A general limit lifting theorem for 2-dimensional monad theory (but - - PowerPoint PPT Presentation

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A general limit lifting theorem for 2-dimensional monad theory (but - - PowerPoint PPT Presentation

Jul 19, 2023 28 likes •400 views

Limit lifting results Unifying morphisms Unifying limits Our result References A general limit lifting theorem for 2-dimensional monad theory (but dont let the long title scare you!) Martin Szyld University of Buenos Aires - CONICET,

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SLIDE 1

Limit lifting results Unifying morphisms Unifying limits Our result References

A general limit lifting theorem for 2-dimensional monad theory (but don’t let the long title scare you!)

Martin Szyld University of Buenos Aires - CONICET, Argentina CT 2017 @ UBC, Vancouver, Canada

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SLIDE 2

Limit lifting results Unifying morphisms Unifying limits Our result References

Limit lifting along the forgetful functor

K is a category, T is a monad on K (K

T

− → K , id

i

⇒ T, T 2 m ⇒ T) T-Alg

U

  • A

F

  • F
  • K

U creates limF ≡ we can give limF a T-algebra structure such that it is limF (we lift the limit of F along U)

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SLIDE 3

Limit lifting results Unifying morphisms Unifying limits Our result References

Limit lifting along the forgetful functor

K is a category, T is a monad on K (K

T

− → K , id

i

⇒ T, T 2 m ⇒ T) T-Alg

U

  • A

F

  • F
  • K

U creates limF ≡ we can give limF a T-algebra structure such that it is limF (we lift the limit of F along U) Previous results

1 (from the V-enriched case) T-Alg

U

− → K creates all limits.

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SLIDE 4

Limit lifting results Unifying morphisms Unifying limits Our result References

Limit lifting along the forgetful functor

K is a 2-category, T is a 2-monad on K ( K

T

− → K, id

i

⇒ T, T 2 m ⇒ T) T-Alg

U

  • A

F

  • F
  • K

U creates limF ≡ we can give limF a T-algebra structure such that it is limF (we lift the limit of F along U) Previous results

1 (from the V-enriched case) T-Alg

U

− → K creates all limits.

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SLIDE 5

Limit lifting results Unifying morphisms Unifying limits Our result References

Limit lifting along the forgetful functor

K is a 2-category, T is a 2-monad on K ( K

T

− → K, id

i

⇒ T, T 2 m ⇒ T) T-Algs

U

  • A

F

  • F
  • K

U creates limF ≡ we can give limF a T-algebra structure such that it is limF (we lift the limit of F along U) The subindex (s, p, ℓ) indicates (strict, pseudo, lax) algebra morphisms Previous results

1 (from the V-enriched case) T-Alg

U

− → K creates all limits.

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SLIDE 6

Limit lifting results Unifying morphisms Unifying limits Our result References

Limit lifting along the forgetful functor

K is a 2-category, T is a 2-monad on K ( K

T

− → K, id

i

⇒ T, T 2 m ⇒ T) T-Algs

U

  • A

F

  • F
  • K

U creates limF ≡ we can give limF a T-algebra structure such that it is limF (we lift the limit of F along U) The subindex (s, p, ℓ) indicates (strict, pseudo, lax) algebra morphisms Previous results

1 (from the V-enriched case) T-Algs

U

− → K creates all limits.

slide-7
SLIDE 7

Limit lifting results Unifying morphisms Unifying limits Our result References

Limit lifting along the forgetful functor

K is a 2-category, T is a 2-monad on K ( K

T

− → K, id

i

⇒ T, T 2 m ⇒ T) T-Algp

U

  • A

F

  • F
  • K

U creates limF ≡ we can give limF a T-algebra structure such that it is limF (we lift the limit of F along U) The subindex (s, p, ℓ) indicates (strict, pseudo, lax) algebra morphisms Previous results

1 (from the V-enriched case) T-Algs

U

− → K creates all limits.

2 T-Algp

U

− → K creates lax and pseudolimits [BKP,89].

slide-8
SLIDE 8

Limit lifting results Unifying morphisms Unifying limits Our result References

Limit lifting along the forgetful functor

K is a 2-category, T is a 2-monad on K ( K

T

− → K, id

i

⇒ T, T 2 m ⇒ T) T-Algℓ

U

  • A

F

  • F
  • K

U creates limF ≡ we can give limF a T-algebra structure such that it is limF (we lift the limit of F along U) The subindex (s, p, ℓ) indicates (strict, pseudo, lax) algebra morphisms Previous results

1 (from the V-enriched case) T-Algs

U

− → K creates all limits.

2 T-Algp

U

− → K creates lax and pseudolimits [BKP,89].

