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 ◦ ✁ ❆ ✁ ☛ ❆ ❯ X Y B ◦ A = A × Y B bicategory

IDEA Consider the topos Set B op of Set -valued presheaves on B Replace Set by a bicategory S and consider lax or pseudo-functors B op → 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?

Lax( B op , Span ) OBJECTS: lax functors X : B op → Span • a set X b , for every object b → X b , for every β : b → b ′ • a span X β : X b ′ − ◦ with η : X b → X id b and µ : X β ′ × X b ′ X β → X β ′ β s.t. . . . f b ′ ✲ X b ′ Y b ′ MORPHSIMS: families f b : X b → Y b with s.t. . . . ✲ X β ◦ ◦ Y β fβ ❄ ❄ ✲ X b Y b f b Lax( B op , Span ) ≃ Lax N ( B op , Prof ), normal lax functors X : B op → Span exponentiable iff ˆ X : B op → Prof pseudo-functor [S]

ANOTHER VIEW OF Lax( B op , Span ) Lax( B op , Span ) ≃ Cat /B p : X → B is exponentiable iff factorization lifting (FL) holds where (FL) = Giraud-Conduch´ e condition [G],[C] = α ′′ ✲ x x ′′ X ······· x ′ ······· ✒ α ′ α ❘ (WFL) + connectivity condition p pα ′′ ✲ px px ′′ ❄ ❅ � ✒ ❅ B β ′ � β ❘ ❅ b ′

x α ⊔ X b → x ′ α ∈ X β Φ . . − → . X �→ Lax( B op , Span ) pt Cat /B , ← − ❄ b β → b ′ B objects p : X → B subequivalences Φ: Lax( B op , Rel ) ≃ Cat f /B faithful exponentiables : X pres ◦ ↔ WFL [N] Φ: Lax( B op , Set ) ≃ DF /B discrete fibrations exponentiables : all Φ: Pseudo( B op , Span ) ≃ UFL /B unique factorization lifting exponentiables : ? Φ: Pseudo( B op , Rel ) ≃ DWFL( Cat f /B ) discrete WFL exponentiables : ?

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]

UFL /B coreflective in Cat /B ⇒ UFL /B is a topos Theorem 1. = Cat /B ( X, Z Y ) Proof. UFL /B ( X × B Y, Z ) ∼ = Cat /B ( X × B Y, Z ) ∼ = UFL /B ( X, � ∼ Sub UFL /B ( X → B ) ∼ = Sub UFL ( X ) ∼ Z Y ) and = Cat ( X, Ω) ∼ = Cat /B ( X, Ω × B ) ∼ � = UFL /B ( X, Ω × B ), where Ω is the UFL subobject classifier in Cat . Pseudo( B op , Span ) coreflective in Lax( B op , Span ) ⇒ Corollary. Pseudo( B op , Span ) is a topos UFL /B is coreflective in Cat /B ⇐ ⇒ (IG) Theorem 2. later Proof.

Given β : b → b ′ , consider the category [ [ β ] ] · ✑ ✸ ◗ ✲ b ′ ✑ s ◗ Objects: Morphisms: b b ′ b ◗ ✸ ✑ s ◗ ✑ · ◗ ✸ ✑ ❄ ◗ s ✑ · The interval glueing condition (IG) [ β ′ ] [ [1 b ′ ] ] ✲ [ ] (*) ❄ ❄ [ β ′ β ] [ [ β ] ] ✲ [ ] → b ′ β ′ β → b ′′ is a pushout in Cat , for all b − −

UFL /B is coreflective in Cat /B ⇐ ⇒ (IG) Theorem 2. 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]. Pseudo( B op , Span ) is coreflective in Lax( B op , Span ) Corollary. ⇒ (IG), and in this case, Pseudo( B op , Span ) is a topos ⇐

Is Pseudo( B op , Rel ) a topos? No, i is mono and epi ( fi = gi ⇒ f = g since p is faithful), but i is not iso in DWFL( Cat f /B ) ≃ Pseudo( B op , Rel ) f i ✲ x ′ x ′ ✲ ✲ x x X ✲ g ❍❍❍❍❍❍❍❍❍❍ ✟ ✟ ✟ ✟ ✟ ✟ p ✟ ✟ ❄ ✟ ❥ ✟ ✙ b ′ b ✲ Is Pseudo( B op , Rel ) cartesian closed? Sometimes A variation of the proof of Theorem 1 shows it is cartesian closed if Pseudo( B op , Rel ) is coreflective in Lax N ( B op , Rel )

REFERENCES [BF] Bunge and Fiori, Unique factorization lifting and categories of processes, Math. Str. Comp. Sci. 10 (2000) 137–163 [BN] Bunge and Niefield, Exponentiability and single universes, JPAA 148 (2000), 217–250 [C] Conduch´ e, Au sujet de l’existance d’adjoints ` a droite aux foncteurs “image r´ eciproque” dans la cat´ egorie des cat´ egories, C. R. Acad. Sci. Paris 275 (1972), 891–894. [G] Giraud, M´ ethode de la descent, Bull. Math. Soc. France, Mem. 2 (1964) [J] Johnstone, A note on discrete Conduch´ e fibrations, TAC 5 (1999), 1–11 [N] Niefield, Change of base for relational variable sets, TAC 12 (2004), 248–261 [S] Street, Powerful functors, expository note (2001) [W] Worytkiewicz, Synchronization from a categorical perspective, arXiv:cs/0411001v1