Categorical Algebra and its Applications: Proceedings of a Conference, held in Louvain-La-Neuve, Belgium, July 26 – August 1, 1987

364 Pages·1988·13.298 MB·English

by  J. Adámek, J. Rosický, V. Trnková (auth.), Francis Borceux (eds.)

ARE ALL LIMIT-CLOSED SUBCATEGORIES OF LOCALLY PRESENTABLE CATEGORIES
REFLECTIVE?
J. Ad~mek, J. Rosick9 and V. Trnkov~
Teeh. Univ. Purkyne Univ. Math. Inst. of the
Suehb~tarova 2 Jan~ckovo n~m. 2a Charles University
Praha 16627 ,6 66295 Brno, CSSR Sokolovsk~ 83
CSSR Praha 18600 ,8 CSSR
Introduction. It is not surprising that the answer to the question
in the title is negative, under "reasonable" set-theoretical
hypothesis (e.g., assuming the non-existence of measurable cardinals).
We even construct two reflective subcategories ~) of the locally present-
able category Graph of graphs whose intersection is not reflective in
Graph (although it is, of course, closed under limits). What might be
surprising is that under other set-theoretical hypothesis, the answer
is affirmative. We introduce a condition called Weak Vop~nka Principle,
consistency of which follows by the existence of huge cardinals and, as
some set-theorists believe, may be added to the usual axioms of set
theory. We prove that assuming Weak Vop~nka Principle, then each locally
presentable category K has the following properties:
(i) Every subcategory of K closed under limits is reflective in
K;
(ii) All reflective subcategories of K form a large complete
lattice (ordered by inclusion);
(iii) The intersection of two reflective subcategories of K is
reflective.
Conversely, assuming the negation of Weak Vop~nka Principle, none of the
above statements holds in K = Graph.
Weak Vop~nka Principle is the following statement:
Ord °p does not have a full embedding into Graph.
(Here Ord °p is the dual of the well-ordered category of all ordinals.)
We have chosen that name because the principle is weaker than the well-
known Vop~nka Principle which, as we show below, can be formulated as
follows:
~)Subcategories are understood to be full throughout our paper.
drO does not have full embedding into Graph.
The position of Vop~nka Principle in set theory is discussed e.g. in
[j]; there are good reasons to believe that Vop~nka Principle does not
contradict the usual axioms of set theory. If so, tb~n we can add
Weak Vop~nka Principle to the usual axioms, and in the resulting set
theory the answer to our title question is affirmative.
On the other hand, by results to be found in [PT] , every concrete
category has a full embedding into Graph provided that we assume the
following
(M) There does not exist a proper class of measurable cardinals.
Thus, (M) implies the negation of both of the above principles. Since
the non-existence of measurable cardinals is certainly not contradict-
ory to the usual axioms of set theory, we conclude that we can add the
negation of Weak Vop~nka Principle to those axioms.In the resulting
set theory, the answer to the above question is negative. A closely
related result has been proved in [AR] where, under (M), a proper class
of reflective subcategories of Graph was presented whose intersection
is not reflective in Graph. The present result is a refinement of the
previous one: the set-theoretical hypothesis is weaker (viz., as weak
as possible) and we have two subcategories in place of a proper class.
The price of that refinement is a much deeper delving into the results
of Prague School. This has made our main construction technically quite
involved. For a reader interested only in the equivalence of our title
question and Weak Vop~nka Principle, it is not necessary to study our
construction (i.e., he can skip parts II and III of our paper) since
J. R. Isbell has made the observation in[Ill that the affirmative
answer easily implies Weak Vop~nka Principle and our Theorem 2 states
the converse. This converse also follows by a result of E. R. Fisher
announced in [FI]. He unfortunately has not published the proof yet but
he kindly sent it to us IF2] .
We finally mention a recent result of M. Makkai and A. Pitts [MP]
which holds without set-theoretical restrictions: let K be a locally
finitely presentable category, then each subcategory of K closed under
limits and filtered colimits is reflective in K.
