Flat functors Starting point for the 2-categorical case Flat 2-functors

On flat 2-functors

Marıa Emilia Descotte 1

University of Buenos Aires

CT 2017Vancouver, Canada

1Joint work with E. Dubuc and M. Szyld

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat functors and main theorem

Definition

AP−→ Set is flat if its left Kan extension

along the Yoneda embedding is left exact(preserves finite limits).

A

P

��

h // Hom(Aop, Set)

P∗

||

ηP=⇒

Set

Theorem∗

For AP−→ Set, the following are equivalent:

1 The category of elements ElP of P is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

∗ [ML,M] Sheaves in Geometry and Logic: a First Introduction to Topos Theory,1992.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat functors and main theorem

Definition

AP−→ Set is flat if its left Kan extension

along the Yoneda embedding is left exact(preserves finite limits).

A

P

��

h // Hom(Aop, Set)

P∗

||

ηP=⇒

Set

Theorem∗

For AP−→ Set, the following are equivalent:

1 The category of elements ElP of P is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

∗ [ML,M] Sheaves in Geometry and Logic: a First Introduction to Topos Theory,1992.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat functors and main theorem

Definition

AP−→ Set is flat if its left Kan extension

along the Yoneda embedding is left exact(preserves finite limits).

A

P

��

h // Hom(Aop, Set)

P∗

||

ηP=⇒

Set

Theorem∗

For AP−→ Set, the following are equivalent:

1 The category of elements ElP of P is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

∗ [ML,M] Sheaves in Geometry and Logic: a First Introduction to Topos Theory,1992.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof1 ElP is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

1 ⇒ 2: P = lim−→ElopP

A(a,−)(=∫ a

Pa×A(a,−)).

2 ⇒ 3: Representable functors are flat.+

Filtered colimit of flat is flat (since filtered colimitscommute with finite limits).

3⇒ 1: CF→ D left exact and C has finite limits⇒ ElF cofiltered.

+ElP∗ cofiltered ⇒ ElP cofiltered.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof1 ElP is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

1 ⇒ 2: P = lim−→ElopP

A(a,−)(=∫ a

Pa×A(a,−)).

2 ⇒ 3: Representable functors are flat.+

Filtered colimit of flat is flat (since filtered colimitscommute with finite limits).

3⇒ 1: CF→ D left exact and C has finite limits⇒ ElF cofiltered.

+ElP∗ cofiltered ⇒ ElP cofiltered.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof1 ElP is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

1 ⇒ 2: P = lim−→ElopP

A(a,−)(=∫ a

Pa×A(a,−)).

2 ⇒ 3: Representable functors are flat.+

Filtered colimit of flat is flat (since filtered colimitscommute with finite limits).

3⇒ 1: CF→ D left exact and C has finite limits⇒ ElF cofiltered.

+ElP∗ cofiltered ⇒ ElP cofiltered.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flatness for V-enriched functors

A notion of flatness for V-enriched functors was already considered byKelly∗.

V = Cat gives us a notion of flatness for 2-functors.

There is no known generalization of the main theorem with thisnotion of flatness.

∗[K] Structures defined by finite limits in the enriched context, I, 1982.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flatness for V-enriched functors

A notion of flatness for V-enriched functors was already considered byKelly∗.

V = Cat gives us a notion of flatness for 2-functors.

There is no known generalization of the main theorem with thisnotion of flatness.

∗[K] Structures defined by finite limits in the enriched context, I, 1982.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flatness for V-enriched functors

A notion of flatness for V-enriched functors was already considered byKelly∗.

V = Cat gives us a notion of flatness for 2-functors.

There is no known generalization of the main theorem with thisnotion of flatness.

∗[K] Structures defined by finite limits in the enriched context, I, 1982.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Expressions for A P−→ Cat

There was no known expressionof P as a (conical) colimit of

representable functors.

P ≈ p∫ A

PA×A(A,−).

Key Fact

This pseudo-coend can be expressed as a special kind of (conical)colimit over the 2-category of elements ElP associated to P .

Flat functors Starting point for the 2-categorical case Flat 2-functors

Expressions for A P−→ Cat

There was no known expressionof P as a (conical) colimit of

representable functors.P ≈ p

∫ APA×A(A,−).

Key Fact

This pseudo-coend can be expressed as a special kind of (conical)colimit over the 2-category of elements ElP associated to P .

Flat functors Starting point for the 2-categorical case Flat 2-functors

Expressions for A P−→ Cat

There was no known expressionof P as a (conical) colimit of

representable functors.P ≈ p

∫ APA×A(A,−).

Key Fact

This pseudo-coend can be expressed as a special kind of (conical)colimit over the 2-category of elements ElP associated to P .

