Axioms for a category of spaces (Birmingham)

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Axioms for a category of spaces (Birmingham)

• Dr Christopher Townsend
• (Open University)

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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.

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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 Sierpinski axiom.
• The double coverage axiom.
• Pullback stability results
• Regularity of (the category of) compact Hausdorff
  and discrete spaces
• Proper/open duality

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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.
• Locopposite 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(OZ, OY)Nat(Loc(_xZ,),Loc(_xY,))
• where Loc(_xZ,)Locop-gtSet is the presheaf,
  Nat(_) the collection of natural transformations
  and the Sierpinski locale.
• In fact, Loc(_xY,)Y.

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Topology for Locales

• Topology works for locales.
• For example, we have the following definition of
  a proper map pZ-gtY. It is one such that there
  exists a triquotient assignment p OZ-gtOY such
  that (I) p a join semilattice hom. (II) Id lt Op
  p.

Definition Triquotient Assignment. If pZ-gtY is
a locale map then a triquotient assignment for p
is a dcpo map pOZ-gt OY, satisfying a mixed
Frobenius/coFrobenius condition with Op
pc/\(d\/Op(e)(pc/\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 Xgt-gtXxX and !X-gt1
are proper.
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Axioms for C

• Order enriched with lax finite limits and
  colimits. The coproducts are stable under
  pullback.
• (Sierpinski 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-gtSet. 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
• XxY(Z)C(ZxX,)xC(ZxY,)
• C(ZxXZxY,)C(Zx(XY),)XY(Z)

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More on the Kleisli (CP )op

• If CLoc 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 qZ-gtY is an epimorphism of locales then Oq 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 fX-gtY is a map of
  spaces then the pullback adjunction extends to
  (CP )op

• Sf(g)fog
• f pullback
• Sf (g )(Sf(g))
• f (g )(f (g))

Sf
C/X
C/Y
gW-gtX
f
Note axiom slice stable
Sf
(C/X)Pop
(C/Y)Pop
g
Proof via co/units
f
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Consequence of the axiom.

• In a topos O has special properties (e.g. it is
  discrete compact etc), since it is a set of truth
  values. We show similar special properties for
  
• O is the initial frame, so there is a unique
  frame hom. O! O-gt OX for any locale X. It
  follows that any dcpo hom. p OX-gt O is a
  triquotient assignment for O!. I.e.
• pc/\(d\/O!(i)(pc/\i)\/p(c/\d)
• holds for free.

THEOREM In a category of spaces any natural
transformation pX-gt is a triquotient
assignment for the unique map !X-gt1.
Proof Show that pc/\!(i) lt(pc/\i) and
pd\/!(i)gti\/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.
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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 eEgt-gtX
  is an equalizer of locales then Oe OX-gt OE 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\/)1x1xf
e
XxXxY
X
E
/\(1x\/)1x1xg
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 Tierneys description, and we want this
  property for the category of spaces.

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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
Open
Proper

• Pullback stability of proper and open is via
  pullback stability of maps with t.a.

Proof Outline A t.a. on pZ-gtY is exactly a map
pZ-gtY, s.t. pc/\(d\/p (a)(pc/\a)\/p(c
/\d) But Y S!(1 ) where !Y-gt1. But ! is left
adjoint to S! 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 pZ-gtY are
are exactly maps p -gt in C/Y. Pullback
stability then follows since f preserves for
any fX-gtY.
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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.

Lemma i a proper/open subspace then ix1 epi.
Sierpinski axiom, coverage axiom and
distributivity axiom
Maps with t.a. are p.back stable and
Beck-Chevalley holds
f proper/open factors as surjection/subspace
Axiom f iso. implies f iso.
Open/Proper maps pullback stable Beck Chevalley
Open/Proper surjection are coequalizers
Full subcat. of KHaus/ discrete (i.e. finite
diagonals proper/open) are regular
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Proper Open Duality

• The duality between compact and open was implicit
  in Vermeulens 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. Sierpinski 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.

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Further Work
Stone Lattice duality, e.g. Pos to Stone DLat
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)
CoLocales. These are opposite to P-algebras.
Behave like top. D.lattices
Grothendiecck topos version
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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 Sierpinski 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.