1
(2.10) COROLLARY. rf F: r~A has abso1ute weak sups and if
there exist natural transformations sand t such that
then G has abso1ute weak sups also. • 57.
§3. ABSOLUTE SUPREMA
Let l be a small category and A an additive category. If
F: I-A is a functor then (A, u) ~s an absolute sup of F if and only if (A, u) ~s an absolute weak sup of F (see (2.6)) and
{U(I)\I E I!.I} is an absolute joint epi (see (1.2)). The following results tell us even more.
(3.1) THEOREM. Let F: I~A be a functor from a small category to an additive one. Then (A, u) ~s an absolute sup of
F if and only if (A, u) ~s an absolute weak sup of F and
{u(I) Ile Ill} is a joint epi.
Proof. The necessity of the condition is obvious.
Now assume that (A, u) is an absolute weak sup and
{u(I)1 l E II b is a joint epi. By theorem (2.6) there exists an
A-column (al) such that for each J~! Il there exist a finite number of I-morphisms b , b , ... , b , and as many F(J)-columns l 2 n n B , B , ••• , Bn such that (aI)u(J) = E(J) + L: M(b. )B.. Thus l 2 i=l ~ ~ n [u(I) ](aI)u(J) = [u(I) ]E(J) + L: [u(I) ]M(bi)B .. Since u(I) ~s i=l ~ natural, by lemma (2.5) [u(I) ]M(b.) = [0] thus we have ~ (f u(I)aI)u(J) = u(J)
But since {u(J)} ~s a joint epi we get
i u(I)aI = A. This shows that {u(I)} is an absolute joint epi and thus proves the theorem. • 58.
sups in We now state the characterization of additively absolute its final forme
(3.2) THEOREM. Let l be a small category, A an additive absolute sup of category, and F: I~A a functor. (A, u) is an that for F if and only if there exists an A-column (al) such b , b , ••• , b , and every J ~ I!.I there exist I-morphisms l 2 n
F(J)-columns BI' B ,···, Bn' such that 2 n (a )u(J) = E(J) ~ L M(b. )B. l i=l 1 1 and
L u(I)a = A. l l •
and has (3.3) COROLLARY. If F: I~A has absolute weak sups a (strong) sup then this (strong) sup is absolute. • (3.4) THEOREM. Let F: I--->A have absolute weak sups. If the notation (A, u) is one of the absolute weak sups of F, then using are equivalent: of (2.6), L u(I)a is an idempotent and the following l I (i) L u(I)a is a split idempotent, l l (ii) F has an absolute sup,
(iii)F has a sup.
Proof. Assume that (a )u(J) = E(J) + LM(b. )B .• l 1 1 + L [u(I) lM(b. )B. and so Then [u(I) l(aI)u(J) = [u(I) lE(J) 1 1
[u(I)l(aI)u(J) = u(J) i.e. (ïu(I)aI)u(J) = u(J). Therefore
iU(I)aI is ~ idempotent. 59.
has a sup Now to see that (iii) implies (i) assume that F the following (A', v). By (2.6) we see that f = ~v(I)a is such that l l diagram commutes:
A'---A such that Since (A, v) = sup F, there exists a unique g:
A' V(J~A
F(I)I~
commutes and by the usual arguments fg = A'. But
= l: u(I)a • l l
This shows that our idempotent is split.
)a = gf Now to see that (i) implies (iii) assume that l:u(I I in 1. Then such that fg= A'. Let weI): F(I)-B be natural 1) = w( 1) . Consider there exists x: A~B such that xu( l:u(I)a • u(I) xg: A'-B. Then xgfu(I) = x. I = x.u(I) = w(I).
Thus (A', fu) is a weak sup of F.
Assume that xfu(I) = yfu(I) for all 1. Then 60.
}:xfu(I)a = }:yfu(I)a thus l l l xf }:u(I)a = yf }:u(I)a Le. xfgf = yfgf thus I I xf = yf
xfg = yfg
x = y
Therefore {fu(I)} is a joint epi and (A',fu) = sup F.
Notice that (ii) is equivalent to (iii) by (3.3) and this finishes the proof. •
94. ABSOLUTE COEQUALIZERS
We now show how these rather complicated conditions simplify in the case of coequalizers.
ao a (4.1) THEOREM. Let )0 A -...... ,'!>-A be maps of A. Then a o is an absolute weak cokernel of a if and only if there exist d o 0 d dl A ~Ao-----lI>o-Al such that
doa + aodl = Ao
Moreover, a 1S an absolute cokernel of ao if and only if there
exist do nnd dl satisfying the above conditions as well as ado A.
Proof. To say that a 1S a weak cokernel of ao means that a
1S a weak coequalizer of ao and O. Assume that a is an absolute weak cokernel of ao. Then 61.
aao = O. Theorem (2.6) implies that there exist maps cl' c2, .•• ' c6 such that
= A cla 0 + aoc4· Taking d and d = we get the required equation. 0 = cl l -(:4 Now assume that there exist d and such that aa = 0 0 dl 0
Ao. Let x: Ao~X be such that xao = o.
