Isotropic Intercategories Robert Par e (with Marco Grandis) - PowerPoint PPT Presentation

Isotropic Intercategories Robert Par´ e (with Marco Grandis) Halifax August 2016 Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 1 / 21

� � � � � � � Intercategory A kind of lax triple category A f � B A A A ◦ B B b α a v f ′ • A ′ A ′ A ′ � B ′ B ′ B ′ a : ◦ φ C C C v ′ • w ′ • σ ′ c C ′ C ′ C ′ D ′ D ′ ◦ g ′ Transversal: · , 1 A , strictly unitary and associative Horizontal: ◦ , id A , associative and unitary up to isomorphism Vertical: • , Id A , associative and unitary up to isomorphism Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 2 / 21

Interchange � χ : σ 1 | σ 2 → σ 1 σ 2 � σ 3 | σ 4 − � σ 3 σ 4 � δ : Id f 1 | f 2 − → Id f 1 | Id f 2 µ : id v 1 − → id v 1 id v 2 v 2 τ : Id id A − → id Id A Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 3 / 21

�� �� � � Weak category object in L x D bl d 0 � B � C � A ◦ id d 1 Laxity of ◦ - χ , δ Laxity of id - µ , τ Equivalently: a weak category object in C x D bl � X 1 � X 2 � X 0 • Id Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 4 / 21

� � � � � � � � � � � � � Spans of spans A a category with pullbacks S pan 2 A · � · · � · · · � · · � · � · · · · � · · · · · · · � · · · · · · · · · · · · · · · · · · � · · � · · · � · · · � · · � χ, δ, µ, τ all isomorphisms Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 5 / 21

� � � � � � � � � � � � � Spans of cospans A category with pullbacks and pushouts S pan C osp A · � · · � · · · � · · � · · · · · · · · · · · · � � · · · · � · · � · · · � · · � · · � · · � · · · · · · · · · · · · · χ is not an isomorphism It’s the canonical comparison from a pushout of pullbacks to a pullback of pushouts δ, µ, τ are isomorphisms Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 6 / 21

� � � Gray categories A category A enriched in (2- Cat , ⊗ , 1 ) � Z 2-functors Φ : X ⊗ Y � Z quasi-functors of two variables Ψ : X × Y � Z , � Z 2-functors Ψ( X , − ) : Y Ψ( − , Y ) : X Ψ( x , y ) is not defined, but Ψ( x , Y ) � Ψ( X ′ , Y ) Ψ( X ′ , Y ) Ψ( X , Y ) Ψ( X , Y ) ψ ( x , y ) � Ψ( X ′ , y ) Ψ( X , y ) Ψ( X , Y ′ ) Ψ( X , Y ′ ) Ψ( X ′ , Y ′ ) Ψ( X ′ , Y ′ ) Ψ( x , Y ′ ) Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 7 / 21

� � �� For a Gray category, composition will be a quasi-functor � A ( A , C ) A ( A , B ) × A ( B , C ) There is no horizontal composition of 2-cells, only whiskering If we define Ψ( x , y ) = Ψ( x , Y )Ψ( X ′ , y ) we get a lax functor � Z Ψ : X × Y satisfying: � Ψ( xx ′ , y ) is an identity • Ψ( x , 1)Ψ( x ′ , y ′ ) � Ψ( x , yy ′ ) is an identity • Ψ( x , y )Ψ(1 , y ′ ) � Ψ(1 , 1) is an identity • 1 We can put all of the homs of a Gray category together to get � � A ( A , B ) � A ( A , B ) × A ( B , C ) � Ob A ◦ id A , B , C A , B a category object in L x D bl , i.e. an intercategory A l Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 8 / 21

� � � � � � � General cube looks like f � B A A A B B f a � f � B a : A A A A B B B A A B B σ σ ′ g A A A • σ ′ A A A B B g � � � � ∗ ∗ ∗ ∗ ∗ ∗ Id Id χ is not an isomorphism, but χ and χ are identities Id ∗ ∗ ∗ ∗ ∗ ∗ Id δ, µ, τ are identities Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 9 / 21

� � � If instead we define Ψ( x , y ) = Ψ( X , y )Ψ( x , Y ′ ) we get a colax functor, which gives a different intercategory A c A A A A A A f a : A A A A A A A A σ ′ � B B B g f B B B B B Now � id ∗ ∗ � � ∗ ∗ ∗ ∗ � id χ and χ are identities ∗ ∗ ∗ ∗ ∗ ∗ id id Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 10 / 21

