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Axioms for a category of spaces (Birmingham)
Dr Christopher Townsend
(Open University)
Main Idea
• THESIS:
Just as axioms exist for the category of Sets, axioms can also
be found for a category of topological spaces.
•We use locales to model spaces rather than usual definition.
•This is a hard thing to do. Essentially there are classes of
spaces that behave like set theories (discrete, compact
Hausdorff) but the category of spaces is not cartesian closed.
•The key idea for the axiomatization is to use an external
representation of dcpo homomorphism as a natural
transformation.
Outline talk objectives
• Locales as the “correct’’ category for topology.
• Initial axioms for a category of spaces C (double power
space monad).
• Change of base results.
• The Sierpiński axiom.
• The double coverage axiom.
• Pullback stability results
• Regularity of (the category of) compact Hausdorff and
discrete spaces
• Proper/open duality
Locales
• Use locales as the model for a category of spaces.
• Locales are slice stable (i.e. Loc/Y is a category of locales in
SY - Joyal and Tierney). Topological spaces are not.
• Loc=opposite category of frames. Frames are complete
Heyting algebras.
• Frame hom. = distributive lattice hom. + Scott continuous
(i.e. directed join preserving, aka dcpo hom.).
• Recent result (with Vickers): If Z, Y are locales then
dcpo(ΩZ, ΩY)=Nat(Loc(_xZ,$),Loc(_xY,$))
where Loc(_xZ,$):Locop->Set is the presheaf, Nat(_) the collection of
natural transformations and $ the Sierpiński locale.
• In fact, Loc(_xY,$)=$Y.
Topology for Locales
• Topology works for locales.
• For example, we have the following definition of a proper map p:Z->Y.
It is one such that there exists a triquotient assignment p#: ΩZ->ΩY such
that (I) p# a join semilattice hom. (II) Id <= Ωp p#.
Definition: Triquotient Assignment.
If p:Z->Y is a locale map then a triquotient assignment for p is a dcpo
map p#:ΩZ-> ΩY, satisfying a mixed Frobenius/coFrobenius condition
with Ωp:
p#[c/\(d\/Ωp(e)]=(p#c/\e)\/p#(c/\d).
This is the usual definition of proper (i.e. gives usual topological notion).
For example, a space is compact Hausdorff iff X>->XxX and !:X->1 are
proper.
Axioms for C
•
•
•
Order enriched with lax finite limits and colimits. The coproducts are stable
under pullback.
(Sierpiński axiom) There exists an order internal distributive lattice $
which classifies open and closed subspaces via pullback along its top and
bottom elements respectively.
(Double power space) For any Y, $^($^Y) exists as a representable functor
in Cop->Set. Recall C(_xY,$)=$Y. Use PY to denote $^($^Y).
Theorem. Given these initial axioms, P defines a monad on C.
Proof: This is true for any exponentiating object.
•
NOTICE: The (opposite) Kleisli category (CP )op is the full subcategory of [Cop,Set]
consisting of functors C(_xY,$)=$Y . This category is order enriched. It is also has
finite products:
[$Xx$Y](Z)=C(ZxX,$)xC(ZxY,$)=
C(ZxX+ZxY,$)=C(Zx(X+Y),$)=$X+Y(Z)
More on the Kleisli (CP )op
• If C=Loc then (CP )op has as objects $Y (opens of Y) and has as
morphisms nat. transformations. These nat. transformations are known
to be dcpo homomorphism. This is a good visualisation of (CP )op.
• If q:Z->Y is an epimorphism of locales then Ωq is a monomorpism
(injection of dcpos). Similarly in this new setting if q is an epi. of
spaces then $q is a monic in (CP )op . (Easy diagram chase using
exponentiation.)
• Change of base works: If f:X->Y is a map of spaces then the pullback
adjunction extends to (CP )op
•Σf(g)=fog
Σf
g:W->X C/X
C/Y
•f * =pullback
Note: $
axiom slice
stable
•Σf ($g )=$^(Σf(g))
f*
Σf
$g
(C/X)Pop
(C/Y)Pop
f#
•f #($g )=$^(f *(g))
Proof via co/units
Consequence of the $ axiom.
•In a topos Ω has special properties (e.g. it is discrete compact etc), since it is a
set of truth values. We show similar ‘special properties’ for $
• Ω is the initial frame, so there is a unique frame hom. Ω!: Ω-> ΩX for any
locale X. It follows that any dcpo hom. p# : ΩX-> Ω is a triquotient assignment
for Ω!. I.e.
p#[c/\(d\/Ω!(i)]=(p#c/\i)\/p#(c/\d)
holds ‘for free’.
THEOREM: In a category of spaces any natural transformation p#:$X->$
is a triquotient assignment for the unique map !:X->1.