Flat functors Starting point for the 2-categorical case Flat 2-functors

Diamonds vs cones (dimension 1)A

P−→ Set, AopF−→ Set.

Dicones for A×Aop P×F−−−→ Set: Cones for ElopP♦opP−→ Aop

F−→ Set:

Pa× Faθa

$$Pa× Fb ≡

id×Ff88

Pf×id&&

Z

Pb× Fbθb

::

Faλ(x,a)

≡ Z

Fb

Ff

OO

λ(Pf(x),b)

>>

θa(x, y) = λ(x,a)(y)

As a corollary,∫ a

Pa× Fa = lim−→ElopP

Fa.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Diamonds vs cones (dimension 1)A

P−→ Set, AopF−→ Set.

Dicones for A×Aop P×F−−−→ Set: Cones for ElopP♦opP−→ Aop

F−→ Set:

Pa× Faθa

$$Pa× Fb ≡

id×Ff88

Pf×id&&

Z

Pb× Fbθb

::

Faλ(x,a)

≡ Z

Fb

Ff

OO

λ(Pf(x),b)

>>

θa(x, y) = λ(x,a)(y)

As a corollary,∫ a

Pa× Fa = lim−→ElopP

Fa.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Diamonds vs cones (dimension 2)A

P−→ Set, AopF−→ Set.

Dicones for A×Aop P×F−−−→ Set: Cones for ElopP♦opP−→ Aop

F−→ Set:

Pa× Faθa

$$Pa× Fb ≡

id×Ff88

Pf×id&&

Z

Pb× Fbθb

::

Faλ(x,a)

≡ Z

Fb

Ff

OO

λ(Pf(x),b)

>>

θa(x, y) = λ(x,a)(y)

As a corollary,∫ a

Pa× Fa = lim−→ElopP

Fa.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Diamonds vs cones (dimension 2)A P−→ Cat, Aop F−→ Cat.

Dicones for A×Aop P×F−−−→ Set: Cones for ElopP♦opP−→ Aop

F−→ Set:

Pa× Faθa

$$Pa× Fb ≡

id×Ff88

Pf×id&&

Z

Pb× Fbθb

::

Faλ(x,a)

≡ Z

Fb

Ff

OO

λ(Pf(x),b)

>>

θa(x, y) = λ(x,a)(y)

As a corollary,∫ a

Pa× Fa = lim−→ElopP

Fa.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Diamonds vs cones (dimension 2)A P−→ Cat, Aop F−→ Cat.

Pseudodicones for A×Aop P×F−−−→ Cat: Cones for ElopP♦opP−→ Aop

F−→ Set:

PA× FAθA

$$PA× FB ⇑ θf

id×Ff77

Pf×id''

Z

PB × FBθB

::

Faλ(x,a)

≡ Z

Fb

Ff

OO

λ(Pf(x),b)

>>

θa(x, y) = λ(x,a)(y)

As a corollary,∫ a

Pa× Fa = lim−→ElopP

Fa.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Diamonds vs cones (dimension 2)A P−→ Cat, Aop F−→ Cat.

Pseudodicones for A×Aop P×F−−−→ Cat: σ-cones for ElopP♦opP−→ Aop F−→ Cat:

PA× FAθA

$$PA× FB ⇑ θf

id×Ff77

Pf×id''

Z

PB × FBθB

::

FAλ(x,A)

!!⇑ λ(f,ϕ) Z

FB

Ff

OO

λ(x′,B)

==

θa(x, y) = λ(x,a)(y)

As a corollary,∫ a

Pa× Fa = lim−→ElopP

Fa.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Diamonds vs cones (dimension 2)A P−→ Cat, Aop F−→ Cat.

Pseudodicones for A×Aop P×F−−−→ Cat: σ-cones for ElopP♦opP−→ Aop F−→ Cat:

PA× FAθA

$$PA× FB ⇑ θf

id×Ff77

Pf×id''

Z

PB × FBθB

::

FAλ(x,A)

!!⇑ λ(f,ϕ) Z

FB

Ff

OO

λ(x′,B)

==

θA(x, y) = λ(x,A)(y)

As a corollary,∫ a

Pa× Fa = lim−→ElopP

Fa.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Diamonds vs cones (dimension 2)A P−→ Cat, Aop F−→ Cat.

Pseudodicones for A×Aop P×F−−−→ Cat: σ-cones for ElopP♦opP−→ Aop F−→ Cat:

PA× FAθA

$$PA× FB ⇑ θf

id×Ff77

Pf×id''

Z

PB × FBθB

::

FAλ(x,A)

!!⇑ λ(f,ϕ) Z

FB

Ff

OO

λ(x′,B)

==

θA(x, y) = λ(x,A)(y) ,(θf )(x,y) = (λ(f,idPf(x)))y

As a corollary,∫ a

Pa× Fa = lim−→ElopP

Fa.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Diamonds vs cones (dimension 2) A P−→ Cat, Aop F−→ Cat.

