Seeing double (https://www.mscs.dal.ca/ pare/FMCS2.pdf) Robert Par - PowerPoint PPT Presentation

Seeing double (https://www.mscs.dal.ca/ ∼ pare/FMCS2.pdf) Robert Par´ e FMCS Tutorial Mount Allison June 1, 2018 Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 1 / 34

� � � � � Before we start Double functors � S lice ( B ) S lice ( A ) are in bijection with natural transformations F A A B B t G The associated double functor is given (on the objects) by FA A Ff FA ′ FA ′ f � �− → A ′ tA ′ GA ′ Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 2 / 34

Words of wisdom If you want something done right you have to do it yourself. And, you have to do it right. Micah McCurdy Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 3 / 34

The plan • The theory of restriction categories is a nice, simply axiomatized theory of partial morphisms • It is well motivated with many examples and has lots of nice results • But it is somewhat tangential to mainstream category theory • The plan is to bring it back into the fold by taking a double category perspective • Every restriction category has a canonically associated double category • What can double categories tell us about restriction categories? • What can restriction categories tell us about double categories? • References - R. Cockett, S. Lack, Restriction Categories I: Categories of Partial Maps, Theoretical Computer Science 270 (2002) 223-259 - R. Cockett, Introduction to Restriction Categories, Estonia Slides (2010) - D. DeWolf, Restriction Category Perspectives of Partial Computation and Geometry, Thesis, Dalhousie University, 2017 Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 4 / 34

Restriction categories Definition A restriction category is a category equipped with a restriction operator ¯ f f � B � A � A A satisfying R1. f ¯ f = f R2. ¯ g ¯ f ¯ g = ¯ f R3. g ¯ g ¯ f = ¯ f R4. ¯ gf = f gf Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 5 / 34

� � � � � � Example Let A be a category and M a subcategory such that (1) m ∈ M ⇒ m monic (2) M contains all isomorphisms (3) M stable under pullback: for every m ∈ M and f ∈ A as below, the pullback of m along f exists and is in M ¯ f � B P P B � m ′ m C C A A f m ∈ M ⇒ m ′ ∈ M Par M A has the same objects as A but the morphisms are isomorphism classes of spans A 0 A 0 m f A A B B with m ∈ M Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 6 / 34

� � � � � � � ✤ � Composition is by pullback The restriction operator is ( m , f ) = ( m , m ) A 0 A 0 A 0 A 0 ¯ ( ) m f m m A A B B A A A A Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 7 / 34

� � � The double category Let A be a restriction category Definition � B is total if ¯ f : A f = 1 A Proposition The total morphisms form a subcategory of A The double category D c ( A ) associated to a restriction category A has • The same objects as A • Total maps as horizontal morphisms • All maps as vertical morphisms f � B A A B • There is a unique cell if and only if gv = wf ¯ v v ⇒ • m • C C D D g Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 8 / 34

� � � Theorem D c ( A ) is a double category Remark � B, f ≤ g ⇔ f = g ¯ C & L define an order relation between f , g : A f Makes A into a 2 -category. They say “seems to be less useful than one might expect” There is a cell f � B A A B v ⇒ • w • C C D D g if and only if gv ≤ wf . So our D c ( A ) is not far from that 2-category. Perhaps it will turn out to be more useful than they might expect! Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 9 / 34

� � � � � � � � � Example In D c Par M ( A ) there is a cell if and only if there exists a (necessarily unique) morphism h f A A A B B B m n h A 0 A 0 A 0 B 0 B 0 B 0 w v C C C D D D g Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 10 / 34

� � � � � Companions Proposition In D c ( A ) every horizontal arrow has a companion, f ∗ = f Proof. f � B A A B 1 · f = 1 · f · ¯ f � ⇒ f • • 1 B B B B 1 1 � A A A A f · 1 = f · 1 · ¯ ⇒ 1 1 • • f A A B B f Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 11 / 34

� � � � � Conjoints Proposition In D cPar M ( A ) , f has a conjoint if and only if f ∈ M Proof. Assume f has conjoint ( m , g ), then there are α, β f � B A A B B B B B m m β � B α � B 0 and A A A A B 0 B 0 B 0 B 0 B 0 B 0 B 0 B B B g g A A A A A A B B f So m α g = fg = β = m which implies α g = 1 Thus α is an isomorphism and f = m α ∈ M Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 12 / 34

