Towards a Characterization of the Double Category of Spans

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

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'Span(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 Map(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 : Span(Map(B)) → B is an equivalence of bicategories.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 3 / 29 Examples 1. The bicategory Rel(E) of relations over a regular category E. 2. The bicategory Span(E) of spans over a ﬁnitely complete category E.

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 Map(B) has finite products ii. Each category B(A, B) has finite products. iii. Certain derived lax functors ⊗ : B × B → B and I : 1 → B, extending the product structure of Map(B), are pseudo.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 4 / 29 2. The bicategory Span(E) of spans over a ﬁnitely complete category E.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Cartesian Bicategories

Examples 1. The bicategory Rel(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

Examples 1. The bicategory Rel(E) of relations over a regular category E. 2. The bicategory Span(E) of spans over a ﬁnitely complete category E.

For a ﬁnitely complete category E, can we characterize the double category Span(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 D0, representing the objects and the vertical arrows.

ii. A category D1, representing the horizontal arrows and the cells written as M A | / B f α g C | // D N

S,T / U / / iii. Functors D1 D0, D0 D1 and D1 ×D0 D1 D1 such that S(UA) = A = T (UA), S(M N) = SN and T (M N) = TM. iv. Natural isomorphisms (M N) P a / M (N P), l r UTM M / M and M USM / M, satisfying the pentagon and triangle identities.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 7 / 29 The double categories together with the double functors and the vertical natural transformations form a 2-category

DblCat.

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 Examples

1. The double category Rel(E) of relations over a regular category E 2. The double category Span(E) of spans over a ﬁnitely complete category E 3. The double category V − Mat, for a Cartesian monoidal category V with coproducts such that the tensor distributes over them

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

∆ ! ' " 1 De ⊥ D × D and D a ⊥ in DblCat. × I

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 9 / 29 2. The double category Span(E) of spans over a ﬁnitely complete category E 3. The double category V − Mat, for a Cartesian monoidal category V with coproducts such that the tensor distributes over them

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

∆ ! ' " 1 De ⊥ D × D and D a ⊥ in DblCat. × I

Examples

1. The double category Rel(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 3. The double category V − Mat, for a Cartesian monoidal category V with coproducts such that the tensor distributes over them

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

∆ ! ' " 1 De ⊥ D × D and D a ⊥ in DblCat. × I

Examples

1. The double category Rel(E) of relations over a regular category E 2. The double category Span(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

∆ ! ' " 1 De ⊥ D × D and D a ⊥ in DblCat. × I

Examples

A C f g B | // D M

∗ there is a horizontal arrow g Mf∗ : A | // C and a cell

∗ g Mf∗ A | / C f ζ g B | // D, M

0 M0 0 A | / C so that every cellfh gk can be factored uniquely through ζ. B | / D, M

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 10 / 29 1. Rel(E), E a regular category 2. Span(E), E ﬁnitely complete category 3. V − Mat, V a Cartesian monoidal category with coproducts such that the tensor distributes over them

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 2. Span(E), E ﬁnitely complete category 3. V − Mat, V a Cartesian monoidal category with coproducts such that the tensor distributes over them

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Examples

1. Rel(E), E a regular category

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 11 / 29 3. V − Mat, V a Cartesian monoidal category with coproducts such that the tensor distributes over them

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Examples

1. Rel(E), E a regular category 2. Span(E), E ﬁnitely complete 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. Rel(E), E a regular category 2. Span(E), E ﬁnitely complete category 3. V − Mat, V a Cartesian monoidal category with coproducts such that the tensor distributes over them

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 11 / 29 Theorem Consider a fibrant double category D such that: i. the vertical category D0 has finite products ×, p, r, I , ii. every H(D)(A, B) has finite products ∧, > and iii. the lax double functors × and I are pseudo. Then D is Cartesian.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Theorem Consider a fibrant double category D such that: i. the vertical category D0 has finite products ×, p, r, I and ii. every H(D)(A, B) has finite products ∧, >. ∗ ∗ Then the formula M × N = (p Mp∗) ∧ (r Nr∗) and the terminal horizontal arrow >I ,I extend the product of D0 to lax double functors

× : D × D → D and I : 1 → D.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 12 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

× : D × D → D and I : 1 → D.

Theorem Consider a fibrant double category D such that: i. the vertical category D0 has finite products ×, p, r, I , ii. every H(D)(A, B) has finite products ∧, > and iii. the lax double functors × and I are pseudo. Then D is Cartesian.

