Relating the Effective Topos to Homotopy Type Theory Giuseppe - - PowerPoint PPT Presentation

Relating the Effective Topos to Homotopy Type Theory

Giuseppe Rosolini

Università di Genova joint work with Steve Awodey and Jonas Frey, Carnegie Mellon Category Theory 2019 Edinburgh, 8-13 July

Overview of the talk

• Pseudo-equivalence relations and exact completions
• The effective topos Eff
• The pseudo-equivalence relations in Asm
• The cubical assemblies Asm Cop
• The embedding of Eff into a homotopy quotient of Asm Cop

Dedicated to the memory of

Aurelio Carboni

A category of pseudo-equivalence relations

A: a category

Gph(A)

def

= A0 B0 A1 d1

• d2

ƒ0

• ƒ1
• ƒ0
• B1

e1

• e2

A0 B0 PsER(A)

def

= full subcategory of Gph(A) on the ps.-equivalence relations, i.e. A0 A1 d1

• d2
• such that there are arrows A0

r

A1

s

• A2

d′

1

• d′

2

• t
• s.t.

A2 d′

1

• d′

2

A1

d1

• A1

d2

A0

p.b. A0 r

• idA0
• A1

d1

• d2
• A0

A1 d1

• d2
• A1

s

• d2
• d1
• A0

A1 d1 A2 d′

2

• d′

1

• t
• A0

A1 d2 d1

• A0

A1 d2

The exact completion of a category with finite limits

A: a category with finite limits Aex is a quotient category of PsER(A):

two arrows A0 B0 A1 d1

• d2

ƒ0

• ƒ1
• ƒ0
• g0
• g1
• g0
• B1

e1

• e2

A0 B0

• f PsER(A) are equivalent

if there is a “half-homotopy” B0 A0 h ƒ0

• g0
• B1

e1

• e2

B0 in A

Carboni, A., Celia Magno, R. The free exact category on a left exact one. J. Aust. Math. Soc. 1982 Carboni, A., Vitale, E. Regular and exact completions. J. Pure Appl. Alg. 1998

Graphs are internal presheaves

A: a lextensive category

G

def

= the internal category b idb

• ∂1
• ∂2



id

• i.e. G0

def

= T T + T T G1

def

= T T + T T + T T + T T . . . e.g. G(b, ) = 2, G(, b) = 0 So PsER(A)

full

Gph(A)

AGop

and, for instance, PsER(PAsm)

full

❴

• PAsm Gop

PAsm ex≡ Eff

Partitioned assemblies, assemblies, and the effective topos

PAsm

def

= S σ

• S′

σ′

• N

N ƒ

• p.r.
• PAsm

full Asm full

Eff

PAsm reg

• PAsm ex

Hyland, J.M.E. The effective topos. The L.E.J. Brouwer Centenary Symposium, North Holland 1982 Hyland, J. M. E., Johnstone, P . T., Pitts, A. M. Tripos Theory. Math. Proc. Camb. Phil. Soc. 1980 Carboni, A., Freyd, P . J., Scedrov, A. A categorical approach to realizability and polymorphic types. M.F.P.S. 1988 Hyland, J.M.E., Robinson, E., Rosolini, G. The discrete objects in the effective topos. Proc. Lond. Math. Soc. 1990 Robinson, E., Rosolini, G. Colimit completions and the effective topos. J. Symb. Logic 1990 Carboni, A. Some free constructions in realizability and proof theory. J. Pure Appl. Alg. 1995 van Oosten, J. Realizability: An Introduction to its Categorical Side. North Holland 2008

The internal category of cubes

A: a locally cartesian closed category with a natural number object N

N C

def

= the internal category 0

• 1
• .

. . . . .

• 2
• .

. . . . .

• · · ·
• .

. .

• .

. . n

• .

. .

• .

. . . . .

• · · ·
• .

