Double Categories The best thing since slice categories (https://www.mscs.dal.ca/ ∼ pare/FMCS1.pdf) Robert Par´ e FMCS Tutorial Mount Allison May 31, 2018 Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 1 / 35
Double categories • A double category is a category with two kinds of morphisms Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 2 / 35
Double categories • A double category is a category with two kinds of morphisms Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 2 / 35
Double categories • A double category is a category with two kinds of morphisms • A double category is two categories with the same objects Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 2 / 35
Double categories • A double category is a category with two kinds of morphisms • A double category is two categories with the same objects Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 2 / 35
Double categories • A double category is a category with two kinds of morphisms • A double category is two categories with the same objects • A double category is a category object in Cat Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 2 / 35
Double categories • A double category is a category with two kinds of morphisms • A double category is two categories with the same objects • A double category is a category object in Cat Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 2 / 35
� � � Double categories • A double category is a category with two kinds of morphisms • A double category is two categories with the same objects • A double category is a category object in Cat p 1 d 0 � A 1 � A : A 2 � A 0 ◦ 1 p 2 d 1 A has • objects A , A ′ , . . . the objects of A 0 f � A ′ , the objects of A 1 • morphisms A f ′ � A ′′ = A f ′ ◦ f f � A ′ � A ′′ • composition A � A . • identities 1 A : A Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 2 / 35
Double categories (cont.) • A 0 also has morphisms – another kind, internal v � ¯ • A A • v • v � ˜ v ¯ v ¯ � ¯ � ˜ • Composition A A = A A A • • • � A • Identities id A : A • Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 3 / 35
� � Double categories (cont.) • A 0 also has morphisms – another kind, internal v � ¯ • A A • v • v � ˜ v v ¯ ¯ � ¯ � ˜ • Composition A A = A A A • • • � A • Identities id A : A • • A 1 has morphisms too – morphisms between external morphisms – cells f � B A A B α ¯ ¯ ¯ ¯ A A B B ¯ f Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 3 / 35
� � � � Double categories (cont.) • A 0 also has morphisms – another kind, internal v � ¯ • A A • v • v � ˜ v v ¯ ¯ � ¯ � ˜ • Composition A A = A A A • • • � A • Identities id A : A • • A 1 has morphisms too – morphisms between external morphisms – cells f f � B � B A A A A B B v � • w • α ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ A A A A B B B B ¯ ¯ f f Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 3 / 35
� � � � � � � � � � � � � � � � � � Double categories (cont.) Cells compose in A 1 f � B A A B f � B v � • w A A B • α ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ v • v � A A A A B B B B = ¯ • ¯ w • w • α • α ¯ ¯ f ˜ ˜ ˜ ˜ v � ¯ • ¯ w A A B B • α ¯ ˜ f ˜ ˜ ˜ ˜ A A B B ˜ f • Also have an “external” composition given by ◦ � A 1 A 2 g g ◦ f f � B � C � C A A B B B C A A C v � v � α • w • x = • x β β ◦ α • • ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ A A B B B B C C A A C C ¯ ¯ g g ◦ ¯ f ¯ f • ◦ and • are associative and unitary on arrows and cells Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 4 / 35
� � � � � � � � � � � � Double categories (cont.) • Interchange � B � C A A B B B C • • • α β ¯ ¯ ¯ ¯ � ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ � ¯ ¯ ¯ ¯ A A A A B B B B B B B B C C C C • • • ¯ α ¯ β ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A A B B B B C C (¯ α ) • ( β ◦ α ) = (¯ β ◦ ¯ β • β ) ◦ (¯ α • α ) • Also identity interchange laws 1 F • 1 v = 1 ¯ id g ◦ id f = id g ◦ f 1 id A = id 1 A v • v Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 5 / 35
Double categories � and � and cells ⇓ tying • So a double category has two kinds of morphisms • them together Many instances of this: • External/internal • Total/partial • Deterministic/stochastic • Classical/quantum • Linear/smooth • Classical/intuitionistic • Lax/oplax • Strong/weak • Horizontal/vertical Double categories formalize this Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 6 / 35
Double categories � and � and cells ⇓ tying • So a double category has two kinds of morphisms • them together Many instances of this: • External/internal • Total/partial • Deterministic/stochastic • Classical/quantum • Linear/smooth • Classical/intuitionistic • Lax/oplax • Strong/weak • Horizontal/vertical Double categories formalize this • Double categories are categories with two related kinds of morphisms Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 6 / 35
� � � � 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) Double Categories May 31, 2018 7 / 35
� � ���� �� � � Slices · · · A 3 A 2 A 1 A 0 A category, A , has a nerve Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 8 / 35
� � � � �� � � � � � �� ���� � Slices · · · A 3 A 2 A 1 A 0 A category, A , has a nerve A 3 A 2 A 1 • Drop the bottom arrows and we get a new category - objects are arrows of A x � C B B C x � ( g ) are commutative triangles - morphisms ( f ) g f A A Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 8 / 35
�� � � � � � � � � � �� ���� � Slices · · · A 3 A 2 A 1 A 0 A category, A , has a nerve A 3 A 2 A 1 • Drop the bottom arrows and we get a new category - objects are arrows of A x � C B B C x � ( g ) are commutative triangles - morphisms ( f ) g f A A • It is the disjoint union of all slices � A A / A Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 8 / 35
� ���� � � � �� � � � � � �� � � � � Slices · · · A 3 A 2 A 1 A 0 A category, A , has a nerve A 3 A 2 A 1 • Drop the bottom arrows and we get a new category - objects are arrows of A x � C B B C x � ( g ) are commutative triangles - morphisms ( f ) g f A A • It is the disjoint union of all slices � A A / A • By dropping the top arrows, we also get a category whose objects are again arrows y � (¯ of A but morphisms ( f ) f ) now are commutative triangles • A A f B B y ¯ ¯ ¯ f A A • We get the disjoint union of all coslices � B B / A Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 8 / 35
� � � � � � � � � Slices (cont.) • We get a double category S lice A • Objects are morphisms of A • Horizontal arrows are slice morphisms (converging triangles) • Vertical arrows are coslice morphisms (diverging triangles) • Cells x � ( g ) ( f ) ( f ) ( g ) y • • z (¯ (¯ f ) f ) (¯ (¯ g ) g ) x ¯ are commutative tetrahedra: need x = ¯ x , z = y and x � C B B B C C g ¯ ¯ f g f ¯ ¯ ¯ A A A A A A y Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 9 / 35
� � � � � � � � � � � � 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 A ¯ ¯ A A 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 • ¯ ¯ ¯ ¯ A A B B s 1 t 1 ¯ ¯ ¯ ¯ ¯ ¯ g A A A B B B g • vertical composition uses pullbacks Robert Par´ e (Dalhousie University) Double Categories May 31, 2018 10 / 35
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