Define y: A~X to be xd . Then xdoa = x(A - aod ) = x - xaod = x. o o l l This shows that the conditions imply that we have a weak cokernel, there- fore an absolute weak cokernel.
If we have the extra relation ado = A then a ~s an absolute epi, thus we have an absolute cokernel.
Finally if aa a is an absolute cokernel of a o then o = o and doa + aod = A Thus adoa + aaod = ad a = a. Since a ~s l 0 l 0 epi we have ad = A. This completes the proof of the theorem. 1 0
The dual of (4.1) ~s the following:
a a o A -A • A ~s an absolute weak kernel diagram if and only if o l d d there exist maps 1.A __o...... A such that o = 0
A dlaO + ado= o Moreover, a is an absolute kernel of if and only if there exist do and dl satisfying the above equations as well as
d a= A. o 62.
We note that a ~s an absolute weak kernel of a if and only o
if ao is an absolute weak cokernel of a.
In Vlew of the fact that if and only if
a a we see that o~ A--..... A ~s a (weak) coequalizer if and only if ~ 0 al
a is a (weak) cokernel of a - al' and thus we state the following o result. a (4.2) THEOREM. Let )0 A be a diagram ~n A.
Then this ~s an absolute weak coequalizer diagram if and only if there
exist do and dl
a a Q Al !IoA~A \ al iU d dl 0 such that aao = a~
Moreover this is an absolute coequalizer if and only if do satisfies
the extra condition
ad = A. 1 o
In the case of coequalizers it is easy to see how the conditions
for total absoluteness reduce to those for additive absoluteness.
a o a Let Al >= A ~A be a totally absolute coequalizer. Then )0 0 al
(l, 4.1) b: A--A and a finite number of by theorem there exist o
maps bi : Ao --~ such that aa = aa o l ab = A
ba = a b v(l) l av (2)bl = av (3)b2 = av (4)b2 av (5)b3
a b = A v(2n) n 0 where v(i) = 0 or l, n ~ o. case The first two equations are the same as in the additive
(do = b). Now, adding the remaining equations we get
But
= {
where Ei = v(2i-l) - v(2i). Thus ba + (ao - al)(~ Eibi) = Ao
Therefore dl = ~Eibi.
up absolute The following theorem is interesting in that it ties because coequalizers with contractible ones. It is also interesting a diagram ensure it is an example where certain simple conditions on
that additive absoluteness implies total absoluteness.
THEOREM. Let be a reflexive diagram in A, 64.
~s b: such an additive category. That ~s to say there Ao--+Al
a o a Then A ~ A ~s an addi ti vely that aob = Ao = alb. Al •)0 0 al coequalizer. absolute coequalizer if and only if it ~s a contractible
then it is Proof. Obviously if Ao ;Al---+-A ~s contractible additively absolute. we have do and Assume now that it ~s additively absolute. Thus dl such that as = aa 0 l
ado = A .... doa + aodl = Ao aldl ·
Ao--+-Al· Define d' l = b + baldl - dl: aod'l = aob + aobaldl - aodl
ald'l = alb .... a1baldl - aldl
- ald Ao = Ao .... aldl l =
Thus we have aa aa o = l A ado = d a = 0 aod'l
ald' 1= Ao
coequalizer. ~.e. AI : Ao--+-A is a contractible •
the proof, thus we could He never used the relation ado = A in this case a have stated the result for weak coequalizers. In e contractible weak coequalizer is the same as the contractible pairs
of Manes [18].
§5. ABSOLUTE PUSHOUTS
(5.1) THEOREM. Consider the following diagra~ 1n A
This is an absolute weak pushout diagram if and only if there exist
yAl~ Ao~ /,p
A 2 such that
Plal= P2a 2
:::; dlPl alcl T Al = -a c d 2Pl 2 l = -a c dl P 2 l 2 = a c ... d 2P2 2 2 A2 ·
Furthermore, this diagram 1S an absolute pushout if and only if we have 66.
the extra relation
Then Proof. Assume that we have an absolute weak pushout. , cl' c'l' c theorem (2.6) implies that we have maps do' dl' d2 2 c ' 2 such that
d o
and o 0 -A o o o P2 = 0 + A o [:~ 2 We get
dOPl = -cl - c'l
dlPl = Al + alcl
dlP2 = alc 2 c' d2P2 = A2 + a2 2' equations and for c' and c' in the first and the fotrrth Solving l 2 substituting in the third and sixth we get
1" cl dlPl = Al al )P -a c (d2 + a2do l = 2 l = al c dl P2 2 67.
(d + a d )P2 = A - a c 2 2 O 2 2 2 Now set d'2 = d + a d and c"2 = -c . The eCluations now look like 2 2 o 2
dlPI = Al + alcl
d'2PI = -a2c I
d IP2 - -alc " 2
d'2P2 = A2 + a 2c"2 the reCluired result.