� � � � � “Symmetric” case A better way of representing a Gray category as an intercategory A s A general cube � B A A A B B � B a : A A A A B B B C C C σ ′ � C C C D D The basic cells are co-quintets (No choice!) - The horizontal composition has to be the lax one - The vertical composition is the oplax one - The (co-)quintet composition determines these � � � � � � � � id ∗ ∗ ∗ ∗ ∗ ∗ Id ∗ ∗ ∗ ∗ ∗ ∗ id Id We have χ , χ , χ , χ all identities ∗ ∗ ∗ ∗ ∗ id ∗ ∗ ∗ ∗ ∗ ∗ Id ∗ id Id Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 11 / 21

� � � � � � � � � � � � Transversal invariance An intercategory A is transversally invariant if for every open box of cells � B A A A B B α � B ′ A ′ A ′ A ′ B ′ B ′ σ φ ψ � D C C C D D β C ′ C ′ C ′ D ′ D ′ D ′ with α, β, φ, ψ transversal isomorphisms, there exist a basic cell � B ′ A ′ A ′ B ′ σ ′ C ′ C ′ D ′ D ′ � σ ′ filling it closing the box and a transversally invertible cube s : σ I.e. basic cells are transportable along isomorphisms Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 12 / 21

� � � � � � � � � � � � Cylindrical intercategories A is (horizontally) cylindrical if its vertical arrows and cells are identities � B A A A B B b α a � D C C C D D A A B B α � A A A • θ • θ C C C D D D a C C C D D Proposition If A is horizontally cylindrical, it is transversally invariant if and only if all transversally invertible horizontal cells have basic companions Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 13 / 21

� � � � � � � Cylindrification Theorem If A is a transversally invariant intercategory, then there is a cylindrical intercategory ZA gotten by taking the vertical arrows to be Id A and vertical cells to be Id a and the rest full � A is strict-pseudo. Furthermore ZA is transversally on this. The inclusion Φ : ZA invariant Proof. A general cube looks like � B A A A B B b α a Id A • � D C C C D D Id a A A A • Id C • Id D θ a C C C D D These compose transversally and horizontally as in A There is no choice for the vertical composite Id A • Id A : it has to be Id A , so the inclusion Φ only preserves composition up to isomorphism λ ′ = ρ ′ : Id A • Id A � Id A . Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 14 / 21

� � � � � � � For vertical composition of basic cells � B A A B Id A � • Id B • σ � B A A A A B B B Id A � • Id B • θ A A B B we use transversal invariance to choose (arbitrarily) a basic cell σ ∗ θ and an invertible cube g ( σ, θ ) � σ ∗ θ g ( σ, θ ) : σ • θ � B A A A B B Id A � • � B A A A B B A A A λ ′ Id A � • σ ∗ θ • Id A • Id B A A A A A B B Vertical composition of cubes is by conjugation a ∗ b = g − 1 · ( a • b ) · g Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 15 / 21

� � � � � � � � � � � � Quintets Let A be (horizontally) cylindrical and transversally invariant. We wish to construct a new intercategory QA whose basic cells are quintets A general cube in QA will look like ◦ � B A A A B B α ◦ A ′ A ′ A ′ ◦ � B ′ B ′ B ′ a : φ C C C ◦ ◦ σ ′ C ′ C ′ C ′ D ′ D ′ ◦ � y f � B f � B � D A A ◦ B A A ◦ B B B ◦ D A basic cell ◦ ◦ y is a cell in A x σ σ ◦ ◦ ◦ C C D D A A C C C C D D g x g Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 16 / 21

� � � � � � � � � � � Composition of arrows (transversal, horizontal, vertical) and of horizontal and vertical cells is performed in A Horizontal composition of basic cells f � B h � E A A B B B E y ? x z σ θ C C D D D D F F g k If horizontal composition were strict it would be: f � B h � E z � F A A B B B E E E F Id f θ � B � D � F A A A A B B B B B B B D D D D D D F F F y f k σ Id k A A C C C C D D D D F F x g k Robert Par´ e (Dalhousie University) Isotropic Intercategories Halifax August 2016 17 / 21