In contrast to locally presentable categorie~ in the category poT
of topological spaces we have found, without additional set-theoretical
hypothesis, two reflective subcategories with non-reflective intersect-
ion, see [TAR].
I. Assumin~ Weak Vop~nka Principle
Convention. We work, throughout our paper, within the usual
G~del-Bernays theory of sets with AC.
We will heavily use the category Graph of graphs (directed), i.e.,
pairs (V,E) where V is a set (of vertices) and E C V x V i~ a set (of
edges). Morphisms f:(V~E) ~ (V',E'), called homomorphisms, are maps
f:V - V' with (fxf)(E) ~ E'. An important property of that category is
that every graph is a coproduct of its connected components (=maximal
indecomposable subobjects). The following result explains why Graph
can be used as a representative locally presentable category:
Theorem .i Every locally presentable category can be fully embedd-
ed into Graph.
Proof. For each small category ,A the category S~t A has a full
embedding into Graph, see [PT] (II.5.3 and Ex. 1.7.1). Consequently,
every locally presentable category is equivalent to a subcategory of
Graph, see [GU]. It is obvious that a category equivalent to a sub-
category of Graph is also isomorphic to (another) subcategory of Graph.
A large discrete category is defined by having a proper class of
objects and no other morphisms than identities.
Lemma .i (Formulations of Vop~nka Principle.) The following
statements are equivalent:
(i) Vop~nka Principle: for each first-order language, every
class of models such that none of them has an elementary
embedding into another one is a set}
(ii) No locally presentable category has a large discrete
subcategory)
(iii) Graph does not have a large discrete subcategory;
(iv) Ord cannot be fully embedded into Graph.
Proof. i ~ iv: Suppose that, on the contrary, Ord has a full
embedding into Graph (denoted by (Vi,Ei) on objects and by
aij:(Vi,E i) -- (Vj,Ej) on morphisms, i ~ j). There clearly exists a
proper class K --COrd such that for each j E K the maps ~ij for i 6 K,
i < j, are not collectively onto~ let us choose a vertex xj E Vj -
- ieK,i<jU (v~ ij i). Now consider the language of one binary relation
and one nullary operation (and no axioms). Then each (vi,Ei,xi),i 6 K,
is a model, and there are no elementary extensions in the resulting
class. This contradicts (i).
iv ~ iii: The category K of connected graphs has the property
that Graph can be fully embedded into K, see [PT] (I.l.ll). Thus, if
(iii) would be false, there would exist a large discrete category of
connected graphs G , x E X. For any ordinal i, X. = {x 6 X \ the rank
x l
of x is smaller than i} is a set (see [J] .) (Using the rank, which is
based on the axiom of regularity, we avoid the axiom of choice for
classes.) There is an isotone map F:Ord ~ Ord such that XF(i) # XF(j)
for i ~ j. The functor A:Ord ~ Graph defined on objects by
i A = ~ G and on morphisms by the obvious coproduct injections
x
XeXF(i)
is full. (In fact, given a homomorphism f:A ~ A then, since f
i 3
preserves connected components, for each x 6 XF(i) there is y 6 XF(j)
such that f maps Gx inside Gy. It follows that x = y and f(v) = v for
each vertex v of G . Thus, i ~ j and F(v) = v for each vertex v of A..)
x 1
This contradicts (iv).
iii ~ ii by Theorem i.
ii - i : Suppose that, on the contrary, there is a first-order
language L and a proper class Mi, i @ X of L-models such that none of
them has an elementary embedding into another. Add a new binary
relation symbol to L and denote by ~ the resulting first order language.
By [PT] II.3.12., the underlying set IMil of i M always carries a
rigid binary relation r.1. Hence (Mi,r i), i E X form a large discrete
subcategory of the category Mod(~) of ~-models and elementary embeddings.
By the downward Lowenheim-Skolem theorem (cf. [CK] , 3.1.6.), Mod(~) has
a small dense subcategory. Hence Mod(~) can be fully embedded into a
category of algebras (cf. [I2] , 4.2.), i.e. into a locally presentable
category. It contradicts (ii).
