Quotient completion for preordered fibrations G. Rosolini joint - - PowerPoint PPT Presentation

Quotient completion for preordered fibrations

• G. Rosolini

joint work with Maria Emilia Maietti Category Theory 2008, Calais, 22-29 June 2008

A regular category from a fibration

A fibration P T

• B

with finite products in B, preordered fibers, and fibred finite limits ∧, ⊤ left adjoints ∃f ⊣ f ∗ satisfying Beck-Chevalley and Frobenius full comprehension P T ⊣

• (–)

⊣

• B

⊤

• ⊤P
• ⊤h
• ⊤X

f

P ∼

→ ∃γP⊤P

• γP
• P
• h

X T (f)

B

• T (γP)
• Rel(T ): B

| R

C

| S

D

• |

∃π1,π3[π1,π2∗R∧π2,π3∗S]

with T (R) = B × C, T (S) = C × D. Rel(T ) is a preordered cartesian bicategory. All objects are discrete and Frobenius. Moreover Map(Rel(T )) is regular and P T

• Mono(Map(Rel(T )))

cod

• B

Map(Rel(T ))

is a change of base. R.F.C. Walters, R.J. Wood, Frobenius objects in cartesian bicategories, T.A.C. 20 (2008)

An The only Xx example that comes to mind

B with finite limits and a stable factorization (E, M) Take P T = cod

• = M→

B Then Rel(cod) is Rel(B, E, M) as in G.M. Kelly, A note on relations relative to a factorization system, CT’90 In particular, B

Map(Rel(B, E, M))

is the reflection from the 2-category of categories with a stable factorization system to that of regular categories, inverting precisely the monos in E.

Manufacturing other examples

fibration with fibred fin.lim’s E p

• Vert(p)
• c(p) ⊣
• ⊣ dom
• fibration of

vertical arrows A E

• id
• comprehension

is full

• is the left biadjoint to forgetting comprehension in fibrations with fibred finite limits.

Moreover

• if A has finite products, then E has finite products
• if

E p

• A

has left adjoints to reindexing with BCC and FR, then so does Vert(p) c(p)

• E
• if

E p

• A

is preordered, then Vert(p) c(p)

• E

is preordered

Manufacturing other examples, 2

• a tripos

T p

• Set

as in

• J. Hyland, P. Johnstone, A. Pitts, Tripos Theory, Math.Proc.Camb.Phil.Soc. 88 (1980)
• for a geometric theory T, the fibration of formulas

WFFT p

• SortT

as in

• M. Makkai, G. Reyes, First Order Categorical Logic, LNM 611, 1977
• for a category A with fin.products and weak equalizers, the fibred preordered reflection

(A→)po cod

• A

essentially as in

• A. Carboni, E. Vitale, Regular and exact completions, J.Pure Appl.Alg. 125 (1998)
• for a left exact functor F: A → C, a suitable fibred preordered reflection

(pbC(F))po cod

• A

as in

• P. Hofstra, Relative completions, J.Pure Appl.Alg. 192 (2004)

Quotient completion

E p

• Vert(p)

c(p)

• Mono(Map(Rel(c(p))))

cod

• Mono(Map(Rel(c(p)))ex/reg)

cod

• A

E Map(Rel(c(p))) Map(Rel(c(p)))ex/reg

||| Map(Splitequiv(Rel(c(p)))) For A with finite products and weak equalizers (A→)po cod

• M(Fr(A))
• Mono(Areg)
• Mono(Aex)
• A

Fr(A) Areg Aex