Eilenberg-Moore Objects

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 13 / 29 • Vertical arrows the comonad morphisms, i.e. vertical arrows f : X → Y together with a cell ψ:

P X | / X f f Y | / Y R which is compatible with the comonad structure. • Horizontal arrows the horizontal comonad maps, i.e. horizontal arrows 0 F : X | // X , together with a cell α:

P F 0 X | / X | / X

0 0 X | / X | / X , F P0 compatible with the counit and the comultiplication.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Consider the double category Com(D) with: • Objects the comonads in D:(X , P : X | // X ), equipped with globular cells δ : P → P P and : P → UX satisfying the usual conditions.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 14 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions Consider the double category Com(D) with: • Objects the comonads in D:(X , P : X | // X ), equipped with globular cells δ : P → P P and : P → UX satisfying the usual conditions. • Vertical arrows the comonad morphisms, i.e. vertical arrows f : X → Y together with a cell ψ:

P X | / X f f Y | / Y R which is compatible with the comonad structure. • Horizontal arrows the horizontal comonad maps, i.e. horizontal arrows 0 F : X | / X , together with a cell α:

P F 0 X | / X | // X

0 0 X | / X | / X , F P0 compatible with the counit and the comultiplication.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 14 / 29 Deﬁnition We say that the double category D has Eilenberg-Moore objects for comonads if the inclusion double functor

J : D → Com(D)

has a right adjoint. Example The double category Span(E), for a ﬁnitely complete category E, has Eilenberg-Moore objects for comonads.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

• Cells that are compatible with the horizontal and the vertical structure.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 15 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

• Cells that are compatible with the horizontal and the vertical structure.

Deﬁnition We say that the double category D has Eilenberg-Moore objects for comonads if the inclusion double functor

J : D → Com(D)

has a right adjoint. Example The double category Span(E), for a ﬁnitely complete category E, has Eilenberg-Moore objects for comonads.

Deﬁnition If a double category D has Eilenberg-Moore objects, then for every comonad (X , P) there is an object EM and a universal comonad morphism

U EM | // EM e e X | / X . P

∼ ∗ If D is ﬁbrant and P = e e∗, we say that D has strong Eilenberg-Moore objects.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 16 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Towards the characterization of spans

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 17 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Deﬁnition We say that a ﬁbrant and Cartesian double category has the Frobenius property if for every object X and its pullback diagram of vertical arrows

X d / XX d d1 XX / XXX 1d

the cell d∗ U d∗ XX | // X | // X | // XX d d U U XX | / XX XX | / XX 1d d1 | // | / | // XX XXX XXX ∗ XX (1d)∗ U (d1) is invertible.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 18 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Deﬁnition (Pare, Grandis) A ﬁbrant double category D has tabulators if for every horizontal arrow F : X | // Y there is an object T and a cell

U T | / T

t1 τ t2 X | // Y F

such that for every other object H and every cell β : UH → F , there is a unique vertical arrow f : H → T such that β = τUf . ∼ ∗ We say that the tabulators are strong if F = t2∗t1 .

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 19 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Conjecture If a double category is ﬁbrant, Cartesian and it satisﬁes the Frobenius property, then for every cell of the form

P X | // X

X | // X UX

there exists a comonad structure (P, , δ), unique up to isomorphism.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 20 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 21 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Corollary (of the conjecture) In a fibrant and Cartesian double category D, that satisfies the Frobenius property, for every horizontal arrow F : X | // Y , the cartesian filling of the niche X × Y X × Y d×Y X ×d X × X × Y | / X × Y × Y X ×F ×Y admits a comonad structure, unique up to isomorphism.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 22 / 29 Moreover, if the Eilenberg-Moore objects are strong, the tabulators are strong.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Corollary (of the conjecture) A ﬁbrant and Cartesian double category D that satisﬁes the Frobenius property and has Eilenberg-Moore objects, has tabulators.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 23 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Corollary (of the conjecture) A ﬁbrant and Cartesian double category D that satisﬁes the Frobenius property and has Eilenberg-Moore objects, has tabulators. Moreover, if the Eilenberg-Moore objects are strong, the tabulators are strong.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 23 / 29 • D is ﬁbrant and Cartesian. • D satisﬁes the Frobenius property • D has strong Eilenberg-Moore objects • D0 has pullbacks • For every composable horizontal arrows F and G, the tabulator of G F is given by the pullback of the tabulators of F and G.

• For every pair of vertical arrows r0 : R → A and r1 : R → B, the ∗ tabulator of r1∗r0 is R.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Corollary (of the conjecture) If a double category D satisﬁes all of the following, then D ' Span(D):

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 24 / 29 • D satisﬁes the Frobenius property • D has strong Eilenberg-Moore objects • D0 has pullbacks • For every composable horizontal arrows F and G, the tabulator of G F is given by the pullback of the tabulators of F and G.