Now assume that we have dl' d2 , cl' c 2 satisfying the given equations. Let xl: AI~X and x 2 : A2~X be such that xlal = x2a2. Define P~X to be xldl + x2d2. + d (xldl X2 2 )PI = xldlPI ~ x 2d 2PI = xl(alc + Al) + x (-a c ) l 2 2 l
= xlalcl + xl - x 2a 2c I = xl
(xld + x d )P2 x d P + x d P l 2 2 = l l 2 2 2 2 = c + xI(-al 2) + x2 (a2c 2 A2 ) = -x a c + x a c + x l l 2 2 2 2 2 = x 2 Thus the equations force the given diagram to be a weak pushout, there- fore an absolute weak pushout.
joint epi, therefore this extra condition ensures that we have an absolute pushout.
Finally, assume again the equations ~n the statement and suppose that the diagram is a pushout. Then 68.
Pl
P d P = -P a c + P a c + P2 P l d l P 2 + 2 2 2 l l 2 2 2 2 =
P } ~s a joint epi, we get Since {Pl' 2
This completes the proof of our theorem. •
We now state the dual result.
if there exist dl' is an absolute weak pullback diagram if and only
c such that d2 , cl' 2 = alPl a 2P2 = Pldl cla1 ... Al = -c a P l d2 l 2 = a P2d l -c2 l P d = c2a + A · 2 2 2 2 only if we have the Furthermore, this is an absolute pullback if and
extra relation
the c· with the d. and by changing We see, by interchanging ~ ~
the sign, that if it ~s an absolute weak ~s an absolute weak pushout if and only pullback.
From this we see that if
A---Ao~Al
then it ~s also an absolute ~s an absolute weak cokernel pair diagram weak equalizer diagram.
96. ABSOLUTE COPRODUCTS
never In the case of total absoluteness, a disconnected diagram ban on disconnected had an absolute sup. For additive absoluteness the particularly diagrams is lifted. In fact disconnected diagrams are long as there are well disposed to having additively absolute sups, as
not too many components.
objects are Trivially the zero object is absolute, thus initial empty coproduct). absolute (initial object = .::3Up of the empty diagram = is trivially so, Thus weak initial objects are also absolute, but this initial. because all objects in an additive category are weak 70.
only if (6.1) THEOREM. A (weak) coproduct is absolute if and there are only finitely many non-zero terms.
{A.} be a family of objects of A and let (A, u) Proof. Let J. the index category be an absolute weak coproduct of the A·J. • since M(id) = zero matrix. l J.s discrete the only maps are identities but u) J.S an Therefore the conditions of theorem (2.6) become: (A, a family of maps absolute weak coproduct if and only if there exists
only fini tely many non-zero, such that a·:J. A --A.,]. (a. )u. = E· J. J J or if i i= j a.u. = é .. ={O J. J J.J lA. i = J. J. if many finitely many a. are non-zero, then only finitely Since only J. non-zero. A·J. can be
and let Now assume that only finitely many ~ are non-zero we have a family (A, u) be a weak coproduct of {Ai}. For a fixed J. __ A. such A· __A· thus there exists a map aJ.': A '"1. of maps é.J., J': J ]. that
almost all a.u· = é .. and since almost all A. are zero then Thus J. J J.J J. 71.
(A, u) is an ai are zero. Then by the above characterization,
weak coproduct of the {A. }. absolute ~ •
for reference. We state the characterization in terms of equations of A. Let (6.2) THEOREM. Let {Ai} be a family of objects Then (A, u) an object of A and u.: A._A a family of maps. A be ~ ~
~ if and only if almost all ~s an absolute weak coproduct of the
A. are zero, and there exist maps ai: A-~ such that ~
0 if ~ =f-j a·u. = { ~ J 'ij~ A· if ~ = j . ~ only if we have the Moreover (A, u) is an absolute coproduct if and
extra condition Eu.a· = A. ~ ~ •
is self- We remark that the notion of absolute (weak) coproduct coproduct of {A.} if and only dual. Thus A is an absolute (weak) ~
(weak) product of {A. }. if A is an absolute ~
(6.3) THEOREM. Let F: I~A be a functor into an additive we have category. Assume that l = l' + l''. Then by restriction sup then so do F' F': l'-----+-A and F": l''---A. If F has a weak and F". If F' and Fil. If F has absolute weak sups so do F' u") respectively and if and F" have weak sups (A', u ' ) and (A", weak sup. If, further- A' and A" have a weak coproduct then F has a so does F. more, F' and F" have absolute weak sups then
be u Proof. Let (A, u) be a weak sup of F. Let u ' 72.
u') is a weak restricted to -l' , then it is easy to see that (A, sups then so does sup of F'. It follows that if F has absolute weak
F'. We have the seme results for F". and (A",u") Assumethat F' and F" haveweaksups (A',u') respectively. Let A be a weak coproduct with injections A. by a' : A'_A and a": A" ___ Define u(I) : F(I)~A
{ s'u'(I) if l E: II' 1 u(I) = a"u"(I) if l E': I!." 1
F. u ~s natural and (A, u) ~s a weak sup of and F" Since finite weak coproducts are absolute then if F' have absolute weak sups, so does F. •
coproducts Most of the interesting additive categories have finite and then we have the following result. (6.4) COROLLARY. If A has finite coproducts then a dis- has absolute weak connected diagrem with a finite number of components sups. sups if and only if each component has absolute weak •
§7. REMARKS
that the As we see from the proofs, it is not necessary to demand additive full properties be preserved by all functors but only by
embeddings. by It is possible to characterize additively absolute sups functors. For demanding only that they be preserved by representable this can be done this one must know what sups are in Ab. We show how 73.