. . i.e. C0

def

= N N C1

def

=

• n,m:N

N n + 2m . . .

e.g. C(n, m) = n + 2m The internal category C in A is the free binary-product completion of the internal category R T T idT

T

• ∂1
• ∂2

I

idI

• !
• Note that

G

• 
• R

C

and

AGop

∗

ACop

∗

• ⊥

Awodey, S. A cubical model of homotopy type theory. Stockholm 2016 Grandis, M., Mauri, L. Cubical sets and their sites. J. Pure Appl. Alg. 2003

Cubical assemblies

Asm Cop

is the quasitopos of cubical assemblies. Note that

Asm Gop

∗

• !
• Asm Cop

∗

• ⊥

⊥ G ✚

• n → Asm Gop

(I In, G)

• where I

I is an interval object which induces a cubical structure in Asm Gop I I0 = T T I I1 = I I I I2 I I3 . . .

• Orton, I., Pitts, A. M.

Axioms for modelling cubical type theory in a topos. Computer Science Logic 2016

Kan fibrations of cubical assemblies An n-box is b: B

In, n > 0, obtained by taking a decidable subobject

D

In−1, and glueing I × D In to D In−1 In.

A map ƒ: E → F has the right lifting property with respect to b: B

In

if, in every diagram B b p

E

ƒ

• In

q

F

there is a diagonal filler B b p

E

ƒ

• In

q

• d
• F

A map ƒ: E

F is a Kan fibration if it has the r.l.p. with respect to all boxes.

Kn Asm Copdef = the full subcategory of Asm Cop

• n the (Kan) fibrant objects,

i.e. objects C such that C !

T

T is a Kan fibration.

Awodey, S., Warren, M. A. Homotopy theoretic models of identity types. Math. Proc. Camb. Phil. Soc. 2009 Bezem, M., Coquand, T., Huber, S. A model of type theory in cubical sets. Types for Proofs and Programs 2014 Uemura, T. Cubical assemblies and the independence of the propositional resizing axiom. arXiv 2018

Ps.-equivalence relations of assemblies as cubical assemblies

PAsm Gop

full Asm Gop

∗

Asm Cop

PsER(PAsm)

• full
• full PsER(Asm)
• full
• Kn

Asm Cop

• full

Ps.-equivalence relations of assemblies as cubical assemblies

PAsm Gop

full Asm Gop

∗

Asm Cop

PsER(PAsm)

• full
• full PsER(Asm)
• full
• Kn

Asm Cop

• full
• Theorem.

The functor PsER(Asm)

Asm Gop

∗

Asm Cop

(i) is faithful

Ps.-equivalence relations of assemblies as cubical assemblies

PAsm Gop

full Asm Gop

∗

Asm Cop

PsER(PAsm)

• full
• full PsER(Asm)
• full
• Kn

Asm Cop

• full
• Theorem.

The functor PsER(Asm)

Asm Gop

∗

Asm Cop

(i) is faithful

Ps.-equivalence relations of assemblies as cubical assemblies

PAsm Gop

full Asm Gop

∗

Asm Cop

PsER(PAsm)

• full
• full PsER(Asm)
• full
• Kn

Asm Cop

• full
• full

up to homotopy

• Theorem.

The functor PsER(Asm)

Asm Gop

∗

Asm Cop

(i) is faithful (ii) is full “up to homotopy”, in the sense that for every g: ∗(G) → ∗(H) in Asm Cop there is ƒ: G → H in PsER(Asm) such that ∗(G) g

• 〈⊥!, id〉

I × ∗(G) h

∗(H)

∗(G) ∗(ƒ)

• 〈⊤!, id〉
• commutes for some

h: I × ∗(G) → ∗(H).

Ps.-equivalence relations of assemblies as cubical assemblies

PAsm Gop

full Asm Gop

∗

Asm Cop

PsER(PAsm)

• full
• full PsER(Asm)
• full
• Kn

Asm Cop

• full
• full

up to homotopy

• Theorem.

The functor PsER(Asm)

Asm Gop

∗

Asm Cop

(i) is faithful (ii) is full “up to homotopy” (iii) maps a ps.-equivalence relation G in PsER(Asm) to a Kan fibrant

• bject in Asm Cop

.

Ps.-equivalence relations of assemblies as cubical assemblies

PAsm Gop

full Asm Gop

∗

Asm Cop

PsER(PAsm)

• full
• full PsER(Asm)
• full
• Kn

Asm Cop

• full
• full

up to homotopy

• Theorem.