3 T-Algℓ

U

− → K creates oplax limits [Lack,05].

slide-9
SLIDE 9

Limit lifting results Unifying morphisms Unifying limits Our result References

Limit lifting along the forgetful functor

K is a 2-category, T is a 2-monad on K ( K

T

− → K, id

i

⇒ T, T 2 m ⇒ T) T-Alg?

U

  • A

F

  • F
  • K

U creates limF ≡ we can give limF a T-algebra structure such that it is limF (we lift the limit of F along U) The subindex (s, p, ℓ) indicates (strict, pseudo, lax) algebra morphisms Previous results

1 (from the V-enriched case) T-Algs

U

− → K creates all limits.

2 T-Algp

U

− → K creates lax and pseudolimits [BKP,89].

3 T-Algℓ

U

− → K creates oplax limits [Lack,05]. Note: All these limits are weighted, and the projections of the limit are always strict morphisms.

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SLIDE 10

Limit lifting results Unifying morphisms Unifying limits Our result References

Limit lifting along the forgetful functor

K is a 2-category, T is a 2-monad on K ( K

T

− → K, id

i

⇒ T, T 2 m ⇒ T) T-Alg?

U

  • A

F

  • F
  • K

U creates limF ≡ we can give limF a T-algebra structure such that it is limF (we lift the limit of F along U) The subindex (s, p, ℓ) indicates (strict, pseudo, lax) algebra morphisms Previous results

1 (from the V-enriched case) T-Algs

U

− → K creates all limits.

2 T-Algp

U

− → K creates lax and pseudolimits [BKP,89].

3 T-Algℓ

U

− → K creates oplax limits [Lack,05]. Note: All these limits are weighted, and the projections of the limit are always strict morphisms. We will present a theorem which unifies and generalizes these results.

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Limit lifting results Unifying morphisms Unifying limits Our result References

Ω-morphisms of T-algebras

A lax morphism A

f

− → B between T-algebras has a structural 2-cell TA

T f a

  • ⇓f

TB

b

  • A

f

B

1 lax (ℓ) morphism: f any 2-cell.

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SLIDE 12

Limit lifting results Unifying morphisms Unifying limits Our result References

Ω-morphisms of T-algebras

A lax morphism A

f

− → B between T-algebras has a structural 2-cell TA

T f a

  • ⇓f

TB

b

  • A

f

B

1 lax (ℓ) morphism: f any 2-cell. 2 pseudo (p) morphism: f invertible.

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SLIDE 13

Limit lifting results Unifying morphisms Unifying limits Our result References

Ω-morphisms of T-algebras

A lax morphism A

f

− → B between T-algebras has a structural 2-cell TA

T f a

  • ⇓f

TB

b

  • A

f

B

1 lax (ℓ) morphism: f any 2-cell. 2 pseudo (p) morphism: f invertible. 3 strict (s) morphism: f an identity.

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Limit lifting results Unifying morphisms Unifying limits Our result References

Ω-morphisms of T-algebras

A lax morphism A

f

− → B between T-algebras has a structural 2-cell TA

T f a

  • ⇓f

TB

b

  • A

f

B

1 lax (ℓ) morphism: f any 2-cell. 2 pseudo (p) morphism: f invertible. 3 strict (s) morphism: f an identity.

Fix a family Ω of 2-cells of K. f is a Ω-morphism if f ∈ Ω.

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Limit lifting results Unifying morphisms Unifying limits Our result References

Ω-morphisms of T-algebras

A lax morphism A

f

− → B between T-algebras has a structural 2-cell TA

T f a

  • ⇓f

TB

b

  • A

f

B

1 lax (ℓ) morphism: f any 2-cell. 2 pseudo (p) morphism: f invertible. 3 strict (s) morphism: f an identity.

Fix a family Ω of 2-cells of K. f is a Ω-morphism if f ∈ Ω. Considering Ωℓ = 2-cells(K), Ωp = {invertible 2-cells}, Ωs = {identities}, we recover the three cases above.

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Limit lifting results Unifying morphisms Unifying limits Our result References

A general notion of weighted limit. The conical case (Gray,1974)

We fix A, B 2-categories, Σ ⊆ Arrows(A), Ω ⊆ 2-cells(B)

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Limit lifting results Unifying morphisms Unifying limits Our result References

A general notion of weighted limit. The conical case (Gray,1974)

We fix A, B 2-categories, Σ ⊆ Arrows(A), Ω ⊆ 2-cells(B)

  • σ-ω-natural transformation: A

F

  • θ⇓

G

B, θ is a lax natural transformation FA

θA

  • F f
  • ⇓θf

GA

Gf

  • FB

θB

GB such that θf is in Ω when f is in Σ.

