Kan extensions and descent theory Fernando Lucatelli Nunes 1 1 PhD - - PowerPoint PPT Presentation

Pseudo-Kan extension Facets of Descent

Kan extensions and descent theory

Fernando Lucatelli Nunes1

1PhD program at University of Coimbra, under supervision of Maria Manuel

Clementino

Category Theory - CT 2015, University of Aveiro

Pseudo-Kan extension Facets of Descent

Setting A : Cop → CAT p ∈ Mor(C), DescA(p) and (assuming that A(p)! ⊣ A(p)) the category of (Eilenberg Moore) algebras of A(p)(A(p)!)

[Janelidze and Tholen 1997] Facets of descent II

Pseudo-Kan extension Facets of Descent

Setting A : Cop → CAT p ∈ Mor(C), DescA(p) and (assuming that A(p)! ⊣ A(p)) the category of (Eilenberg Moore) algebras of A(p)(A(p)!)

[Ross Street 1976] Limits indexed by category-valued 2-functors [Ross Street 1980] Correction to: “Fibrations in bicategories”

Pseudo-Kan extension Facets of Descent

Aim Work Investigate whether formal methods and commuting properties

• f bilimits are useful in proving theorems of descent theory in

the context of Facets of Descent II Talk Give a proof of Bénabou-Roubaud Theorem using this approach

Pseudo-Kan extension Facets of Descent

Outline

1

Pseudo-Kan extension Definition Factorization Eilenberg Moore Algebras Descent Object Commuting property

2

Facets of Descent (Usual context of) Facets of Descent Bénabou-Roubaud Theorem

Pseudo-Kan extension Facets of Descent Definition

Right pseudo-Kan extension Fully Faithful v : S → ˙ S Obj( ˙ S) = {e} ∪ Obj(S) 2-category H

• ˙

S, H

• PS

[v,H]PS

• [S, H]PS

Pseudo-Kan extension Facets of Descent Definition

Right pseudo-Kan extension Fully Faithful v : S → ˙ S Obj( ˙ S) = {e} ∪ Obj(S) 2-category H

• ˙

S, H

• PS

[v,H]PS

• ⊣

[S, H]PS

Ps-Ranv

• ε

: [v, H]PS ◦ Ps-Ranv → Id η : Id → Ps-Ranv ◦ [v, H]PS s : IdL ∼ = (εL) ◦ (Lη) t : (Uε) ◦ (ηU) ∼ = IdU plus coherence

Pseudo-Kan extension Facets of Descent Definition

Pointwise pseudo-Kan extension Theorem (Pointwise pseudo-Kan extension) Given a pseudofunctor D : S → H, Ps-RanvD(a) =

• ˙

S(a, v−), D

• bi ,

provided that these weighted bilimits exist in H.

Pseudo-Kan extension Facets of Descent Definition

Pointwise pseudo-Kan extension Theorem (Pointwise pseudo-Kan extension) Given a pseudofunctor D : S → H, Ps-RanvD(a) =

• ˙

S(a, v−), D

• bi ,

provided that these weighted bilimits exist in H. We get actually extensions (up to pseudonatural equivalences): the counit of the biadjunction is a pseudonatural equivalence (since v is fully faithful) We always assume that the pointwise right pseudo-Kan extensions exist.

Pseudo-Kan extension Facets of Descent Factorization

Factorization and Comparison

D : ˙ S → H Fully faithful v : S → ˙ S, Obj( ˙ S) = {e} ∪ Obj(S) v-comparison: ηDe : D(e) → Ps-Ranv (D ◦ v) (e) v-“factorizations”: For each morphism f : e → a of ˙ S, D(e)

ηDe

• D(f)
• Ps-Ranv (D ◦ v) (e)
• D(a)

∼ =

Pseudo-Kan extension Facets of Descent Factorization

Factorization and Comparison

D : ˙ S → H Fully faithful v : S → ˙ S, Obj( ˙ S) = {e} ∪ Obj(S) v-comparison: ηDe : D(e) → Ps-Ranv (D ◦ v) (e) v-“factorizations”: For each morphism f : a → e of ˙ S, D(e)

ηDe

Ps-Ranv (D ◦ v) (e) D(a)

• D(f)
• ∼

=

Pseudo-Kan extension Facets of Descent Factorization

v-effective v : S → ˙ S, Obj( ˙ S) = {e} ∪ Obj(S) D : ˙ S → H Comparison: ηDe : D(e) → Ps-Ranv (D ◦ v) (e) Definition Herein, if ηDe : D(e) → Ps-Ranv (D ◦ v) (e) is an equivalence, we say that D is v-effective. This means that D is v-effective if it is “in the image” of the global right pseudo-Kan extension.