The functor PsER(Asm)

Asm Gop

∗

Asm Cop

(i) is faithful (ii) is full “up to homotopy” (iii) maps a ps.-equivalence relation G in PsER(Asm) to a Kan fibrant

• bject in Asm Cop

. Moreover, if the graph G is in PAsm, and ∗(G) is Kan fibrant, then G is a ps.-equivalence relation.

Ps.-equivalence relations of assemblies as cubical assemblies

PAsm Gop

full Asm Gop

∗

Asm Cop

PsER(PAsm)

• full
• full PsER(Asm)
• full
• Kn

Asm Cop

• full
• full

up to homotopy

• Theorem.

The functor PsER(Asm)

Asm Gop

∗

Asm Cop

(i) is faithful (ii) is full “up to homotopy” (iii) maps a ps.-equivalence relation G in PsER(Asm) to a Kan fibrant

• bject in Asm Cop

. Moreover, if the graph G is in PAsm, and ∗(G) is Kan fibrant, then G is a ps.-equivalence relation.

Homotopies for pseudo-equivalence relations Theorem. Consider G0 H0 G1 d1

• d2

ƒ0

• ƒ1
• ƒ0
• g0
• g1
• g0
• H1

e1

• e2

G0 H0 in PsER(Asm). The following are equivalent: (i) ƒ and g represent the same arrow in Asm ex, i.e. there is h: G0 → H1 in Asm such that H0 G0 h ƒ0

• g0
• H1

e1

• e2

H0 (ii) the maps ∗(G) ∗(ƒ)

• ∗(g)

∗(H) are homotopically equivalent in

C.

Homotopies for pseudo-equivalence relations

PAsm Gop

full Asm Gop

∗

Asm Cop

PsER(PAsm)

• full
• full PsER(Asm)
• full
• full

up to homotopy

• Kn

Asm Cop

• full
• PAsm ex

❴

• full Asm ex

❴

• full Ho
• Kn
• C
• ❴
• Eff

Ext

van den Berg, B., Moerdijk, J. Exact completion of path categories and algebraic set theory. J. Pure Appl. Alg. 2017

References

Awodey, S. A cubical model of homotopy type theory. Stockholm 2016 Awodey, S., Warren, M. A. Homotopy theoretic models of identity types. Math. Proc. Camb. Phil. Soc. 2009 van den Berg, B., Moerdijk, J. Exact completion of path categories and algebraic set theory. J. Pure Appl. Alg. 2017 Bezem, M., Coquand, T ., Huber, S. A model of type theory in cubical sets. Types for Proofs and Programs 2014 Carboni, A. Some free constructions in realizability and proof theory. J. Pure Appl. Alg. 1995 Carboni, A., Celia Magno, R. The free exact category on a left exact one. J. Aust. Math. Soc. 1982 Carboni, A., Freyd, P . J., Scedrov, A. A categorical approach to realizability and polymorphic types. M.F.P.S. 1988 Carboni, A., Vitale, E. Regular and exact completions. J. Pure Appl. Alg. 1998 Grandis, M., Mauri, L. Cubical sets and their sites. J. Pure Appl. Alg. 2003 Hyland, J.M.E. The effective topos. The L.E.J. Brouwer Centenary Symposium, North Holland 1982 Hyland, J. M. E., Johnstone, P . T ., Pitts, A. M. Tripos Theory. Math. Proc. Camb. Phil. Soc. 1980 Hyland, J.M.E., Robinson, E., Rosolini, G. The discrete objects in the effective topos. Proc. Lond. Math. Soc. 1990 Menni, M. A characterization of the left exact categories whose exact completions are toposes. J. Pure Appl. Alg. 2003 van Oosten, J. Realizability: An Introduction to its Categorical Side. North Holland 2008 Orton, I., Pitts, A. M. Axioms for modelling cubical type theory in a topos. Computer Science Logic 2016 Robinson, E., Rosolini, G. Colimit completions and the effective topos. J. Symb. Logic 1990 Uemura, T . Cubical assemblies and the independence of the propositional resizing axiom. arXiv 2018 van Oosten, J. Extensional realizability. Ann. Pure Appl. Logic 1997