Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Towards a Characterization of the Double Category of Spans Evangelia Aleiferi Dalhousie University Category Theory 2017 July, 2017 Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 1 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Motivation Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 2 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Theorem (Lack, Walters, Wood 2010) For a bicategory B the following are equivalent: i. There is an equivalence B ≃ S pan ( E ) , for some finitely complete category E . ii. B is Cartesian, each comonad in B has an Eilenberg-Moore object and every map in B is comonadic. iii. The bicategory M ap ( B ) is an essentially locally discrete bicategory with finite limits, satisfying in B the Beck condition for pullbacks of maps, and the canonical functor C : S pan ( M ap ( B )) → B is an equivalence of bicategories. Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 3 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Cartesian Bicategories Definition (Carboni, Kelly, Walters, Wood 2008) A bicategory B is said to be Cartesian if: i. The bicategory M ap ( B ) has finite products ii. Each category B ( A , B ) has finite products. iii. Certain derived lax functors ⊗ : B × B → B and I : ✶ → B , extending the product structure of M ap ( B ), are pseudo. Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 4 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Cartesian Bicategories Definition (Carboni, Kelly, Walters, Wood 2008) A bicategory B is said to be Cartesian if: i. The bicategory M ap ( B ) has finite products ii. Each category B ( A , B ) has finite products. iii. Certain derived lax functors ⊗ : B × B → B and I : ✶ → B , extending the product structure of M ap ( B ), are pseudo. Examples 1. The bicategory R el ( E ) of relations over a regular category E . Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 4 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Cartesian Bicategories Definition (Carboni, Kelly, Walters, Wood 2008) A bicategory B is said to be Cartesian if: i. The bicategory M ap ( B ) has finite products ii. Each category B ( A , B ) has finite products. iii. Certain derived lax functors ⊗ : B × B → B and I : ✶ → B , extending the product structure of M ap ( B ), are pseudo. Examples 1. The bicategory R el ( E ) of relations over a regular category E . 2. The bicategory S pan ( E ) of spans over a finitely complete category E . Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 4 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Question For a finitely complete category E , can we characterize the double category S pan ( E ) of • objects of E • arrows of E vertically • spans in E horizontally as a Cartesian double category? Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 5 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Cartesian double categories Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 6 / 29
� Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Definition A (pseudo) double category D consists of: i. A category D 0 , representing the objects and the vertical arrows. ii. A category D 1 , representing the horizontal arrows and the cells written as M � A B | � α g f � � D C | � N S , T � D 0 , D 0 ⊙ U � D 1 and D 1 × D 0 D 1 � D 1 such iii. Functors D 1 that S ( U A ) = A = T ( U A ), S ( M ⊙ N ) = SN and T ( M ⊙ N ) = TM . a � M ⊙ ( N ⊙ P ) , iv. Natural isomorphisms ( M ⊙ N ) ⊙ P l r � M and M ⊙ U SM � M , satisfying the pentagon U TM ⊙ M and triangle identities. Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 7 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions The objects, the horizontal arrows and the cells with source and target identities form a bicategory H ( D ) . Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 8 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions The objects, the horizontal arrows and the cells with source and target identities form a bicategory H ( D ) . The double categories together with the double functors and the vertical natural transformations form a 2-category DblCat . Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 8 / 29
� � � � Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Definition A double category D is said to be Cartesian if there are adjunctions ∆ ! D ⊥ D × D and D ⊥ in DblCat . ✶ × I Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 9 / 29
� � � � Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Definition A double category D is said to be Cartesian if there are adjunctions ∆ ! D ⊥ D × D and D ⊥ in DblCat . ✶ × I Examples 1. The double category R el ( E ) of relations over a regular category E Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 9 / 29
� � � � Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Definition A double category D is said to be Cartesian if there are adjunctions ∆ ! D ⊥ D × D and D ⊥ in DblCat . ✶ × I Examples 1. The double category R el ( E ) of relations over a regular category E 2. The double category S pan ( E ) of spans over a finitely complete category E Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 9 / 29
� � � � Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Definition A double category D is said to be Cartesian if there are adjunctions ∆ ! D ⊥ D × D and D ⊥ in DblCat . ✶ × I Examples 1. The double category R el ( E ) of relations over a regular category E 2. The double category S pan ( E ) of spans over a finitely complete category E 3. The double category V − M at , for a Cartesian monoidal category V with coproducts such that the tensor distributes over them Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 9 / 29
� � � � � � Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Definition A double category is called fibrant if for every niche of the form A C g f � � D B | � M | � C and a cell there is a horizontal arrow g ∗ Mf ∗ : A g ∗ Mf ∗ � C A | ζ g f � � D , B | M M ′ � C ′ A ′ | � so that every cell can be factored uniquely through ζ . gk fh � � D , B | � M Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 10 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Examples Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 11 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Examples 1. R el ( E ), E a regular category Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 11 / 29
Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Examples 1. R el ( E ), E a regular category 2. S pan ( E ), E finitely complete category Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 11 / 29
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