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 C o p • The embedding of Eff into a homotopy quotient of Asm C op

Dedicated to the memory of Aurelio Carboni

� � � � � � � � � � � � � � � � � � � � � � � � � � � A category of pseudo-equivalence relations A : a category A 0 ƒ 0 B 0 e 1 d 1 def Gph ( A ) ƒ 1 = A 1 B 1 e 2 � d 2 � A 0 ƒ 0 B 0 def PsER ( A ) = full subcategory of Gph ( A ) on the ps.-equivalence relations , i.e. s d ′ d 1 1 A 0 A 1 such that there are arrows A 0 r � A 1 A 2 s.t. t d ′ d 2 2 id A 0 d ′ A 0 d ′ A 1 A 2 A 2 � A 1 1 2 d 1 d 2 d 1 d 1 � t d 1 d ′ d ′ p.b. A 0 r A 1 A 1 s A 1 A 0 A 1 d 1 1 2 d 2 � d 2 d 1 d 2 d 2 A 1 d 2 � A 0 A 0 A 1 A 0

� � � � � � � � � � � The exact completion of a category with finite limits A : a category with finite limits A ex is a quotient category of PsER ( A ) : ƒ 0 A 0 B 0 g 0 e 1 d 1 ƒ 1 of PsER ( A ) are equivalent two arrows A 1 B 1 g 1 e 2 � d 2 � ƒ 0 A 0 B 0 g 0 B 0 ƒ 0 e 1 in A if there is a “half-homotopy” A 0 h � B 1 e 2 � g 0 B 0 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 ∂ 1 def G = the internal category id b b �  id  ∂ 2 def def i.e. G 0 = T T + T T G 1 = T T + T T + T T + T T . . . e.g. G ( b,  ) = 2 , G ( , b ) = 0 PsER ( A ) � � � Gph ( A ) A G op So full PAsm G op PsER ( PAsm ) � � full � and, for instance, ❴ PAsm ex ≡ Eff

� � � � � � Partitioned assemblies, assemblies, and the effective topos full � Asm � � � Eff ƒ PAsm � � S ′ S � � full def PAsm = σ ′ σ PAsm reg � � PAsm ex N N p.r. 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 . . . . . . . . . def C = the internal category  0  1  2 · · ·  n · · · . . . . . . . . . . . . . . . . . . . . . � N n + 2 m . . . def def e.g. C (  n ,  m ) = n + 2 m i.e. C 0 = N N C 1 = n,m :N The internal category C in A is the free binary-product completion of ∂ 1 the internal category R id T T T � I id I T ∂ 2 !  ∗ � C A G op � A C op G � � R � � Note that and ⊥ � �  ∗  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 C o p is the quasitopos of cubical assemblies .  ! ⊥  ∗ Asm G op Asm C op Note that ⊥ � � n �→ Asm G op  ∗ I n , G ) G ✚ ( I I is an interval object which induces a cubical structure in Asm G op where I I 0 = T I 1 = I I 2 I 3 I T I I I I . . . � • • � • � • • • • • � • • • � • • • • Orton, I., Pitts, A. M. Axioms for modelling cubical type theory in a topos. Computer Science Logic 2016

� � � � Kan fibrations of cubical assemblies � I n , n > 0 , obtained by taking a decidable subobject An n -box is b : B � � I n to D � � I n − 1 , and glueing I × D � � I n − 1 � � I n . D � � I n A map ƒ : E → F has the right lifting property with respect to b : B p p � E � E if, in every diagram B there is a diagonal filler B d ƒ ƒ b � b � I n � F I n F q q A map ƒ : E � F is a Kan fibration if it has the r.l.p. with respect to all boxes. � Asm C op � def = the full subcategory of Asm C op Kn on 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 G op � � full � Asm G op � Asm C op  ∗ full full full � Asm C op � full � PsER ( Asm ) PsER ( PAsm ) � � Kn

� � � � � � � � � Ps.-equivalence relations of assemblies as cubical assemblies PAsm G op � � full � Asm G op � Asm C op  ∗ full full full � Asm C op � full � PsER ( Asm ) PsER ( PAsm ) � � Kn � Asm G op � Asm C op The functor PsER ( Asm ) � � Theorem.  ∗ (i) is faithful

� � � �� � � � � � � � Ps.-equivalence relations of assemblies as cubical assemblies PAsm G op � � full � Asm G op � Asm C op  ∗ full full full � Asm C op � full � PsER ( Asm ) PsER ( PAsm ) � � Kn � Asm G op � Asm C op The functor PsER ( Asm ) � � Theorem.  ∗ (i) is faithful

� � � � � �� � � � � � � � � Ps.-equivalence relations of assemblies as cubical assemblies PAsm G op � � full � Asm G op � Asm C op  ∗ full full full full up to homotopy � Asm C op � � � full � PsER ( Asm ) PsER ( PAsm ) Kn � Asm G op � Asm C op The functor PsER ( Asm ) � � Theorem.  ∗ (i) is faithful (ii) is full “up to homotopy”, in the sense that for every g :  ∗ ( G ) →  ∗ ( H ) in Asm C op there is ƒ : G → H in PsER ( Asm )  ∗ ( G ) g 〈⊥ ! , id 〉 � commutes for some h �  ∗ ( H ) I ×  ∗ ( G ) such that h : I ×  ∗ ( G ) →  ∗ ( H ) . 〈⊤ ! , id 〉  ∗ ( ƒ )  ∗ ( G )

� � � � �� � � � � � � � Ps.-equivalence relations of assemblies as cubical assemblies PAsm G op � � full � Asm G op � Asm C op  ∗ full full full full up to homotopy � Asm C op � full � PsER ( Asm ) PsER ( PAsm ) � � � � Kn � Asm G op � Asm C op The functor PsER ( Asm ) � � Theorem.  ∗ (i) is faithful (ii) is full “up to homotopy” (iii) maps a ps.-equivalence relation G in PsER ( Asm ) to a Kan fibrant object in Asm C op .

� � � � � � �� � � � � � Ps.-equivalence relations of assemblies as cubical assemblies PAsm G op � � full � Asm G op � Asm C op  ∗ full full full full up to homotopy � Asm C op � full � PsER ( Asm ) PsER ( PAsm ) � � � � Kn � Asm G op � Asm C op The functor PsER ( Asm ) � � Theorem.  ∗ (i) is faithful (ii) is full “up to homotopy” (iii) maps a ps.-equivalence relation G in PsER ( Asm ) to a Kan fibrant object in Asm C op . 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 G op � � full � Asm G op � Asm C op  ∗ full full full full up to homotopy � Asm C op � full � PsER ( Asm ) PsER ( PAsm ) � � � � Kn � Asm G op � Asm C op The functor PsER ( Asm ) � � Theorem.  ∗ (i) is faithful (ii) is full “up to homotopy” (iii) maps a ps.-equivalence relation G in PsER ( Asm ) to a Kan fibrant object in Asm C op . 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 ƒ 0 G 0 H 0 g 0 e 1 d 1 ƒ 1 in PsER ( Asm ) . Theorem. Consider G 1 H 1 g 1 e 2 � d 2 � ƒ 0 G 0 H 0 g 0 The following are equivalent: H 0 (i) ƒ and g represent the same arrow in Asm ex , i.e. ƒ 0 e 1 there is h : G 0 → H 1 in Asm such that G 0 h � H 1 g 0 e 2 � H 0  ∗ ( ƒ ) �  ∗ ( H ) are homotopically equivalent in � (ii) the maps  ∗ ( G ) C .  ∗ ( g )