• For every pair of vertical arrows r0 : R → A and r1 : R → B, the ∗ tabulator of r1∗r0 is R.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Corollary (of the conjecture) If a double category D satisﬁes all of the following, then D ' Span(D): • D is ﬁbrant and Cartesian.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 24 / 29 • D has strong Eilenberg-Moore objects • D0 has pullbacks • For every composable horizontal arrows F and G, the tabulator of G F is given by the pullback of the tabulators of F and G.

• For every pair of vertical arrows r0 : R → A and r1 : R → B, the ∗ tabulator of r1∗r0 is R.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Corollary (of the conjecture) If a double category D satisfies all of the following, then D ' Span(D): • D is fibrant and Cartesian. • D satisfies the Frobenius property

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 24 / 29 • D0 has pullbacks • For every composable horizontal arrows F and G, the tabulator of G F is given by the pullback of the tabulators of F and G.

• For every pair of vertical arrows r0 : R → A and r1 : R → B, the ∗ tabulator of r1∗r0 is R.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 24 / 29 • For every composable horizontal arrows F and G, the tabulator of G F is given by the pullback of the tabulators of F and G.

• For every pair of vertical arrows r0 : R → A and r1 : R → B, the ∗ tabulator of r1∗r0 is R.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 24 / 29 • For every pair of vertical arrows r0 : R → A and r1 : R → B, the ∗ tabulator of r1∗r0 is R.

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Corollary (of the conjecture) If a double category D satisfies all of the following, then D ' Span(D): • D is fibrant and Cartesian. • D satisfies the Frobenius property • D has strong Eilenberg-Moore objects • D0 has pullbacks • For every composable horizontal arrows F and G, the tabulator of G F is given by the pullback of the tabulators of F and G.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 24 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

• For every pair of vertical arrows r0 : R → A and r1 : R → B, the ∗ tabulator of r1∗r0 is R.

Further Questions

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 25 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

FbrCatQ: The full sub-2-category of DblCat determined by the ﬁbrant double categories D in which every category H(D)(A, B) has coequalizers and preserves them in both variables.

Q If D is a double category in FbrCat , then we can deﬁne the double category Mod(D) of • monads, • monad morphisms vertically, • modules horizontally and • equivariant maps. Moreover, Mod(D) is ﬁbrant.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 26 / 29 Since every 2-functor preserves adjunctions, we have the following: Corollary If D is a fibrant Cartesian double category in which every category H(D)(A, B) has coequalizers and preserves them in both variables, then Mod(D) is a fibrant Cartesian double category too. In particular, for a finitely complete and cocomplete category E, since

Prof(E) ' Mod(Span(E)),

the double category Prof(E) is Cartesian. Question By using the above construction of modules, can we characterize the double category of profunctors as a Cartesian double category?

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Theorem (Shulman 2007) Q Q Mod deﬁnes a 2-functor FbrCat → FbrCat .

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 27 / 29 In particular, for a ﬁnitely complete and cocomplete category E, since

Prof(E) ' Mod(Span(E)),

the double category Prof(E) is Cartesian. Question By using the above construction of modules, can we characterize the double category of profunctors as a Cartesian double category?

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Theorem (Shulman 2007) Q Q Mod defines a 2-functor FbrCat → FbrCat . Since every 2-functor preserves adjunctions, we have the following: Corollary If D is a fibrant Cartesian double category in which every category H(D)(A, B) has coequalizers and preserves them in both variables, then Mod(D) is a fibrant Cartesian double category too.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 27 / 29 Question By using the above construction of modules, can we characterize the double category of profunctors as a Cartesian double category?

Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Prof(E) ' Mod(Span(E)),

the double category Prof(E) is Cartesian.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 27 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Prof(E) ' Mod(Span(E)),

the double category Prof(E) is Cartesian. Question By using the above construction of modules, can we characterize the double category of profunctors as a Cartesian double category?

M Grandis and R Paré. Limits in double categories. Cahiers de topologie et géométriedifférentiellecatégoriques, 40(3):162–220, 1999. Stephen Lack, R. F C Walters, and R. J. Wood. Bicategories of spans as cartesian bicategories. Theory and Applications of Categories, 24(1):1–24, 2010. Susan Niefield. Span, Cospan, and other double categories. Theory and Applications of Categories, 26(26):729–742, 2012. Michael Shulman. Framed bicategories and monoidal fibrations. 20(18):80, 2007.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 28 / 29 Motivation Cartesian double categories Eilenberg-Moore Objects Towards the characterization of spans Further Questions

Thank you!

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 29 / 29