A double category take on restriction categories Robert Par´ e CT2018 Azores, Portugal July 13, 2018 Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 1 / 31
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 [CL] R. Cockett, S. Lack, Restriction Categories I: Categories of Partial Maps, Theoretical Computer Science 270 (2002) 223-259 [C] R. Cockett, Introduction to Restriction Categories, Estonia Slides (2010) [DeW] D. DeWolf, Restriction Category Perspectives of Partial Computation and Geometry, Thesis, Dalhousie University, 2017 Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 2 / 31
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) A double category take on restriction categories July 13, 2018 3 / 31
Double categories • There are many instances where we have two kinds of morphism between the same kind of objects: • External/internal • Total/partial • Deterministic/stochastic • Classical/quantum • Linear/smooth • Classical/intuitionistic • Lax/oplax • Strong/weak • Horizontal/vertical Double categories formalize this � and � , and • A double category is a category with two kinds of morphisms, • cells ⇓ relating them Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 4 / 31
� � � � The usual suspects • R el – Sets, functions, relations f � B A A B a ∼ R c ⇒ f ( a ) ∼ S g ( c ) R � • • S ≤ g � C C D D If A is a regular category we can also construct R el ( A ) • � A – A any category – the double category of commutative squares in A f � B A A B h � k g � C C D D There is a subdouble category of pullback squares P b � A f � B A A B α • Q A – A is a 2-category – the double category of quintets in A h � k C C g � D D Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 5 / 31
� � � � � � � � � � � Spans A a category with pullbacks S pan( A ) has same objects as A • horizontal arrows are morphisms of A A A A s 0 • vertical arrows are spans S S • s 1 � C C C f A A A B B B f � B A A B s 0 t 0 α � α • cells S � are commutative diagrams S S S T T T • • T t 1 C C D D s 1 g C C C D D D g • vertical composition uses pullbacks S pan( A ) is a weak double category Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 6 / 31
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. f g = gf R3. gf = gf R4. gf = f gf Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 7 / 31
� � � � � Example Let A be a category. A stable system of monics M is 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 g � B P P B � m ′ m C C A A f m ∈ M ⇒ m ′ ∈ M Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 8 / 31
� � ✤ � � � � � � � � � M -partial morphisms 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 Composition is by pullback (like for spans) The restriction operator is ( m , f ) = ( m , m ) A 0 A 0 A 0 A 0 ¯ ( ) f m m m A A B B A A A A Example Let A = Top and M given by the open subspaces. Then Par M ( Top ) is the category of topological spaces with continuous functions defined on an open subspace Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 9 / 31
Properties (cribbed from [CL]) 2 = f P1. f P2. f gf = gf P3. gf = gf P4. f = f P5. gf = gf P6. f g = f ⇒ f = f g Definition f is total if f = 1 T1. Monos are total T2. f , g total ⇒ gf total T3. gf total ⇒ f total Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 10 / 31
Order (also from [CL]) Definition � B For f , g : A define f ≤ g iff f = gf Theorem ≤ is an order relation compatible with composition. This makes A into a (locally ordered) 2 -category “...seems to be less useful than one might expect” – [CL] Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 11 / 31
� � � The double category Let A be a restriction category 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 (iff gv = wf v ) v • w • C C D D g “Perhaps this will turn out to be more useful than one might expect!” – Me Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 12 / 31
� � � � � � � � � � � � Companions Definition f � B and A v � B are companions if there are given cells (binding cells) A • 1 A � A f � B A A B A A A v � and • id B id A � • v • ǫ • η f � B B B B A A B B 1 B such that 1 A � A A A A 1 A � A 1 A � A f � B f � B A A A A A B A A B A A A id A � • v • η f � = and A A A A B B B = v � id A � • v • id B id A � • id B • v • • id f • 1 v η ǫ f � f � A A B B B B B B A A B B v � B B B B • id B • ǫ 1 B 1 B B B B B 1 B In D c ( A ) every horizontal arrow has a companion, f ∗ = f Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 13 / 31
� � � � � Conjoints Definition f u � B and B � A are conjoints if there are given cells (conjunctions) A • 1 B � B f � B A A B B B B id A � • u u • id B • • α β A A A A A A B B 1 A f such that βα = id f and α • β = 1 u Proposition In D c Par M ( A ) , f has a conjoint if and only if f ∈ M Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 14 / 31
Conjoints, companions, adjoints Definition v u � B is left adjoint to B � A if it is so in V ert A A • • Theorem (1) If f has a companion (conjoint) it is unique up to globular isomorphism (2) If f has companion (conjoint) v and g has companion (resp. conjoint) w then gf has companion w • v (resp. conjoint v • w) (3) Any two of the following conditions imply the third • v is a companion for f • u is a conjoint for f • v is left adjoint to u in V ert A Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 15 / 31
� � � � � � Tabulators v � B in A its tabulator , if it exists, is an object T and a cell τ Given a vertical arrow A • A A t 0 T T • v τ t 1 B B such that for any other cell A A c 0 C C • v γ c 1 B B � T such that γ = τ c there exists a unique horizontal morphism c : C The tabulator is effective if t 0 has a conjoint t ∗ 0 and t 1 has a companion t 1 ∗ and the � v is an isomorphism canonical cell induced by τ , t 1 ∗ • t ∗ 0 Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 16 / 31
� � � � � � � Tabulators Proposition � B has an effective tabulator if and only if v splits In D c ( A ) , v : A • Corollary D c Par M ( A ) has tabulators and they are effective � B is: The tabulator of ( m , v ) : A • A A m m 1 A A 0 A 0 A 0 A 0 A 0 A 0 A 0 A 0 v v B B Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 17 / 31
� � � Double functors and all that � B is a function taking elements of A to similar ones of B A double functor F : A Ff � FA ′ f � A ′ A ′ FA ′ A A FA FA v • v ′ �− → Fv � • Fv ′ • • α F α C ′ C ′ FC ′ FC ′ g � Fg � C C FC FC preserving all compositions and identities There is a category Doub of double categories and double functors Theorem Doub is cartesian closed This tells us that between double functors there are canonically defined horizontal and vertical transformations as well as double modifications relating them Robert Par´ e (Dalhousie University) A double category take on restriction categories July 13, 2018 18 / 31
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