by working out a typical example.
a o a ~n an ~ A be an absolute coequalizer ~, Let Al • A " 0 al additive category. Apply (A, -) to this diagram. (A, a) (A, Al)~ (A, A ) -----!~- (A, A) -- 0 Ab, but epi is is a coequalizer in Ab. Thus (A, a) ~s epi in
b E (A, A ) such the same as onto in Ab. Thus there exists a map o
that (A, a) (b) = ab = A.
We now apply the functor (Ao ' -). ' a) (Ao , A) (Ao' Al)-=::(Ao ' Ao)----.,,-(Ao
~s a coequalizer in Ab. Let Coequalizers are constructed as follows ~n Ab:
be two abelian group homomorphisms. Let gl coequalizer ~s H = {go(x) - gl(x)IX~Gl}' H ~s a subgroup and the
Go/R with the canonical surjection.
Since (A ' a)(ba) = aba = a = (A ' a)(A ) , thus o o o Therefore we have a map xe:(A ' Al) ba - AoE {aox - alx lx E:-(Ao ' Al)}' o
such that
ba - Ao = aox - alx.
This is the required characterization. 74.
CHAPTER IV
Cat - ABSOLUTENESS
In this chapter we are concerned with those properties of functors which are preserved by all hyperfunctors from Cat to itself. Here we take Cat to be the hypercategory (see Eilenberg-Kelly [5] ) of small categories with functors as morphisms and natural transformations as hypermorphisms. A hyperfunctor from Cat to itself sends categories to categories, functors to functors between the corresponding categories, and natural transformations to natural transformations between the corresponding functors. Furthermore a hyperfunctor is required to pre- serve both kinds of identities and the four kinds of composition.
One can also take --Cat to be Lawvere's hypercategory of all cate- gories.
We shall see that Cat-absolute properties can be expressed in terms of equations involving functors, natural transformations, and identities. The Cat-absolute property derived from some property is, in a sense, the closest one can come to defining the given property in terms of the hyperstructure of --Cat. The fact that a functor is an equivalence of categories ~s a --Cat-absolute property. To say that (T, n, ~) ~s a triple is a Cat- absolute property (we used this fact in chapter II, where we applied A the hyperfunctor )- to a given triple).
The example which motivated this study is the following. Clearly, the fact that a functor U has a left adjoint ~s a --Cat-absolute 75.
thus they pre- property. Functors with left adjoints preserve monos, mild serve monos absolutely. As we shall see, assuming certain functors which completeness properties, this is the whole story; i.e.
preserve monos absolutely usually have left adjoints.
From now on, when we talk of absoluteness we mean Cat-absoluteness.
We also make the convention to underline hyperfUnctors.
§l. ABSOLUTE PRESERVATION
two small (1.1) THEOREM. Let U: B~A be a functor between
categories. Then the following statements are equivalent:
(i) U preserves weak infs absolutely
(ii) U preserves (strong) infs absoluteiy
(iii)U preserves monos absolutely (only the (iv) U preserves those monos which are also epis absolutely
property of being mono lS to be preserved) transformations (v) There exist a functor F: A~B and two natural
E:: FU-B and n: A~UF such that
nU Ue: U --~-UFU ---"-U = U.
Proof. Plan of proof:
i) (v) ~ (iii)=? (iv)9- (v) ~(ii )::::----T is a To see that (v) implies (i) assume that G: I~B as in (v). functor with a weak inf (B, v) and let F, E:, n be 76.
Then we have Assume that \l(1): A-.,...UG(l) 1.S natural in 1.
F\l(l) EG(I) F A -----?~- FUG (1) --""":!"- G (1)
map b: FA~B which is also natural in 1. Thus there exists a such that the following diagram commutes:
Applying U we get
Thus
Uv(I) .Ub.nA = UEG(I) .UF\l(I) .nA
= UEG(I).nUG(I).\l(I)
= \leI)
and thus (v) implies This shows that (UB, Uv) 1.S a weak inf of UG an absolute that U preserves weak infs. But (v) 1.S obviously
property therefore U preserves weak infs absolutely. follow from That (i) implies (iii) and (ii) implies (iii) B b if B==::B~B' the fact that a map b: B---B' is mono if and only B B b 1.S 1.S if and only if B~B ~B' 1.S a weak kernel pair, and this so B
a strong kernel pair.
That (iii) implies (iv) is obvious. shall construct We now show that (iv) implies (v). For this we 77.
X and morphisms a hypercategory ~ by adding to Cat an extra object and hypermorphisms as described be1ow. where There are three kinds of morphisms X--.-~: (F, 1)
F: A-C and (G, 2) and (G, 3) where G: B--C.
