Van Kampen diagrams are bicolimits Pawel Sobocinski, University of - PowerPoint PPT Presentation

Van Kampen diagrams are bicolimits Pawel Sobocinski, University of Southampton (joint work with Tobias Heindel) CT ’08, Calais, 23/06/08

and Span( C ) C (Benabou) in this talk category = category with pullbacks Γ : C → Span( C ) f f C → D C ← C → D − �− → − • Internal cat in = monad in Span( C ) C • takes internal functors to monad Γ morphisms

Plan of talk • Extensive cats, adhesive cats, VK diags • Bicolimits • The theorem & rough proof • Example and future work

� � � � � Extensive Cats (Lawvere; Schanuel; Carboni, Lack & Walters) • A category is extensive when it has coproducts and, for any coproduct diag and comm diag below TFAE: � Z X Y � A + B A B i 1 i 2 i: top row is a coproduct diagram ii: the two squares are pullbacks “coproducts exist and are well-behaved”

� � � � � � � Adhesive Cats (Lack & Sobocinski) • A category is adhesive when, pushouts along monos exist, and for any pushout along mono, and comm cube with rear faces pullbacks, TFAE: C ′ f ′ m ′ � ��������� � � � c A ′ B ′ i: top row is a pushout diagram � ��������� � � g ′ � � D ′ n ′ a b ii: the two squares are pullbacks C m f ���� � � � � � ���� A B d � ��������� � � � � g n D “pushouts along monos exist and are well-behaved”

� � � � � � � Van Kampen diagrams (Lack & Sobocinski, Cockett & Guo) a functor F : J → C is a VK diagram when a cocone κ : F → C for all functors F ′ : J → C , cocones κ ′ : F ′ → C ′ and cartesian nat. trans. γ : st F ′ F → F ′ u � F ′ F ′ TFAE j κ ′ i j � � � � � i. is a colimit diagram κ ′ � C ′ κ ′ γ i i ii. are all γ j F ′ i C ′ C F i � F j F i c pullback diagrams F u κ j � � � � � � C κ i

� � � � � � � VK diagrams F ′ u � F ′ F ′ tfae j κ ′ i j � � � � � i. is a colimit diagram κ ′ � C ′ κ ′ γ i i ii. are all γ j F ′ i C ′ C F i � F j F i c pullback diagrams F u κ j � � � � � � C κ i ii ⇒ i: stability i ⇒ ii: “ co-stability ”

Simple observations • VK diagrams are colimit diagrams • sums in extensive cats are VK cospans • pushouts along monos in adhesive cats are VK squares • strict initial objects are VK empty diagrams

� � Bicolimits (Kelly & Street) (pseudo) functor M : J → B bic M ∈ B & pseudo-cocone κ : M → bic M M u � M j M i κ u κ id i = 1 κ i � � � � κ i � � � � � κ v ◦ u = ( κ v ◦ M u ) • κ u κ j bic M

� � � � � � � Universal property & ∃ ! for all pseudo cocones ∃ λ : M → X exists pseudo mediating morphism : & isos ϕ : λ i ⇒ ( ∆ h ) ◦ κ h : bic M → X M u � M j M u � M j M i M i � � � � � λ u � ϕ j � � � � ����������� st � � � � � � κ u � κ j κ j � � κ i � ϕ i � = λ i bic M λ i bic M λ j � ������ � ������ h h X X h, h ′ : bic M → X given , any modification ! ψ : ∆ h ◦ κ → ∆ h ′ ◦ κ is for unique ξ : h ⇒ h ′ ( ∆ ξ ) ◦ κ

Univ prop restated a pseudo cone κ : M → C is universal for when h : C → D κ , D for all , ψ : ∆ h ◦ κ ⇒ ∆ h ′ ◦ κ h ′ : C → D exists ! ξ : h ⇒ h ′ st ψ = ( ∆ ξ ) ◦ κ is a bicolimit diagram iff : κ for all pseudo cocones i. λ : M → D exists pseudo mediating morphism { h : C → D, ϕ } where is universal for h κ ii. all arrows h : C → D are universal for κ , D

The theorem Γ : C → Span( C ) Suppose that has -colimits. C J is Van Kampen in C κ : F → C is a bicolimit diagram in Span( C ) Γ κ

Vague idea • There is a 1-1 correspondence between: • 2-dimensional “pseudo” diagrams in Span( C ) • 3-dimensional diagrams (with some pullbacks) in C

Rough proof In the presence of -colimits: J stability ⇔ for all pseudo cocones λ : M → D exists pseudo mediating morphism { h : C → D, ϕ } where is universal for h κ co-stability ⇔ all arrows h : C → D are universal for κ , D

VK diags are bicolimits • Being a VK diagram in is a universal C property, in ! (a bigger universe) Span( C ) • Corollaries: • init. object in is strict iff preserves it Γ C • is extensive iff it has coproducts and Γ C preserves them • ...

� � � � � � � � � � � � � � Little example - symmetries 2 2 � ( 1 ( 1 1 ) 1 ) � � � ! ! � � � � “ id � � � � � � � � ” Γ � � � � � � id 1 1 1 1 � �− → � � � � � � � � � � � � � � � � � � � 1 1 � � 1 1 is not VK in Ord f 2 2 • Consider ( 1 ( 1 ( 1 ( 1 1 ) 1 ) 1 ) 1 ) � � � � � � “ id � � � � � � � � ” Mat( Ord f ) ( tw � id ) � � � � � � � � id 1 1 1 1 � � � � • objects are natural � � � � � � � � � � � � � � 2 2 2 2 � � � � 1 1 2 2 ( 1 ( 1 ( 1 ( 1 1 ) 1 ) 1 ) 1 ) � � � � � � numbers � � � � � � � � ( id tw ) � � ( tw � tw ) � � � � � � � 1 1 1 1 � � � � � � • arrows are � � � � � � � � � � � � 2 2 2 2 � � � � k × l k → l 1 1 But there are only matrices 2 mediating morphisms!

Future work • Quasiadhesive categories don’t seem to be common in the wild (Johnstone, Lack & S) Γ ′ : C → Par ( C ) • It seems that categories in certain pushouts are preserved by are more common Γ ′ ( is a smaller universe than ) Span( C ) Par ( C )