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Categories whose objects are sets: morphisms
Set
f: X → Y functions
Rel
R: X− →Y relations R ⊆ X × Y
2−category
Span
A: X −
- → Y
spans A X Y
✁ ✁ ☛ ❆ ❆ ❯
Lax and Pseudo Presheaves and Exponentiability Susan Niefield, Union - - PowerPoint PPT Presentation
Lax and Pseudo Presheaves and Exponentiability Susan Niefield, Union - - PowerPoint PPT Presentation
Lax and Pseudo Presheaves and Exponentiability Susan Niefield, Union College Categories whose objects are sets: morphisms f : X Y functions Set R : X Y relations R X Y Rel 2 category A A : X Y spans Span
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IDEA Consider the topos SetBop of Set-valued presheaves on B Replace Set by a bicategory S and consider lax or pseudo-functors Bop → S When S is Rel or Span, and morphisms are map-valued op-lax transformation, the result is a category When is it a topos?
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Lax(Bop, Span) OBJECTS: lax functors X: Bop → Span
- a set Xb, for every object b
- a span Xβ: Xb′ −
- → Xb, for every β: b → b′
with η: Xb → Xidb and µ: Xβ′ ×Xb′ Xβ → Xβ′β s.t. . . . MORPHSIMS: families fb: Xb → Yb with s.t. . . . Xb′ Xb Yb′ Yb
❄
Xβ ◦
❄
Yβ
- ✲
fb′
✲
fb
fβ
✲
Lax(Bop, Span) ≃ LaxN(Bop, Prof), normal lax functors X: Bop → Span exponentiable iff ˆ X: Bop → Prof pseudo-functor [S]
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ANOTHER VIEW OF Lax(Bop, Span) Lax(Bop, Span) ≃ Cat/B p: X → B is exponentiable iff factorization lifting (FL) holds where (FL) = Giraud-Conduch´ e condition [G],[C] = X
❄
p B x x′′
✲
α′′
·······
❘
α
x′·······
✒
α′
px px′′
✲
pα′′
❅ ❅ ❅ ❘
β
b′
- ✒
β′
(WFL) + connectivity condition
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Lax(Bop, Span)
Φ
− → ← −
pt Cat/B,
X → ⊔Xb
❄
B x α → x′ . . . b β → b′ α ∈ Xβ subequivalences
- bjects p: X → B
Φ: Lax(Bop, Rel) ≃ Catf/B faithful exponentiables : X pres ◦ ↔ WFL [N] Φ: Lax(Bop, Set) ≃ DF/B discrete fibrations exponentiables: all Φ: Pseudo(Bop, Span) ≃ UFL/B unique factorization lifting exponentiables: ? Φ: Pseudo(Bop, Rel) ≃ DWFL(Catf/B) discrete WFL exponentiables: ?
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IS UFL/B A TOPOS? Lamarch (1996): conjectured UFL/B is a topos Bunge/Niefield (1998): UFL/B is a topos, for B with (IG), and coreflective in Cat/B, using “model-generated” categories [BN] Johnstone (1998): UFL/B is not cartesian closed when B is a square and is a topos when B has (CFI), using sheaves [J] Bunge/Fiori (1998): (IG) iff (CFI), and UFL/B is a topos for B with (IG), using sheaves [BF]
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Theorem 1.
UFL/B coreflective in Cat/B ⇒ UFL/B is a topos
- Proof. UFL/B(X×BY, Z) ∼
= Cat/B(X×BY, Z) ∼ = Cat/B(X, ZY ) ∼ = UFL/B(X, ZY ) and SubUFL/B(X → B) ∼ = SubUFL(X) ∼ =
Cat(X, Ω) ∼
= Cat/B(X, Ω × B) ∼ = UFL/B(X,
- Ω × B), where Ω
is the UFL subobject classifier in Cat. Corollary. Pseudo(Bop, Span) coreflective in Lax(Bop, Span) ⇒ Pseudo(Bop, Span) is a topos Theorem 2.
UFL/B is coreflective in Cat/B ⇐
⇒ (IG) Proof. later
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Given β: b → b′, consider the category [ [β] ] Objects: Morphisms: b
✲ b′ ◗ ◗ s ✑ ✑ ✸
· b b′
✑ ✑ ✸ ◗ ◗ s
·
◗ ◗ s ✑ ✑ ✸
·
❄
The interval glueing condition (IG) [ [β] ] [ [1b′] ] [ [β′] ] [ [β′β] ]
❄ ❄ ✲ ✲
(*) is a pushout in Cat, for all b
β
− →b′ β′ − →b′′
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Theorem 2.
UFL/B is coreflective in Cat/B ⇐
⇒ (IG)
- Proof. Suppose UFL/B is coreflective in Cat/B. Then (*) is
a pushout in UFL/B ⇒ (*) is a pushout in Cat/B ⇒ (*) is a pushout in Cat. The converse was proved in [BN]. Corollary. Pseudo(Bop, Span) is coreflective in Lax(Bop, Span) ⇐ ⇒ (IG), and in this case, Pseudo(Bop, Span) is a topos
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Is Pseudo(Bop, Rel) a topos? No, i is mono and epi (fi = gi ⇒ f = g since p is faithful), but i is not iso in DWFL(Catf/B) ≃ Pseudo(Bop, Rel) x x′ x x′
✲ ❄ ✲
i
b b′
✲ ✲ ✲
f g
X
❍❍❍❍❍❍❍❍❍❍ ❥ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙
p
Is Pseudo(Bop, Rel) cartesian closed? Sometimes A variation of the proof of Theorem 1 shows it is cartesian closed if Pseudo(Bop, Rel) is coreflective in LaxN(Bop, Rel)
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REFERENCES
[BF] Bunge and Fiori, Unique factorization lifting and categories of processes,
- Math. Str. Comp. Sci. 10 (2000) 137–163