L~mna 2. Vop~nka Principle implies Weak Vop~nka Principle (in
other words, if Ord °p has a full embedding into Graph then also Ord
has such an embedding).
Proof. Let A:Ord °p ~ Graph be a full embedding denoted by A =
l
= (Vi,E i) on objects and eij:Ai - Aj (j N i) on morphisms. There
clearly exists a proper class K COrd such that ~. is not one-to-one
13
for any i,j E K, i # j. Consider - the theory of two binary relations
(and no axiom). For each i e K we have a model (Vi,Ei,Ei) where E[ =
1
= {(x,y) I x,y 6 Vi, x # y} and there exist no homomorphisms from one
of those models into another one - thus, Vop~nka Principle does not hold.
Open problem: Are the two principles equivalent?
Theorem 2. Assuming that Weak Vop~nka Principle is true, each
subcategory of a locally presentable category K which is closed under
limits is reflective in K.
Proof. We first observe that for each set H of objects of K, the
least subcategory H* of K closed under limits and containing H is
reflective in K. This follows from the Special Adjoint Functor Theorem
[M]: H is a cogenerating set, and since K is wellpowered and complete
(being locally presentable), so is H*.
Let [ be a subcategory of K closed under limits. For j @ Ord, let
[j be the set of all objects of i having the rank smaller than j
(compare the proof of Lemma I; we are again avoiding the axiom of
choice for classes). The categories i~ are reflective. Thus,for each
3
object K of K we have reflections r.:K ~ K. of K in [~ for all j 6
... 3 J J
• Ord. Since i ~ j implies L~ _ C 3 _ i "" C i we have the unique K-morphism
eji: Kj - i K with i = r eji.r j. We are going to prove that the chain
e.. is stationary, i.e., that there exists an ordinal o i such that
31
all eji with io ~ i ~ j are isomorphisms. It then follows that
rio:K ~ KioiS the reflection of K in each L~, i ~ io, and hence, the
reflection in L; this will conclude the proof.
Suppose that, on the contrary, there is an object K such that eji
is not stationary. From thewellpoweredness of K it follows that there
is a transfinite sequence to < I < t .-. i < < t ... of
ordinals such that for i < j the morphism e is not a monomorphism~
tj i t
Now consider the comma-category KSK (of all K-arrows with domain K,
see [M] :) we have a functor
E:Ord °p ~ KiK
rt.
defined on objects by i E = (K i Kt ) and on morphisms by (i ~ j)~
et3 ti" This functor is a full embedding. In fact, let f:Ej ~ i E
be a morphism, i.e., let f:Kt. ~ Kt ' fulfil f.rt. = rt . If i S j,
3 l 3 l
then f.rt t = e .r t and this implies f t = e (by the uniqueness
3 J ti J jti
requirement on reflections). And the case i > j cannot occur since
we would have (f'etitj)'rtl f.rtj = rti, and hence, f.etitj = id,
z
although e is not a monomorphism. It is well-known that since
tit j
K is locally presentable, so is KSK (see, e.g., [MPa]6.1.1.), and hence,
there is a full embedding of KiK in Graph by Theorem i. This embedding
composed with E above, shows that Weak Vop~nka Principle is false -
a contradiction. (There is,a more straightforward way to finish the
proof because E induces a full embedding of KSK in G~Graph for G = EK
and GiGraph has a full embedding into a locally presentable category
of graphs with constants c indexed by vertices of G (sending f:G
x
X to (X,f(V))veV(G))).
Corollary. Assuming Weak Vop~nka Principle, each intersection of
reflective subcategories of a locally presentable category K is
reflective in K. Thus, the lattice of all reflective subcategories of
K is large-complete.
II. Assuming Negation of Weak Vop~nk a Principle
Theorem .3 Assuming that Weak Vop~nka Principle is false, the
category Graph has two reflective subcategories whose intersection is
not reflective.