There are no morphisms C __ X and only X: X-PX. where The hypermorphisms (F, i)~(F', i) are (t, i) t: F----F' and i = l, 2, 3. and (t, v) The hypermorphisms (F, l)~(G, 2) are (t, u) symbo1s. where t: FU ~G and u and v are arbi trary (s, w) The hypermorphisms (G, 2)~(H, 3) are of the form
where s: G ~ H and w i s an arbi trary symbo1. (r, w') The hypermorphisms (F, l)~(H, 3) are of the form
where r: FU---H and w' is an arbitrary symbo1. i > j. There are no hypermorphisms (F, i)---(F', j) for
We define composition by the fo11owing relations: K(F, i) = (KF , i)
K(t, *) = (Kt, *)
t(F, i) = (tF, i)
(s , i) . (t, i) = (s. t, i)
(s, 2). (t, u) = (s. t, u)
(s, 2) . (t, v) = (s.t, v)
(r, 3) • (s, w) = (r. s, w)
(r, 3) • (s.w') = (r.s, w' )
(s. w). (r, 2) =(s.r, w)
(s, w ).( t, u) = (s.t, w')= (s, w).(t, v)
(t, u). (s, 1) = (t.sU, u) 78.
(t, v). (s, 1) = (t.sU, v)
(s, w'). (r, 1) = (s. rU, w')
and the With composition defined this way, A 1S a hypercategory are straight- embedding of Cat in A 1S a hyperfunctor. The details forward and are left to the reader.
of Next we define a congruence relation on the hypermorphisms
A. Every hypermorphism is congruent to itself and only if there (t, u), (t, v): (F, l)~(G, 2) are congruent if and such that exist H: A~~, 1: F--GH, and m: HU--B
GHU
on the hyper- commutes. This indeed defines a congruence relation kinds of com- morphisms, in the sense that it respects all possible
position in A. objects Form the quotient hypercategory B which has the same classes and morphisms as A, and whose hypermorphisms are congruence of A. Cat is still embedded in B and this of hypermorphisms - embedding is a hyperfunctor.
Define the hyperfunctor F to be the composition (x, -) Cat~B ---~~-Cat. 79.
monos which F(U): F(B)---F(A). Therefore F(U) preserves those are epis. F: A~B The objects of F(~) = (X, B) are (F, 1) where the map and (G, 2) and (G, 3) where G: B-+-B. Consider
(B, w): (B, 2)---(B, 3). w) (B, w) is epi in F(B) for if (s, 3).(B, w) = (s', 3).(~,
then (s, w) = (s', w)
thus
s = s'
therefore (s, 3) = (s', 3) w).(r', 2) (B, w) is mono ~n F(B) for if (B, w). (r, 2) = (~,
then (r, w) = (r', w)
so r = r'
therefore (r, 2) = (r', 2).
w).(t', u ) for some t, t': KU-B If (B, w).(t, u1 ) = (~, 2 then
thus t = t'.
Since 80.
t KU ~B ~;'t KU
= (t', u ) in commutes then (t, ~) :: (t', u2) ~n A, ~ . e. (t, ~) 2 is mono. B where 11ï = u or v. This shows that (B, w) in K(A). Consequently r(U)(B, w) = U(~, w) ::: (U, w) ~s mono F: A--A and The objects of F(A) are of the form (F, 1) where
(G, 2) and (G, 3) where G: B---A. Consider (U, u) (U, w) (!, 1) ---:-_~:_ (U, 2) ----=>~(U, 3). (U, v) (U, u) = (U, v) Both compositions are equal to (U, w'). Therefore By definition of -, there ~n F(A), ~.e. (U, u) :: (U, v) ~n A. such that exist F: A~~, n: A~UF, and e:: FU->-B
UFU
commutes. This completes the proof of the theorem. •
We now establish when such a functor ha~ a left adjoint.
U s uch that ( 1. 2 ) THEOREM. As sume that B ~ A are functors G
there exist e: GU-+-B and h: A~UG such that 81.
hU Ue Gh eG U .. UGU .. U = U. Then G --- GUG ~ G ~s an idempotent in ~ and U has a left adjoint F, if and only if eGo Gh is a split idempotent.
Proof. The commutativity of the following diagram shows that eGo Gh is an idempotent.
Gh eG G .. GUG ~ G
Gb lGUGb Gb GhUG eGUG 1 1 ~ ,. GUG GUG GUGUG
~GUG lGueG leG eG GUG .. G
Assume that U has a left adjoint F with adjunctions
E: FU~B and n:~--UF. Then the following diagrams show how eGo Gh spli ts.
Gn eF G ----~>-~ GUF ~ F
Gh lGUFh lFh 1 GnUG eFUG GUG~~:€-G--e-G~~-FI:G
GUG ~ G 82.