Pseudo-Kan extension Facets of Descent Eilenberg Moore Algebras

The 2-category Adj

Free Adjunction (Street and Schanuel) We denote by Adj the 2-category generated by the diagram e

u

a

f

• with 2-cells

η : Ida → uf ε : fu → Ide satisfying the triangular identities. We define the full inclusion m : Mnd → Adj, with Obj(Mnd) = {a}

Pseudo-Kan extension Facets of Descent Eilenberg Moore Algebras

Eilenberg Moore Factorization

Each adjunction in a 2-category H corresponds to a diagram D : Adj → H. The m-factorization gives the Eilenberg Moore factorization (if H is bicategorically complete) and the Eilenberg Moore comparison 1-cell. D(e)

• D(u)
• Ps-Ranv (D ◦ v) (e)
• D(a)

∼ =

Thereby D is m-effective if and only if the right adjoint D(u) is monadic. [S. Schanuel and R. Street 1986] The Free Adjunction

Pseudo-Kan extension Facets of Descent Descent Object

The category ∆

Definition We denote by ˙ ∆ the category generated by the diagram

d

1

d0

• d1

2

s0

• ∂0
• ∂1
• ∂2

3 with the usual cosimplicial identities ∂kdi = ∂idk−1, if i < k s0d0 = Id1 Id1 = s0d1 d1d = d0d We define the full inclusion g : ∆ → ˙ ∆, with Obj( ˙ ∆) = Obj(∆) ∪ {0}

Pseudo-Kan extension Facets of Descent Descent Object

Descent Object We define the full inclusion g : ∆ → ˙ ∆, with Obj( ˙ ∆) = Obj(∆) ∪ {0} Theorem (Descent Object) If D : ∆ → H has a (pointwise) right pseudo-Kan extension along g, then Ps-RangD(0) is indeed the descent object of D.

Pseudo-Kan extension Facets of Descent Descent Object

Descent Object We define the full inclusion g : ∆ → ˙ ∆, with Obj( ˙ ∆) = Obj(∆) ∪ {0} Theorem (Descent Object) If D : ∆ → H has a (pointwise) right pseudo-Kan extension along g, then Ps-RangD(0) is indeed the descent object of D. Main observation: This is a very easy description of the descent object!

Pseudo-Kan extension Facets of Descent Commuting property

Diagram of effective diagrams We consider

v : S → ˙ S Obj( ˙ S) = {e} ∪ Obj(S) j : R → ˙ R Obj( ˙ R) = {z} ∪ Obj(R) Theorem (Main commuting property) Given a pseudofunctor M : ˙ S →

• ˙

R, H

• PS

M is v-effective The image of M ◦v : S →

• ˙

R, H

• PS has only j-effective diagrams.

Then M(e) : ˙ R → H is j-effective as well.

Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent

Factorizations

1

C with pullbacks;

2

A : Cop → CAT;

3

p ∈ C(E, B) ˙ ∆ → C E ×B E ×B E

E ×B E

• E
• p

B

Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent

Factorizations

1

C with pullbacks;

2

A : Cop → CAT;

3

p ∈ C(E, B) A

Desc

p

: ˙ ∆ → CAT (descent diagram induced by p) A(B)

A(p) A(E)

• A(E ×p E)
• A(E ×p E ×p E)

Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent

Factorizations

1

C with pullbacks;

2

A : Cop → CAT;

3

p ∈ C(E, B)

4

A(p)! ⊣ A(p) A

Desc

p

: ˙ ∆ → CAT (descent diagram induced by p) A(B)

A(p) A(E)

• A(E ×p E)
• A(E ×p E ×p E)

A

Adj

p

: Adj → CAT (adjunction induced by p) A(B)

A(p) A(E) A(p)!

Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent

Factorizations

1

C with pullbacks;

2

A : Cop → CAT;

3

p ∈ C(E, B)

4

A(p)! ⊣ A(p) A

Desc

p

: ˙ ∆ → CAT

(g-factorization - natural isomorphism)

A(B)

A(p)

• A(E)

DescA(p)

• A

Adj

p

: Adj → CAT

(m-factorization - natural isomorphism)

A(B)

A(p)

• A(E)

AlgA(p)

Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent

Factorizations A

Desc

p

: ˙ ∆ → CAT

(g-factorization - natural isomorphism)

A(B)

A(p)

• A(E)

DescA(p)

• A

Adj

p

: Adj → CAT

(m-factorization - natural isomorphism)

A(B)

A(p)

• A(E)

AlgA(p)

Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent

Usual definitions Definitions p is of A-effective descent if A

Desc

p

: ˙ ∆ → CAT is g-effective; p is of A-monadic descent if A

Adj

p

: Adj → CAT is m-effective.

Pseudo-Kan extension Facets of Descent Bénabou-Roubaud Theorem

Assumed Lemmas A : Cop → CAT Lemmas Every split epimorphism p is of A-effective descent (proved in Facets of Descent II) ; Every split epimorphism p is of A-monadic descent provided that A(p)! ⊣ A(p) (easy consequence of Beck’s Theorem). They are corollaries of properties of the right pseudo-Kan extensions along m : Mnd → Adj and g : ∆ → ˙ ∆ (respectively).

Pseudo-Kan extension Facets of Descent Bénabou-Roubaud Theorem

Bénabou-Roubaud Theorem

Bénabou-Roubaud Theorem A : Cop → CAT A(q)! ⊣ A(q) for all q ∈ Mor(C) The natural isomorphisms A(B)

A(q)

• A(q)
• A(E)

A(d0)

• ∼

= A(E)

A(d1)

A(E ×q E)

satisfy Beck Chevalley condition

Then p is A-monadic descent if and only if p is A-effective descent

Pseudo-Kan extension Facets of Descent Bénabou-Roubaud Theorem

Proof p ∈ Mor(C) Pseudodiagram Dp : 2 →

• ˙

∆, CAT

• PS

A(B)

A(p)

• A(p)

A(E)

A(d0)

• A(E ×p E)
• A(E ×p E ×p E)
• A(E)

A(d1)

A(E ×p E) A(E ×p E ×p E)

• A(E ×p E ×p E ×p E)

Pseudo-Kan extension Facets of Descent Bénabou-Roubaud Theorem

Proof p ∈ Mor(C) 2 → Adj Dp : Adj →

• ˙

∆, CAT

• PS (doctrinal adjunction)

˙ ∆ → [Adj, CAT]PS (Beck Chevalley Condition)

A(B)

A(p)

• A(p)

A(E)

A(d0)

• A(E ×p E)
• A(E ×p E ×p E)
• A(E)

A(p)!

• A(d1)

A(E ×p E)

A(d0)!

• A(E ×p E ×p E)
• A(E ×p E ×p E ×p E)

Pseudo-Kan extension Facets of Descent Bénabou-Roubaud Theorem

Conclusion of this proof Bénabou-Roubaud Theorem is a (formal) consequence of the following facts: Every split epimorphism p is of A-effective descent (which is a property of ˙ ∆); Every split epimorphism p is of A-monadic descent provided that A(p)! ⊣ A(p) (wich is a property of Adj); Beck Chevalley condition.

Pseudo-Kan extension Facets of Descent

Bibliography

[Janelidze and Tholen 1997] G. Janelidze and W. Tholen Facets of descent II Applied Categ. Structures 5 (1997), no. 3, 229-248, 1997. [Ross Street 1976] Ross Street Limits indexed by category-valued 2-functors

• J. Pure Appl. Algebra (1997), 8 (1976), no. 2, 149-181, 1976.

[Ross Street 1980] Ross Street Correction to: “Fibrations in bicategories” [Cahiers Topologie Géom. Différentielle 21 (1980), no. 2, 111-160]. Cahiers Géom. Différentielle Catég. 28 (1987), no. 1, 53-56. [S. Schanuel and R. Street] Schanuel and Street The Free Adjunction Cahiers de Topologie et Géom. Différentielle Catégoriques. 27: 81-83, 1986.

Pseudo-Kan extension Facets of Descent

Further Applications Most of the (classical) theorems of Descent Theory can be proved using this language (and some finer commuting properties or properties of the particular weighted bilimits involved. We can get A-effective descent morphisms using this approach: mostly, from bilimits of categories for which we already know the effective descent morphisms. More precisely, if we get A as a weighted bilimit of other weighted bilimits. I used this approach to get some criteria (theorems) and to get effective descent morphisms for the category of enriched categories in a monoidal closed category V (provided that V satisfy some hypotheses).

Pseudo-Kan extension Facets of Descent

Thank you!