Pseudodicones for A×Aop P×F−−−→ Cat: σ-cones for ElopP♦opP−→ Aop F−→ Cat:

PA× FAθA

$$PA× FB ⇑ θf

id×Ff77

Pf×id''

Z

PB × FBθB

::

FAλ(x,A)

!!⇑ λ(f,ϕ) Z

FB

Ff

OO

λ(x′,B)

==

θA(x, y) = λ(x,A)(y) ,(θf )(x,y) = (λ(f,idPf(x)))y

As a corollary, p∫ A

PA× FA = σ-lim−−−→ElopP

FA.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat 2-functors and main theorem

Definition

AP−→ Set is flat if its left Kan extension

along the Yoneda embedding is left exact(preserves finite limits).

A

P

��

h // Hom(Aop, Set)

P∗

||

ηP=⇒

Set

Theorem

For a functor AP−→ Set, the following are equivalent:

1 The category of elements ElP of P is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

∗[LN] On biadjoint triangles, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat 2-functors and main theorem

Definition

A P−→ Cat is flat if its left Kan extensionalong the Yoneda embedding is left exact(preserves finite limits).

A

P

��

h // Homp(Aop, Cat)

P∗

||

ηP=⇒

Cat

Theorem

For a functor AP−→ Set, the following are equivalent:

1 The category of elements ElP of P is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

∗[LN] On biadjoint triangles, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat 2-functors and main theorem

Definition

A P−→ Cat is flat if its left bi-Kanextension∗ along the Yoneda embedding isleft exact (preserves finite limits).

A

P

��

h // Homp(Aop, Cat)

P∗

||

ηP=⇒

Cat

Theorem

For a functor AP−→ Set, the following are equivalent:

1 The category of elements ElP of P is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

∗[LN] On biadjoint triangles, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat 2-functors and main theorem

Definition

A P−→ Cat is flat if its left bi-Kanextension∗ along the Yoneda embedding isleft exact (preserves finite weightedbi-limits).

A

P

��

h // Homp(Aop, Cat)

P∗

||

ηP=⇒

Cat

Theorem

For a functor AP−→ Set, the following are equivalent:

1 The category of elements ElP of P is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

∗[LN] On biadjoint triangles, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat 2-functors and main theorem

Definition

A P−→ Cat is flat if its left bi-Kanextension∗ along the Yoneda embedding isleft exact (preserves finite weightedbi-limits).

A

P

��

h // Homp(Aop, Cat)

P∗

||

ηP=⇒

Cat

Theorem

For a 2-functor A P−→ Cat, the following are equivalent:

1 The category of elements ElP of P is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

∗[LN] On biadjoint triangles, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat 2-functors and main theorem

Definition

A P−→ Cat is flat if its left bi-Kanextension∗ along the Yoneda embedding isleft exact (preserves finite weightedbi-limits).

A

P

��

h // Homp(Aop, Cat)

P∗

||

ηP=⇒

Cat

Theorem

For a 2-functor A P−→ Cat, the following are equivalent:

1 The 2-category of elements ElP of P is cofiltered.

2 P is a filtered colimit of representable functors.

3 P is flat.

∗[LN] On biadjoint triangles, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat 2-functors and main theorem

Definition

A P−→ Cat is flat if its left bi-Kanextension∗ along the Yoneda embedding isleft exact (preserves finite weightedbi-limits).

A

P

��

h // Homp(Aop, Cat)

P∗

||

ηP=⇒

Cat

Theorem

For a 2-functor A P−→ Cat, the following are equivalent:

1 The 2-category of elements ElP of P is σ-cofiltered (with respectto the family of co-cartesian arrows).

2 P is a filtered colimit of representable functors.

3 P is flat.

∗[LN] On biadjoint triangles, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Flat 2-functors and main theorem

Definition

A P−→ Cat is flat if its left bi-Kanextension∗ along the Yoneda embedding isleft exact (preserves finite weightedbi-limits).

A

P

��

h // Homp(Aop, Cat)

P∗

||

ηP=⇒

Cat

Theorem

For a 2-functor A P−→ Cat, the following are equivalent:

1 The 2-category of elements ElP of P is σ-cofiltered (with respectto the family of co-cartesian arrows).

2 P is (equivalent to) a σ-filtered σ-colimit of representable2-functors.

3 P is flat.

∗[LN] On biadjoint triangles, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof 1 ElP is σ-cofiltered.

