Retrocells Robert Par e CT2019 Edinburgh, Scotland July 10, 2019 - PowerPoint PPT Presentation

Retrocells Robert Par´ e CT2019 Edinburgh, Scotland July 10, 2019 Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 1 / 33

� � � Bimodules • The bicategory B im has rings R , S , T , . . . as objects, bimodules � S as 1-cells, and S - R -linear maps as 2-cells M : R • Composition is ⊗ S S M • N • R R T T • N ⊗ S M • B im is biclosed, ⊗ has right adjoints in each variable � N � T P M � P N ⊗ S M � P ⊘ R M N N � T P = Hom T ( N , P ), P ⊘ R M = Hom R ( M , P ) Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 2 / 33

Biclosed Many bicategories are biclosed • B im : Rings, bimodules, linear maps • P rof : Categories, profunctors, natural transformations • V - P rof : V − with colimits preserved by ⊗ − biclosed − limits • S pan ( A ) : A with pullbacks and locally cartesian closed Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 3 / 33

Scandal Good bicategories (all of the above) are the vertical part of naturally occurring double categories: R ing , C at , V - C at , S pan A But the internal homs ⊘ and � are not double functors! Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 4 / 33

� � � Double categories • A double category is a “category with two sorts of morphisms” • Example: R ing f � S R R S α � M • • N R ′ R ′ S ′ S ′ f ′ Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 5 / 33

� � � C at • Example: C at F � B A A B P : A op × C � Set Q : B op × D � Set φ � P • • Q � Q ( F − , G =) φ : P ( − , =) C C D D G Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 6 / 33

� � � � � � � S pan • Example: S pan A f A A A B B B σ 0 τ 0 h S S S T T T τ 1 σ 1 C C C D D D g Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 7 / 33

� � Left homs • A has left homs if y • ( ) has a right adjoint y \ • ( ) in V ert A x � B A A B • � z y • x in V ert A • • y � y \ x • z z C C Mike Shulman, “Framed bicategories and monoidal fibrations” (TAC 2008) Roald Koudenburg, “On pointwise Kan extensions in double categories” (TAC 2014) Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 8 / 33

Respecting boundaries • y \ • z is covariant in z and contravariant in y β \ • γ � y ′ \ β γ y ′ � y , z � z ′ • z ′ y \ • z � • Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 9 / 33

� � � � � � Respecting boundaries • y \ • z is covariant in z and contravariant in y β \ • γ � y ′ \ β γ y ′ � y , z � z ′ • z ′ y \ • z � • � y • We have evaluation ǫ : y • ( y \ • z ) A A A A A A A A A A A A A A A A y \ • z • y \ • z • = γ � ǫ � B B B B B B B B B B B B • z • z ′ β � y y ′ • • C C C C C C C C C C C C C C C C � y ′ \ • z ′ y \ • z Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 9 / 33

� � � � � � � Respecting boundaries • y \ • z is covariant in z and contravariant in y β \ • γ � y ′ \ β γ y ′ � y , z � z ′ • z ′ y \ • z � • � y • We have evaluation ǫ : y • ( y \ • z ) a � A ′ A ′ A A A A A A A A A A A A • y \ • z ? γ � b � B ǫ � B ′ B ′ B ′ B ′ B B B B B B B • z • z ′ β � y ′ y • • C ′ C ′ C ′′ C ′′ C C C C C C C C C C C C c c ′ ? ? � y ′ \ • z ′ y \ • z Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 10 / 33

� � � � � � � � � � � � � Globular universal ∀ A A A A A A A A ∃ ! A A A A β � x x • y \ • z • • α � B B B B • z B B B B y • C C C C C C C C s.t. A A A A A A A A A A A A A A A A β � x • y \ • z x • • ǫ � α � B B B B B B B B B B • z = B B B B • z y • y y = • • C C C C C C C C C C C C C C C C C C C C Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 11 / 33