Proof: Show that p#[c/\$!(i)] <=(p#c/\i) and p#[d\/$!(i)]>=i\/p#(d) by exploiting
the classification of closed/open part of the $ axiom (and using the naturality of
p#) then note that $ is a distributive lattice and so these two conditions together
imply the triquotient assignment condition.
The double coverage axiom
•Triquotient assignments (in locale theory) are dcpo maps, and it is the ability to
describe these maps in terms of generators and relations that allows key pullback
stability results to work. The main result needed is that if e:E>->X is an equalizer
of locales then Ωe: ΩX-> ΩE can be calculated as a particular dcpo coequalizer.
•In our context this theorem is taken as an axiom.
f
e
(Double coverage axiom). If E
X
Y is an equalizer in C then
g
/\(1x\/)[1x1x$f]
$e
X
X
Y
$ x$ x$
$X
$E
/\(1x\/)[1x1x$g]
is a coequalizer in (CP )op. Further this is true in every slice of C.
•This implies the $e is an epi. whenever e is a regular monic.
•Slice stability is always true of Loc by Joyal and Tierney’s description, and
we want this property for the category of spaces.
Pullback stability results
•Pullback stability of proper and open maps is key in topology (e.g. descent,
regularity of compact Hausdorff spaces etc).
Maps with
triquotient
assignment
Proper
Open
•Pullback stability of proper and open is via pullback stability of maps with t.a.
Proof Outline: A t.a. on p:Z->Y is exactly a map p#:$Z->$Y, s.t.
p#[c/\(d\/$p (a)]=(p#c/\a)\/p#(c/\d)
But $Y= Σ!($1 ) where !:Y->1. But !# is left adjoint to Σ! and so p# corresponds to a map
to $ internally in C/Y. You then exploit the fact that any map to $ is a triquotient
assignment ‘for free’ (internally in C/Y) and the double coverage axiom (in C/Y) to
show that t.a.s on p:Z->Y are are exactly maps $p -> $ in C/Y. Pullback stability then
follows since f# preserves $ for any f:X->Y.
Compact Hausdorff and Discrete
•With this pullback stability result we can now prove ‘the usual results’ that were
developed by Joyal and Tierney for open maps and Vermeulen for proper maps.
The results here are identical by replacing finite joins for meets and reversing the
order enrichment.
Sierpiński axiom,
Lemma: i a
coverage axiom and
proper/open
Maps with t.a.
distributivity axiom
subspace then $ix1
are p.back stable
epi.
and BeckChevalley holds
$f iso.
Open/Proper
maps
pullback
stable + Beck
Chevalley
Axiom:
implies f iso.
Open/Proper
surjection are
coequalizers
f proper/open
factors as
surjection/sub
space
Full subcat. of KHaus/
discrete (i.e. finite diagonals
proper/open) are regular
Proper Open Duality
•
•
The duality between compact and open was implicit in Vermeulen’s work and
was central to my thesis. There is a remarkable symmetry between the theories
of these two maps.
For example the proof that a proper map is pullback stable is identical in
structure to the proof that an open map is pullback stable.
•
This duality is now a formal order-enriched duality: Theorem: If C is a category of spaces then so is Cko where ‘ko’ denotes taking
the order-enriched dual.
Proof: The axioms are clearly dual under order enrichment. E.g. Sierpiński is a
distributive lattice and so its dual is a distributive lattice.
• Since proper is dual to open, the theories of compact Hausdorff
and discrete are mapped to each other under this duality. So ...
The theories of compact Hausdorff and discrete spaces have equal
status in this setting.
Further Work
Pontryagin
duality (AbGrp to
Khaus AbGrp)
Priestley duality.
(Should be able to
describe set of upper
closed subsets of an
ordered KHaus space
since regular)
Stone Lattice
duality, e.g.
Pos to Stone
DLat
CoLocales.
These are
opposite to Palgebras. Behave
like top.
D.lattices
Grothendiecck
topos version
Summary
•
•
•
•
•
•
•
Dcpo maps (i.e. Scott continuous maps) between frames are natural
transformations and so this aspect of continuity can be modelled with a
categorical axiom
The axioms say that a category of spaces is order enriched, has a Sierpiński
space ($) classifying closed and open subspaces and has double exponentiation
with respect to $.
This allows change of base results to work in the Kleisli category with respect
to the monad induced by the double exponentiation.
The Kleisli category has a concrete representation in [Cop,Set], consisting of
all spaces of the form $X.
Further a double coverage result is needed which expresses $i as a regular
epimorphism in the Kleisli category for any regular monic i.
Proper and open maps are pullback stable (shown via triquotient assignments).
This allows a proof that the categories of compact Hausdorff and discrete
spaces are both regular.
By the proper/open duality the theories of compact Hausdorff and discrete
spcaes have equal status in this settting.