The aim of the following two sections is to prove the theorem. We
shall construct acollection D (i E Ord) of graphs with the following
1
properties:
(a) The subcategory i of Grap~ consisting of those graphs G such
that for each ordinal i there is precisely one homomorphism
from D to G is not reflective in Graph;
1
(b) Let i and L denote the subcategories of Gaaph consisting of
e o
those graphs G satisfying the following conditions:
(bl) Given an even ordinal i (odd ordinal i, respectively)
such that for each t < i there is a homomorphism from t D
to G, then there is also a homomorphism from D to G,
i
(b2) For each ordinal i there is at most one homomorphism fror~
D to G,
1
(b3) There is a homomorphism from O D to G.
Then e L and ° i are both reflective subcategories of Graph.
Since, obviously, L e = N L Lo, this will prove Theorem .3
We will now present a condition guaranteeing that (b) is true. In
the last section we will construct, assuming the negation of Weak
Vop~nka Principle, a collection i D satisfying both that condition and
(a) above. Recall that a collection of objects is rigid if the
corresponding subcategory is discrete.
Proposition i. Let i D (i 60rd) be a rigid collection of
connected graphs such that for each graph G the following holds:
if there is a homomorphism from Di+ 1 to G and a surjective
homomorphism from k<~l Dk to G, then there is also a
homomorphism from D. to G.
Then the above subcategories L and L are reflective in Graph.
e o
Proof. We present a proof that Lo is reflective, the case of e i
is analogous. Let i denote the category of all graphs satisfying (b2)
above. Observe that % is obviously closed under products and
subobjects in Graph, and therefore, ~ is epireflective in Graph. It is
thus sufficient to prove that o L is reflective in .L We denote, for
short, by [p,Q] the set of all homomorphisms from P to Q.
We first show that for each graph G 6 ~ with [Di,G] # ~ for all
even ordinals i there exists an ordinal io with [Di,G] # ~ for all
i ~ io. Without loss of generality, we can assume that each vertex of
G lies in the image of some homomorphism from i, D i E Ord. (For else
we work with the subgraph G' of G consisting of all vertices lying in
those images: proving the statement for G' would clearly prove it for
G too.) We can, then, choose an ordinal io such that each vertex of G
lies in the image of some homomorphism from Di, i < io. Let us show
that for each ordinal j ~ io we have [Dj,G] # 9. Suppose, on the
contrary, that [Dj,G] = 9. It follows that j is odd, and hence j + 1
is even and therefore, [Dj+I,G] # 9- Put
A = {i e Ordli < j, and [Di,S] = 9}.
There is a surjective homomorphism from ~ k D to G + ~Di: for each
k <j i eA
k < j we either have a unique homomorphism from k D to G (if k g A) or
k D is a summand of ~ i D (if k 6 A) and the resulting homomorphism
3<]~k:f Dk - G + ~ i D is surjective by the choice of io, since j ~ io.
i
We now use the above property of the graphs i D to conclude that there
is a homomorphism from Dj to G + ~ i. D But this is a contradiction
ieA
since j ~ A and [Dj,G] = 9: each homomorphism g:Dj ~ G + i~ Di would,
by the connectedness of Dj, map Dj either to G, or to some i D (but then
i = j).
We are ready to describe the reflection of any graph G 6 .i We
denote by io either the least even ordinal with [Dio,G] = @, or (if no
such even ordinal exists) the least ordinal with [Di,G] # ~ for all
i ~ io. We will prove that tne refl~ction of G in io is the following
coproduct injection
r : G ~ G + je~B Dj
where
B j{ E Ordlj < io and [Dj,G] @}.
= =
(A) G + ~ Dj lies in L . Firsts this graph obviously lies in .L
jeB o
Now let i be an odd ordinal with [Dt, G + ~_[ Dj] # @ for all t < i.
We are to show that [D i, G + ~ Dj] # ~. This is obvious whenever
either [Di,G] ~ @, or i E B. Thus, we can suppose that i ~ io.