Fh e:G F )' FUG .. G IFn lFuGn Gn FhUF e:GUF t FUF ~ FUGUF ... GUF ~ IFueF leF e:F FUF .,F F
Assume that eGo Gh splits making the following diagram commute.
eG·Gh G ~G ~/s\ F ~ F F
SU e h Ua. Define e: = FU--''-- GU-----B and n = A~UG~UF. Now the
following diagrams commute showing that F is left adjoint to U with adjunctions n and e:.
nU / hU ua.u~ U .. UGU .. UFU lhU lUGhU ueul hUGU UeGU UGU ;. UGUGU ,)0 UGU Ue: UGU /1Ue " U U 83.
Fn
F/ >- FUG "-.. FUF Fh FUa s1 SUGl SUFl e:F F G )0 GUG ~ GUF Gh GUa al eGl eFi F )0 G F ~ e a/ • F
and (1.3) COROLLARY. Let A and B be small categories has split idempotents assume that B has split idempotents. Then B! if and only if it has a and U: B~A preserves monos absolutely left adjoint. •
monos We now give an example of a functor which preserves absolutely but has no adjoint. maps {O, l}, Let B be the category with one object B and two be the one composition being ordinary multiplication. Let A and G: A~B morphism category j\.. U: B-->-A is the only functor the identity thus is also the only functor. Now UG: A~! must be GU---lo-B by define n: A----UG to be A: A---+-A. Define e::
e:(B) = O. Then we have nU Ue: U .. UGU ~ U = U. hom set in B However there can be no adjoint to U, for the only 1 (they cannot be has 2 elements and the only hom set in A has 84.
isomorphie) •
for example, We obtain the dual statements as fo1lows. Assume, Consider D: Cat __ Cat, that U: B~A preserves epis absolutely. the order D(~) = COI'. ~ is not a hyperfunetor sinee it interehanges desirable of composition of hypermorphisms, but it has al1 the
then Qq~ is also properties. If G: Cat--Cat is any hyperfunetor we see that a hyperfunetor. Thus ~~~(U) preserves epis. Then preserves monos Q~(U) preserves monos, i.e. G(Uop): G(BOp)~Q.(AOp) preserves monos for all hyperfunetors G. Therefore Uop: Bop~Aop
absolutely, and by (1.1) there exist h: Aop~UOpG sueh that hUop UOPe Uop .. UopGUop .. Uop = Uop
p and Putting F = GOp , e:: = hOP, n = eO we get e::: UF~A
n: B--...FU sueh that
Un e::U U ~ UFU ~U = u. adjoint. If B has split idempotents then U has a right
12. ABSOLUTE REFLECTION
P me ans If P is a property, then to say that U refleets are really that U preserves "not pli, thus ref1eetion properties
preservation properties.
smal1 (2.1) THEOREM. Let U: B~A be a funetor between categories. Then the following are e~uivalent:
(i) U reflects isos absolutely,
(H) U reflects monos absolutely,
(iii) U reflects epis absolutely,
(iv) U reflects weak infs absolutely,
(v) U reflects weak sups absolutely,
(vi) U reflects joint monos absolutely,
(vii) U reflects joint epis absolutely,
(viii ) U reflects infs absolutely,
(ix) U reflects sups absolutely,
(x) U reflects totally absolute monos absolutely,
(xi) U reflects totally absolute epis absolutely, by U (xii) U reflects totally absolute infs absolutely (where only reflects totally absolute infs we mean that U not
reflects the inf but also the absoluteness),
(xiii) U reflects totally absolute sups absolutely, trans- (xiv) there exist a functor F: A~B and two natural
formations r: B-FU and s: FU---B such that
r, s as above. Proof. (xiv)~ (iv). Assume that we have F,
H: I~B. Assume Let u(I): B~H(I) be natural in I, where let v(I): B'~H(I) that (UE, Uu) is a weak inf of UR in A. Now that be natural in I. Then there exists a map a such 86.
UB ,"", a Uu(I) j "')UB' /Uv(1) UR(1)
commutes. Therefore
Fa
H. commutes, showing that (B, u) is a weak sup of
By duality, (xiv) ==:> (v).
The following implications are straightforward:
(iv)~ (vi)=? (ii) ~ (xi)
(v)~ (vii)9 (iii) ~ (x)
« ii) 1\ (xi) ) ==? (i )
«iv) /\ (vi))='>- (viii)
«v) A(vii))=:> (ix) u(1): H(1)~B «1,2.10) /\(xiv) /\(ix))=:>(xii). 1ndeed, let be natural in 1 and assume that (liB, Uu) is an absolute sup of
UR. Then (ix) ==9- (B, u) = sup H. (xiv)~ H~FUR~H = H thus
(1, 2.l0)~ H has absolute weak sups, Therefore (B, u) is an
absolute sup of H. (dual of (1,2.10) 1\ (xiv) À(viii»~(xii)
This shows that (xiv) implies all the other statements.
To show the converse we have the following straightforward implications:
(iv)=9 (ii)
(v)=:> (iii)
(vi)=> (ii)
(vii)=;> (iii)
(viii)~ (i)
(ix)==> (i)
(xii)~(i)
(xiii) =?(i)
Now each of (i), (ii), and (x) implies that U reflects isos to monos absolutely (U reflects isos to monos meaning that Um iso~m mono).