� � � � � � � � � � � � � More universal f f � A � A A ′ A ′ A ′ A ′ A ′ A ′ ∀ A A A ∃ ! A β � x x • y \ • z • • α � B B B B • z B B B B y • C C C C C C C C s.t. f f � A � A A ′ A ′ A ′ A ′ A ′ A ′ A A A A A A A A β � x • y \ • z x • • ǫ � α � B B B B B B B B B B • z = B B B B • z y • y y = • • C C C C C C C C C C C C C C C C C C C C Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 12 / 33

� � � � Strong universality Strong universal property: f � A f � A A ′ A ′ A ′ A ′ A A α � β � • • z • • y • x x y \ • z C C B B Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 13 / 33

� � � � � � � � � � � � � � � � � Companions � B is a companion of • In a double category A , a vertical arrow v : A • � B if there are binding cells α and β such that a horizontal arrow f : A 1 A � A f � B f � B A A A A A B A A B β � id f � α � id A • v • id B = id A • id B βα = id f • • A A B B B B B B A A B B 1 A f f 1 A � A A A A 1 A α � � A id A • v A A A • 1 v � � B A A A A B B B = v • v β • α = 1 v • f β � B B B B v • id B • 1 B B B B B 1 A Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 14 / 33

� � � � � � � Properties • Companions, when they exist, are unique up to globular isomorphism We make a choice of companion f ∗ and, following Ronnie Brown, denote the binding cells by corner brackets • We have (1 A ) ∗ ∼ = id A and ( gf ) ∗ ∼ = g ∗ f ∗ • A A A A � A A A A • f ∗ f � B f � B A A B A A A A B B B v � • • f ∗ φ � φ � ψ � �− → = C C C C B B B B v � • w v � • w • • g � D g ∗ � C C g � D D C C C C D D D • w • g ∗ � D D D D • � D D D D gives a bijection between φ ’s and ψ ’s Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 15 / 33

� � � � � � � � � � � � � � � � � Conjoints There is a dual notion of conjoint f ∗ 1 B f � B � B f � B A A B B B B A A B ψ � χ � id f � id A • id B = id A • id B χψ = id f • • f ∗ • A A A A A A B B A A B B 1 A f f 1 B � B B B B 1 B χ � � B B B B f ∗ • id B • 1 f ∗ � A A A A B B B B = ψ • χ = 1 f ∗ f ∗ • • f ∗ f ψ � A A A A id A • f ∗ • 1 A A A A A 1 A Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 16 / 33

� � � � Examples � S • In R ing , f : R f ∗ is S considered as an S - R bimodule f ∗ is S considered as an R - S bimodule � B • In C at , F : A F ∗ = B ( F − , =) and F ∗ = B ( − , F =) � B • In S pan ( A ), f : A A A B B 1 A f and f ∗ is A f ∗ is A A A f 1 A B B A A Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 17 / 33

What strong means • The strong universal property is equivalent to the globular one plus the stability property • ( z • f ∗ ) ∼ y \ = ( y \ • z ) • f ∗ • If every horizontal arrow has a conjoint, then the strong universal property is equivalent to the globular one Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 18 / 33

� � � � � � Left duals • Suppose A left closed � B we can define its left dual • v = v \ � A • For v : A • id B : B • • We have • id B ∼ = id B • v • • w � • ( w • v ) So perhaps we get a lax normal A co � A g f � C � D A A C B B D ? α � • α � v • w � • v • • w • • B B D D A A C C g f Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 19 / 33

� � � � � � � Retrocells A retrocell A A A A f � C A A C • v f ∗ • � α α � v • w is a cell C C C C B B B B in A • • g ∗ B B D D w • g D D D D Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 20 / 33

� � � � � � � � Quintets • Example: In Q ( A ), a cell is a quintet f � B A A B h k C C D D g and a retrocell is a coquintet f � B A A B h k C C C C g Robert Par´ e (Dalhousie University) Retrocells July 10, 2019 21 / 33