2 P is (equivalent to) a σ-filtered σ-colimit ofrepresentable 2-functors.

3 P is flat.

1 ⇒ 2: P = lim−→ElopP

A(a,−).

2 ⇒ 3: Representable functors are flat.+

filtered colimit of flat is flat (since filtered colimitscommute with finite limits).

3 ⇒ 1: CF→ D left exact and C has finite limits⇒ ElF cofiltered .

+ElP∗ cofiltered ⇒ ElP cofiltered .

∗[DDS] A construction of certain weak colimits and an exactness property of the2-category of categories, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof 1 ElP is σ-cofiltered.

2 P is (equivalent to) a σ-filtered σ-colimit ofrepresentable 2-functors.

3 P is flat.

1 ⇒ 2: P ≈ σ-lim−−−→ElopP

A(A,−).

2 ⇒ 3: Representable functors are flat.+

filtered colimit of flat is flat (since filtered colimitscommute with finite limits).

3 ⇒ 1: CF→ D left exact and C has finite limits⇒ ElF cofiltered .

+ElP∗ cofiltered ⇒ ElP cofiltered .

∗[DDS] A construction of certain weak colimits and an exactness property of the2-category of categories, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof 1 ElP is σ-cofiltered.

2 P is (equivalent to) a σ-filtered σ-colimit ofrepresentable 2-functors.

3 P is flat.

1 ⇒ 2: P ≈ σ-lim−−−→ElopP

A(A,−).

2 ⇒ 3: Representable 2-functors are flat.+

filtered colimit of flat is flat (since filtered colimitscommute with finite limits).

3 ⇒ 1: CF→ D left exact and C has finite limits⇒ ElF cofiltered .

+ElP∗ cofiltered ⇒ ElP cofiltered .

∗[DDS] A construction of certain weak colimits and an exactness property of the2-category of categories, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof 1 ElP is σ-cofiltered.

2 P is (equivalent to) a σ-filtered σ-colimit ofrepresentable 2-functors.

3 P is flat.

1 ⇒ 2: P ≈ σ-lim−−−→ElopP

A(A,−).

2 ⇒ 3: Representable 2-functors are flat.+

σ-filtered σ-colimit of flat is flat (since filtered colimitscommute with finite limits).

3 ⇒ 1: CF→ D left exact and C has finite limits⇒ ElF cofiltered .

+ElP∗ cofiltered ⇒ ElP cofiltered .

∗[DDS] A construction of certain weak colimits and an exactness property of the2-category of categories, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof 1 ElP is σ-cofiltered.

2 P is (equivalent to) a σ-filtered σ-colimit ofrepresentable 2-functors.

3 P is flat.

1 ⇒ 2: P ≈ σ-lim−−−→ElopP

A(A,−).

2 ⇒ 3: Representable 2-functors are flat.+

σ-filtered σ-colimit of flat is flat (since σ-filtered σ-colimitscommute with finite weighted bilimits∗).

3 ⇒ 1: CF→ D left exact and C has finite limits⇒ ElF cofiltered .

+ElP∗ cofiltered ⇒ ElP cofiltered .

∗[DDS] A construction of certain weak colimits and an exactness property of the2-category of categories, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof 1 ElP is σ-cofiltered.

2 P is (equivalent to) a σ-filtered σ-colimit ofrepresentable 2-functors.

3 P is flat.

1 ⇒ 2: P ≈ σ-lim−−−→ElopP

A(A,−).

2 ⇒ 3: Representable 2-functors are flat.+

σ-filtered σ-colimit of flat is flat (since σ-filtered σ-colimitscommute with finite weighted bilimits∗).

3 ⇒ 1: C F→ D left exact and C has finite weighted bilimits⇒ ElF σ-cofiltered.

+ElP∗ cofiltered ⇒ ElP cofiltered .

∗[DDS] A construction of certain weak colimits and an exactness property of the2-category of categories, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Idea of the proof 1 ElP is σ-cofiltered.

2 P is (equivalent to) a σ-filtered σ-colimit ofrepresentable 2-functors.

3 P is flat.

1 ⇒ 2: P ≈ σ-lim−−−→ElopP

A(A,−).

2 ⇒ 3: Representable 2-functors are flat.+

σ-filtered σ-colimit of flat is flat (since σ-filtered σ-colimitscommute with finite weighted bilimits∗).

3 ⇒ 1: C F→ D left exact and C has finite weighted bilimits⇒ ElF σ-cofiltered.

+ElP∗ σ-cofiltered ⇒ ElP σ-cofiltered.

∗[DDS] A construction of certain weak colimits and an exactness property of the2-category of categories, 2016.

Flat functors Starting point for the 2-categorical case Flat 2-functors

Thank you!