(AI) If io is even with [Dio,G] = ~ then, since i is odd, we
have i > io and then io is one of the t's above - thus, [Dio, G +
+ ~ Dj] ~ ~. This is a contradiction since [Di0,G] = ~ and io ~ B.
J
(A2) If io is such that [Dk,G] # ~ for all k Z io, we are ready
since i ~ io.
(B) Any ~omomorphism h:G ~ H with H O E i factors uniquely through
r. Since H E ,L it is sufficient to show that [Dj,H] # ~ for all j
E B: then we have a (necessarily unique~ extension of h to G + J~ D..
j~B 3
We will prove by induction on i < io that [Di,H] # @. The case i = O
follows from the definition of Lo" Suppose i < io is such that [Dt,H]#
# ~ for all t < i. Since H ~ io, if i is odd we conclude [Di,H] # @.
If i is even then i < io implies [Di,G] # ~, and hence, [Di,H] # ~.
III. Main Construction
The construction of the graphs i D which would prove Theorem 3 has
several levels, and we start with some auxiliary constructions. We
denote by
Graph o
the subcategory of Graph formed by all non-trivial, acyclic, strongly
connected graphs, i.e., graphs of at least two vertices and such that
for arbitrary two distinct vertices x and y there exists either a
directed path from x to y, or a directed path from y to x, but not both.
The sets of vertices and edges of a graph G are denoted by V(G) and
E(G), respectively. When speaking about subgraphs and quotient graphs,
we understand the strong sense, i.e., regular monos and regular epis,
respectively. A pair (G,@) consisting of a graph G and a vertex a of G
is called a pointed graph; homomorphisms of pointed graphs are those
graph homomorphisms which preserve the chosen vertices.
Construction " . Let G and H be graphs, and let a be a vertex
of H. We denote by a
H ~ G
the graph obtained from G by gluing onto each vertex a copy of the
graph H using the gluing vertex a:
a~ C V V V ~ G
H
a
H~ G
Formally, H ~, G is the quotient of the sum'G + I I H × {x}, where
x~ ]GfT~
H x {x} is a copy of the graph H, under the equivalence ~ with x ~ (a,
x) for each x 6 V(G) and with all other equivalence classes singleton
sets.
Construction #. For each pointed graph (H,b) denote by (H,b) # the
qraph obtained from H by iteratively gluing a copy of H onto vertices
(using b as the gluing point) in such a way that, in the end, each
~ertex X has a separate copy of (H,b) # glued to x.
Thus, (H,b) # is the direct limit of the following c~-sequence
Ho ~ 1 2 ~ ~ H H ... of subgraphs:
Ho is the graph with single vertex b and no edge,
I H = H,
2 H is obtained by gluing a copy of H on each vertex in V(H) - {b},
i.e., 1 H is the quotient of H + ~ H × {x} (where H×{x} is
xeV(H),x~b
a copy of H disjoint with H) under the least equivalence ~ with
x ~ (b,x) for all x E V(H) - b: and, in general, Hn+ 1 is obtained
by gluing a copy of H on each vertex in V(H n) - V(Hn_I), i.e.,
Hn+ 1 is the quotient of
n + H ~ H x {x}
x~V(Hn]-V(Hn_ )I
(where H × {x} is a copy of H disjoint with H n) under the least
equivalence ~ with x ~ (b,x) for all x E V(Hn)-V(Hn_I).
10
H H H H
2 H 3 H
Construction ~. For each graph G we denote by G the following
graph.
I
I
I
4 98
I
I
I
I
G
G without edges
Formally, V(G) is a disjoint union of V(G), V(G) = {~Ix e V(G)} and
{1,2,...,9} (we suppose, for simplicity, that the sets are disjoint),
and E(G) = E(G) U E(K) U {x,~)}xeV(G) O {C~,I)}xeV(G) , where K denotes
the above graph on the vertices 1,2,...,9.
Construction 4. Given graphs G and H, we denote by
G ~'~ H
the graph obtained from H by first gluing a copy of the following
5
graph C3,5 6// ~
\ /
3