Also (iii) is the dual of (ii) and (xi) is the dual of (x). Therefore it will be sufficient to show that (U reflects isos to monos abso- lutely)=::;> (xiv).
We shall construct a hypercategory A containing =Cat, by adding a new object X and morphisms and hypermorphisms as described below.
The morphisms X-->- C are of the form (F, i) where
F: B~C and i = l, 2, 3. There are no morphisms C ~X and only
X: X--X. 880
form (t, i) The hypermorphisms (F, i)~(F', i) are of the where t: F~F' for i = l, 2, 30 (t, u ) The hypermorphisms (F, l)----(G, 2) are of the form l and u are arbi trary and (t, ~) for t: F -.. Gand where ul 2 synlbolso (t, v) The hypermorphisms (F, 1) --»>-(H, 3) are of the form for t: F~H and v an arbitrary symbole (t, m) The hypermorphisms (G, 2)~(H, 3) are of the form for t: G~ and m an arbitrary symbolo classes of The hypermorphisms (H, 3)--+-(G, 2) are equivalence
K, r) r- (s', K', rI) (s, K, r) where r: H~KU and s: KU-Go (s, Ko' Kl' K ,00 if and only if there exist a finite number of functors 2
000, Kn' natural transformations
K o q. such that K= Ko, K'=~, and natural transformations and ].
and
H G
commutes 0 1). There are no hypermorphisms (G, 3)~(H, 1) nor (G, 2)~(H,
Composition of hypermorphisms is defined by the following relations: G(F, i) = (GF, i) G(t, *) = (Gt, *) G(s, K, r) = (Gs, GK, Gr) t(F, i) = (tF, i) (t, *). (r, *) = (t.r, *) codomain and l.n where * on the right is determined by the domain and ce.se of u. the i's are the same as on the left. the l. et, 2).(s, K, r) = (t.s, K, r)
(s, K, r).(t, 3) = (s, K, r.t) (t, m) . ( s , K, r) = (t.s.r, 3) (s, K, r).(t, m) = (s.r.t, 2)
that this The verifications that composition is weIl defined and the reader. We defines a hypercategory structure on A are left to to show why we had work out the only case which is not straightforward,
to take equivalence classes as hypermorphisms.
Let (s,K,r):(H, 3)-'>-(G, 2) and 1: L ---M, then the
following must commute:
L(s, K, r) L( R, 3) > L(G, 2)
G 2) .t(R, 3t ' M(s, K, r) r M(R, 3) )0 M(G, 2) 90.
The following diagram commutes by naturality
Therefore (~G.Ls, LK, Lr) = (Ms, MK, Mr.tH) , 2.e.
(iG, 2).(Ls, LK, Lr) = (Ms, MK, Mr).(lH, 3). Which 1S what we had to prove.
Therefore we have a hypercategory A and Cat 1S contained in it.
Next we define a congruence relation on the hyperroorphisms of
A which identifies (t, ul) with (t, u2) for some t: F~G.
(t, u2 ) if and only if there exist H and rand s such that
t F .. G
\1HU
commutes. We make this relation reflexive by requiring that all hyper- morphisms be in relation with themselves. The details that this is a
congruence relation are left to the reader.
Forro the quotient hypercategory B. Cat 1S embedde~ (hyper-
embedded) in B • 91.
Consider the hyperfunctor
F = Cat~B-----!>~Cat. to monos. By hypothesis F(U): F(B)~F(A) reflects ~sos 3). But (U, m).(U, A, U) ~(U)(B, m) = UCB, m)= (U, m): (U, 2)--(U, CU, m) = CU.U.U, 3) = CU, 3) = identity on (U, 3). Also CU, A, U). F(U)(B, m) = CU.U.U, 2) = CU, 2) = identityon (U,2). Therefore
~s mono in F(~) • Now ~s an iso in FCA) • Thus (B, m)
m) • (B, since both equal (B, v) . This (B, m).(B, u l ) = (B, u2) u ) ~n F(~) , ~.e. (B, u ) - (B, implies that (~, ul ) = (~, u2) l 2 H and natural transforma- ~n A. Therefore there exist a functor
tions sand r such that
This completes the proof of the theorem. •
and functors: Consider the following diagram of small categories
U B ------.,1.... A
\!C
monos Then by "absoll.i.te nonsense" we see that (U" reflects (U preserves monos absolutely) 1\ (U' preserves monos absolutely)~ characterizations. absolutely). We should be able to see this from the
Assume that we have G, F, r, s, n, E such that 92.
nU' U'E U' :> U'FU' >- U' = ü'.
Then for U we get
rU GnU"U sUFU"U UE ---'!II~- U = U U ---'!,,"~ GU"U ---.....- GU"UFU"U ------il...- UFU"U l..e.
