Transcript The Relationship between Topology and Logic

Axioms for a category of spaces
Dr Christopher Townsend
(Open University)
Main Idea
• THESIS:
Just as axioms exist for the category of Sets, axioms can also
be found for the category Top of topological spaces.
•Categorical axioms give the structure of the class of all spaces
and so are ‘higher order’
•These axioms will model the category of LOCALES
(constructive topological spaces)
•Such axiomatizations are powerful unifying tools, e.g. toposes
Outline Objectives
•
•
•
•
Define Category, Functor, Natural Transformation.
Describe the axiomatization of regular categories
Discuss the difficulties with axiomatizing spaces
Introduce locales as the correct category for constructive
topology
• Give the axiomatization of spaces
• Describe compact Hausdorff and discrete spaces
• Discuss proper/open duality
Basic Definitions
• A category is a set of objects and morphisms between
objects. E.g. the category of Sets (with functions), topological spaces (with cts
functions.)
• For any category C there is a category Cop it has morphisms
in the opposite direction.
• For any category C there is a functor category [Cop ,Set]. Its
objects are functors, F
•And morphisms natural
Cop
A
f
B
F
Set
transformations, a:F->G,
which are indexed maps
F(A)
aA :FA->GA
F(f)
for every object A satisfying a
F(B)
naturality condition.
Regular Categories
• Regular categories are those good enough for basic subset
manipulations.
AxBC
C
• Formally defined as those categories such that
• (a) pullbacks exist
A
B
• (b) equalizers exist
• (c) pullback stable image factorizations exists.
E
B
C
•E.g. category of sets and of compact
Hausdorff spaces is regular.
•Posets definable in any reg. cat.
A
A
f
B
Im(f)
B
Order Enriched Categories
• A category is order enriched if for any objects A,B the
collection of morphisms from A to B is a poset (and
function composition preserves order).
• E.g. the category of posets is order enriched since the set
of monotone functions is itself a poset.
In any category with products you can define
distributive lattices. These are objects X with maps
/\:XxX->X, \/:XxX->X satisfying the usual axioms.
In the order-enriched case these maps MUST
agree with background order.
What’s the Problem?
Q: What’s so difficult about axiomatizing a category that
looks like topological spaces? (After all, you’ve said you can
axiomatize Set.)
A: It must be close enough to set theory so as to contain setlike parts (the regular subcategories of compact Hausdorff
and discrete spaces), but is NOT a set theory in structure
(principally there are no function spaces). It is somewhere in
between.
Locales as the category Top
•The axioms to follow cover the category of locales. This is a
category which is well known to be good enough for topological
theory. E.g. Stone-Cech compactification, Stone duality, Priestley duality, Pontryagin
duality etc.
JT84
•The category of Locales is defined as the dual of the category of
frames (complete Heyting algebras).
•Frame homomorphism are particular directed join preserving maps
between frames.
•The categorical structure of locales is slice stable. I.e. anything
true of the category Loc is also true of the slice Loc/Y for any
locale Y. (This is not true of topological spaces.)
X1 g X2
The objects of C/Y are maps f:X->Y and
the morphisms are arrows g such that: -
f2
f1
Y
Axioms I: The Sierpinski Space
Some of the less central axioms omitted. E.g. pullbacks exists, coequalizers
exists.
ALSO: The category of spaces is axiomatized to be ORDER ENRICHED
The Sierpinski space (classically) is the 2 point topological space such that
the singleton of one of the points is open. E.g.
$={0,1} Opens of $={ Empty, $, {1}}. Points are TRUTH VALUES.
Axiom I: There exists an internal distributive lattice $ such
that for every space X the pullbacks:
a
1
¬a
1
1
X
a
$
0
X
a
$
…uniquely define every
open (resp. closed)
subspace of X
Application of Sierpinski Axiom I
•The existence of $ makes the following very simple settheoretic fact true of spaces: THEOREM: If i, j are subsets of {*}=1, then
(i subset of j) if and only if (* in i ) implies (* in j).
Proof: (classically) the subsets of {*} is the poset {0,1}
•Recall that $={0,1} classically, and so the theorem gives us a ‘trick’
for arguing when two ‘truth values’ are less than or equal to each
other.
•Of course, Loc has a Sierpinski space.