� • If we suspect that A is of the form D c Par M ( A ) we can recover M as those horizontal arrows having a conjoint • Is the requirement of stability under pullback of conjoints a good double category notion? � B always has a companion f ∗ , and if it also • In D c ( A ), a horizontal arrow f : A has a conjoint f ∗ then f ∗ ⊣ f ∗ so A f ∗ • f ∗ • A is a comonad, i.e. an idempotent ≤ id A Proposition In D c ( A ) , f ∗ • f ∗ = ¯ f ∗ Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 13 / 34

� � � Tabulators Proposition D cPar M ( A ) has tabulators and they are effective Proof. � B , the tabulator is Given ( m , v ) : A • m � A A 0 A 0 A m A 0 A 0 A 0 A 0 A 0 A 0 A 0 A 0 v v � A 0 A 0 B B � B has a tabulator if and only if ¯ Conjecture: In a general D c ( A ), v : A v splits • Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 14 / 34

Classification of vertical arrows • The original definition of elementary topos was in terms of a partial map classifier � A B • � ˜ B A • In a topos, relations are classifiable � A B • � Ω A B • For profunctors � A B • � ( Set A ) op B provided A is small • How do we formalize this in a general double category? Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 15 / 34

Classification (Beta version) • The desired bijection v � A B • v � ˜ � B A gives eA : ˜ � A and hA : A � ˜ A A • • We express our definition in terms of eA Definition Let A be a double category and A an object of A . We say that A is classifying if we are given an object ˜ A and a vertical morphism eA : ˜ � A with the following A • universal properties: Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 16 / 34

� � � � � � � � � � � � � A there exist a horizontal arrow � � ˜ (1) For every vertical arrow v : B v : B A and • a cell � v � ˜ ˜ B B A A ǫ v • • v eA A A such that for every cell α g v � B � � ˜ ˜ D D B B B B B A A • α • w eA C C C C A A A A f there exists a unique cell ¯ α such that g g v v � B � � ˜ ˜ � B � � ˜ ˜ D D B B B B B A A D D B B B B B A A ǫ v = • α ¯ • • • α • w v w eA eA C C C C A A A A C C C C A A A A f f Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 17 / 34

� � � � � � � � � � � � � (2) For every cell g � B D D B β • • v w A A there exists a unique cell ¯ ¯ β such that g g v � v � � B � ˜ ˜ ˜ ˜ D D B B B B B A A D D B B B B A A ¯ ¯ • • β = β ǫ v id id w � � ˜ ˜ ˜ ˜ D D D D D A A A A A A • • • • eA w ǫ w • • w eA A A A A A A Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 18 / 34

� � � � � � � Complete classification • How do we understand this? • Take a more global approach Assume A is companionable, i.e. every horizontal arrow f has a companion f ∗ Then we get a (pseudo) double functor � A ( ) ∗ : Q H or A f f � B � B A A B A A B α �− → h ∗ h k • • k α ∗ C C D D C C D D g g Exercise! Definition Say that A is classifying if ( ) ∗ has a down adjoint ˜ ( ), i.e. a right adjoint in the vertical direction Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 19 / 34

� � � � Bijections The adjunction can be formalized in terms of bijections B v � � ˜ B A v • A � A there exists a � � ˜ More precisely, for v : B v : B A and an isomorphism • B B B B B B B B ( � v ) ∗ • ˜ ˜ ˜ ˜ A A A A ∼ • v = eA • A A A A A A A A This can be expressed without mention of ( ) ∗ because we have a bijection Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 20 / 34

� � � � � � � � Bijections (cont.) g g � B � B D D D B B g � v � B � ˜ ˜ • ( � v ) ∗ D D B B B B B A A ˜ ˜ ˜ ˜ A A A A w ⇒ ⇒ • • • w eA C C C C A A A A eA f C C C C A A A A f f Yonedafication now yields the single-object definition Robert Par´ e (Dalhousie University) Seeing double June 1, 2018 21 / 34