(sUFU" . GnU" . du UE U ~ UFU"U ~u = u.
monos absolutely) We also see from the diagram that (U" preserves monos absolutely). /\ (U' reflects monos absolutely)~(U reflects U U reflects monos abso We see more from the characterization: if 1 2 • lutely then so does U2
§3. ABSOLUTE FAITHFULNESS
two small (3.1) THEOREM. Let U: B--A be a functor between only if there categories. Then U is absolutely faithful if and rand s exist a functor G: ~B and natural transformations
such that
B-L-GU~B = B.
Since Proof. Assume that U l.S absolutely faithful. faithful)~ (U faithful)::;i> (U reflects monos) then (U absolutely exist G, r, s (U reflects monos absolutely) and by (2.1) there
such that B~GU~B- = -B. 93.
Assume that we have G, r, s as above. Let bl , b2 : B~B' be such that Ubl = Ub 2 • Then sB'. GUbl = sB'. GUb 2 and by naturali ty
and composing with rB we get b 1 = b 2' Thus U is faithful. But the above conditions are absolute thus U 1S absolutely faithful. •
A functor U which is triple able is faithful thus absolute tripleableness implies absolute faithfulne~s. Therefore U must have a left adjoint and there must exist G, r, s such that
B-4GU-4B = B. We saw (II, 3.9) that if B has split idem- potents these conditions imply tripleableness, even V.T.T. Of course this is not absolute tripleableness but it 1S close enough.
No doubt one could find an exact characterization of absolute triple- ableness but it would probably not be worth the effort, considering how weak a condition idempotent splitting is.
§4. ABSOLUTE FULNESS
(4.1) THEOREM. Let U: B ~ A be a functor between small categories. U is absolutely full if and only if there exist functors
G: A-~, (i = l, 2, 3, ... , n-l), and natural trans- formations r: B---GU, s: GU---B,
k k k k n UG J .-K ~ 2 K2-~3~~~ ••.• ~Kn_l ...04Cr--A, p.: U~K.U, and 1 1 1 qi: KiU~U such that the following diagram commutes: 94.
U U
A be the Proof. Assume that U is absolutely full. Let G: Cat ___ Cat hypercategory described in (2.1). Define a hyperfunctor (x, -) to be the composition -Cat~A ~ Cat. G(U)(B, 2) = U(B,2) is full. Now G(U)(B, 3) = U(B, 3) = (U, 3) and a map of G(A). = (U, 2). We have (U, A, U): (U, 3)-+--(U, 2) G(B), say Therefore there exists a map (B, 3)~(B, 2) ~n G, r) U, A, U). ( s, G, r): ( B , 3) ---(B, 2), s uch that G ( U) ( s, = ( To say that But G(U)(s, G, r) = U(s, G, r) = (Us, UG, Ur). functors (Us, UG, Ur) = (U, A, U) means that there exist
~: A--~ and natural transformations 95.
q.: K.U~U such that the diagram in the statement of the theorem ~ ~ commutes.
Now assume that we have functors and natural transformations such as in the statement of the theorem. Let B and BI be objects of
B and a: UB~UB' an A-morphism. We have the following morphism in B:
B rB ~ GUB Ga~ GUll' sB' ~ B'.
Applying U we get
UB UrB.. UGUB UGa.. UGUB' _U_6_B_'"",!~", UB' •
However the following diagram commutes
UGa
UB' 96.
The conditions thus proving that UsB' .UGa.UrB = a, and U ~s full. full. are obviously absolute therefore U is absolutely •
§5. REMARKS
of theorems We include a few remarks to clarifY the constructions following course. (1.1) and (2.1). Effectively we have pursued the to Cat In (1.1) we constructed a hypercategory A by adding as follows a new object X, three new morphisms, ~l' ~2, ~3
X
such that and three new hypermorphisms
that our hyper- Um.u = Um.v, and all consequences. Keeping in mind all pairs of functor F will be essentially (X, -) we identifY of this. Then hypermorphisms coequalized by m and all consequences monos, Um is mono ~n in F(~), m is mono; s~nce F(U) preserves of this we obtain F(A) and thus u = v. Interpreting the consequences
the characterizati0n. X to In (2.1), the idea is the same. We add a new object four new hyper Cat, three new morphisms ~l' ~2' ~3: X~B and
morphisms 97.
u m ~1==:'~2--~3 v
and
such that m.u = m.v and Um.w = U~3 and w.Um = U~2 and all con
sequences. The same idea as 1n (1.1) g1ves the characterization.
This construction should have occurred in (4.1) but luckily it was possible to use the construction of (2.1) again. 98.
CONCLUSION
The methods used ~n this thesis are quite general. Given a certain property, one can find necessary and sufficient conditlons f"or the diagram to possess this property absolutely. The characteri zation takes the form of equations involving aIl the peculiaritles of the extra structure considered.
These characterizations are found by generating categorles with prescribed properties. However, once the characterizations have been found it is fairly easy to find more elegant proofs, eliminating the messiness involved in generating categories.
It is amazing that one has a "mechanical~' means of extracting the absoluteness from a given property. We obtain a "best approxi mation" of a given property by an absolute one.
It ~s our thesis that absoluteness properties ln category theory are fundamental in mathematics and should be studied more extensively. One should be able to recognize these properties in a
diagram and make full use of them. 99.
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