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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

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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

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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

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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

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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 : ✶ → 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

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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 : ✶ → B, extending

the product structure of Map(B), are pseudo.

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

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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 : ✶ → B, extending

the product structure of Map(B), are pseudo.

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.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 4 / 29

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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 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

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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

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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 A

|

M f

B

g

  • α

C

|

N

D

  • iii. Functors D1

S,T D0, D0 U

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),

UTM ⊙ M

l

M and M ⊙ USM

r

M, satisfying the pentagon

and triangle identities.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 7 / 29

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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

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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

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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

!

  • ⊥

✶

I

  • in DblCat.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 9 / 29

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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

!

  • ⊥

✶

I

  • in DblCat.

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

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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

!

  • ⊥

✶

I

  • in DblCat.

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

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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

!

  • ⊥

✶

I

  • in DblCat.

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

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 9 / 29

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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

f

C

g

  • B

|

M

D

there is a horizontal arrow g∗Mf∗ : A

  • | C and a cell

A

f

|

  • g∗Mf∗ C

g

  • ζ

B

|

  • M

D,

A′

fh

|

M′ C ′ gk

  • so that every cell

can be factored uniquely through ζ. B

|

M

D,

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 10 / 29

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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

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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

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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

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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

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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 : ✶ → D.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 12 / 29

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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 : ✶ → 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.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 12 / 29

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

Eilenberg-Moore Objects

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 13 / 29

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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 ǫ : P → UX satisfying the usual conditions.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 14 / 29

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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 ǫ : P → UX satisfying the usual conditions.

  • Vertical arrows the comonad morphisms, i.e. vertical arrows

f : X → Y together with a cell ψ: X

|

P f

X

f

  • Y

|

R

Y

which is compatible with the comonad structure.

  • Horizontal arrows the horizontal comonad maps, i.e. horizontal arrows

F : X

  • | X ′ , together with a cell α:

X

|

  • P X

|

  • F

X ′

X

|

F

X ′

|

P′ X ′,

compatible with the counit and the comultiplication.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 14 / 29

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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

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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.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 15 / 29

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

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 EM

e

|

U EM e

  • X

|

  • P

X.

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

  • bjects.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 16 / 29

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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

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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 d XX d1

  • XX

1d

XXX

the cell XX

|

  • d∗

X

d

  • |
  • U

X

d

|

  • d∗

XX

XX

|

  • U XX

1d

XX

d1

  • |
  • U XX

XX

|

(1d)∗

XXX

|

U

XXX

|

(d1)∗ XX

is invertible.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 18 / 29

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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

T

t1

|

  • U

T

t2

  • τ

X

|

  • F

Y

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∗t∗

1.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 19 / 29

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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 X

|

  • P

X

ǫ X

|

  • UX

X

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

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 20 / 29

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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

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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

d×Y

X × Y

X×d

  • X × X × Y

|

X×F×Y

X × Y × 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

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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

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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

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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

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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

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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

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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

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 24 / 29

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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

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 24 / 29

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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

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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.

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

tabulator of r1∗r∗

0 is R.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 24 / 29

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

Further Questions

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 25 / 29

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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. If D is a double category in FbrCatQ, 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

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

Theorem (Shulman 2007)

Mod defines a 2-functor FbrCatQ → FbrCatQ.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 27 / 29

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

Theorem (Shulman 2007)

Mod defines a 2-functor FbrCatQ → FbrCatQ. 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

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

Theorem (Shulman 2007)

Mod defines a 2-functor FbrCatQ → FbrCatQ. 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.

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 27 / 29

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

Theorem (Shulman 2007)

Mod defines a 2-functor FbrCatQ → FbrCatQ. 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?

Evangelia Aleiferi (Dalhousie University) Towards a Characterization of the Double Category of Spans July, 2017 27 / 29

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

M Grandis and R Par´ e. Limits in double categories. Cahiers de topologie et g´ eom´ etrie diff´ erentielle cat´ egoriques, 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

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Thank you!

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