Power Spaces background
• Just as set theory is axiomatized via power sets, spaces are to be
axiomatized via a power locale.
• The correct notion of power locale for spaces is the double power
locale, P. It is the combination of two more well known power locales,
and its points can be visualised as sets of subsets of points.
• Let C be a category, then recall the category [Cop ,Set]. For any X,
there is a functor C(_xX,$): Cop ->Set, which takes any Y to the set of
morphisms C(YxX,$). It is denoted $X (it is the exponential in [Cop
,Set] )
• Recent result (with Vickers) shows that Loc(X, PY)=
Nat[$Y, $X]
where Nat[_] is the collection of natural transformations... hence the
double power space axiom...
Double Power Space, Axiom II
• Since Loc(X, PY)=Nat[$Y, $X] for locales, we have
Axiom II: For any space Y there exists another space PY
such that (for all X),
C(X, PY)=Nat[$Y, $X]
•In contrast to set theory (the points of the power set are subsets),
the points of the double power space are natural transformations.
•The axiom forces PY=$^($Y) (double exponentiation) since:
C(X, PY)=Nat[$Y, $X]=Nat[Xx$Y, $]=Nat[X, $^($Y)]
Coverage Axiom III
•
(Called ‘coverage’ since it mimics a technical result in locale theory;
Johnstone’s coverage theorem).
Axiom III (coverage): If e:E->X is an equalizer in C then
$e :$X -> $E
is (like a) coequalizer in [Cop ,Set] .
Compare to set theory: If i:B>->A a subset inclusion then
i-1:PA->PB
is a surjection (i.e. coequalizer).
•This axiom is more technical. (And we hope to remove it,
following experience from the monadicity result in topos theory.)
Applications: Open maps.
• Recall from topology that a map f:X->Y is open iff the direct image of
an open subset is open.
• For a category of spaces, f:X->Y is open iff there exists a map
f #:$X -> $Y
in [Cop ,Set] , which satisfies a Frobenius condition (plus others omitted).
• This is the correct notion for C=Loc.
• It can be shown that a space is discrete iff X>->XxX and !:X->1 are
open maps. Therefore we can define discrete spaces in this abstract
context: X is discrete iff X>->XxX and !:X->1 are open maps.
Theorem: The category of discrete spaces is regular.
(In other words good enough for basic theory of subsets, e.g. posets etc)
Applications: Proper maps.
• Recall from topology that a map f:X->Y is proper iff the direct image
of a closed subset is closed and every fibre is compact.
• For a category of spaces, f:X->Y is proper iff there exists a map
f #:$X -> $Y
in [Cop ,Set] , which satisfies a coFrobenius condition (plus others
omitted).
• This is the correct notion for C=Loc.
• It can be shown (Vermeulen) that a space is compact Hausdorff iff X>>XxX and !:X->1 are proper maps. Therefore we can define compact
Hausdorff spaces in this abstract context: X is compact Hausdorff
iff X>->XxX and !:X->1 are proper maps.
Theorem: The category of compact Haus. spaces is regular.
(In other words good enough for basic theory of subsets, e.g. posets etc)
Proper/Open Duality
• The theories of proper and open maps are dual. Recall that C is order
enriched, simply turn the order around and everything compact
Hausdorff is discrete etc. (E.g. $ is a distributive lattice and so its
opposite is again a distributive lattice.)
• The theory of ‘sets’ and the theory of ‘compact Hausdorff spaces’
therefore have equal status in this setting.
• This does not mean that all facts about set theory are true of compact
Hausdorff spaces; you must restrict yourself to working within that
part of set theory determined by the axioms for spaces.
Summary
• It is possible to axiomatize classes of mathematical investigation
categorically. E.g. set theory via toposes, or regular categories for a
basic theory with subsets/relational composition.
• This work attempts to do the same for spaces.
• The key appears to be axiomatizing the notion of power space.
• It has been observed that the points of the (double) power space are
natural transformations, i.e. morphisms of [Locop ,Set], and this is then
taken as an axiom.
• A Sierpinski space is introduced which classifies open and closed
subspaces and so allows basic results about truth values to go through.
• Proper and open maps can be defined leading to regular categories of
discrete and compact Hausdorff spaces.
• These two theories are identical under open/proper duality.