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SLIDE 18

Limit lifting results Unifying morphisms Unifying limits Our result References

A general notion of weighted limit. The conical case (Gray,1974)

We fix A, B 2-categories, Σ ⊆ Arrows(A), Ω ⊆ 2-cells(B)

  • σ-ω-natural transformation: A

F

  • θ⇓

G

B, θ is a lax natural transformation FA

θA

  • F f
  • ⇓θf

GA

Gf

  • FB

θB

GB such that θf is in Ω when f is in Σ.

  • σ-ω-cone (for F, with vertex E ∈ B): is a σ-ω-natural A

△E

  • θ⇓

F

B, i.e. FA

F f

  • E

θA

  • θB
  • ⇓θf

FB such that θf is in Ω when f is in Σ.

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Limit lifting results Unifying morphisms Unifying limits Our result References

  • σ-ω-limit: is the universal σ-ω-cone, in the sense that the following

is an isomorphism B(E, L)

π∗

− → σ-ω-Cones(E, F)

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Limit lifting results Unifying morphisms Unifying limits Our result References

  • σ-ω-limit: is the universal σ-ω-cone, in the sense that the following

is an isomorphism B(E, L)

π∗

− → σ-ω-Cones(E, F) On objects: ϕ θ FA

F f

  • E

⇓θf θA

  • θB
  • L

⇓πf πA

  • πB
  • FB
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SLIDE 21

Limit lifting results Unifying morphisms Unifying limits Our result References

  • σ-ω-limit: is the universal σ-ω-cone, in the sense that the following

is an isomorphism B(E, L)

π∗

− → σ-ω-Cones(E, F) On objects: ϕ θ FA

F f

  • E

∃ !ϕ ⇓θf θA

  • θB
  • L

⇓πf πA

  • πB
  • FB
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SLIDE 22

Limit lifting results Unifying morphisms Unifying limits Our result References

  • σ-ω-limit: is the universal σ-ω-cone, in the sense that the following

is an isomorphism B(E, L)

π∗

− → σ-ω-Cones(E, F) On objects: ϕ θ FA

F f

  • E

∃ !ϕ ⇑θf θA

  • θB
  • L

⇑πf πA

  • πB
  • FB
  • We have the dual notion of σ-ω-opnatural, yielding σ-ω-oplimits,

where the direction of the 2-cells is reversed.

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Limit lifting results Unifying morphisms Unifying limits Our result References

  • σ-ω-limit: is the universal σ-ω-cone, in the sense that the following

is an isomorphism B(E, L)

π∗

− → σ-ω-Cones(E, F) On objects: ϕ θ FA

F f

  • E

∃ !ϕ ⇑θf θA

  • θB
  • L

⇑πf πA

  • πB
  • FB
  • We have the dual notion of σ-ω-opnatural, yielding σ-ω-oplimits,

where the direction of the 2-cells is reversed.

  • The notions of lax, pseudo and strict limits are recovered with

particular choices of Ω (and Σ).

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Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K
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SLIDE 25

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K

Can we give L = σ-ω-limF a structure of algebra such that the projections are strict morphisms? TFA

a

FA TL

T πA

  • ℓ

L

πA

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SLIDE 26

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K

Can we give L = σ-ω-limF a structure of algebra such that the projections are strict morphisms? TFA

a

FA

F f

  • TL

T πB

  • T πA
  • ℓ

⇓θf L ⇓πf πA

  • πB
  • TFB

b

FB

slide-27
SLIDE 27

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K

Can we give L = σ-ω-limF a structure of algebra such that the projections are strict morphisms? We need the 2-cells θf yielding a σ-ω-cone: TFA

a

FA

F f

  • TL

⇓θf T πB

  • T πA
  • TFB

b

FB θf ∈ Ω if f ∈ Σ:

slide-28
SLIDE 28

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K

Can we give L = σ-ω-limF a structure of algebra such that the projections are strict morphisms? We need the 2-cells θf yielding a σ-ω-cone: TFA

T F f

  • F f

⇒ a

FA

F f

  • ⇓θf : TL

⇓T πf T πB

  • T πA
  • TFB

b

FB θf ∈ Ω if f ∈ Σ:

slide-29
SLIDE 29

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K

Can we give L = σ-ω-oplimF a structure of algebra such that the projections are strict morphisms? We need the 2-cells θf yielding a σ-ω-opcone: TFA

T F f

  • F f

⇒ a

FA

F f

  • ⇑θf : TL

⇑T πf T πB

  • T πA
  • TFB

b

FB θf ∈ Ω if f ∈ Σ:

slide-30
SLIDE 30

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K

Can we give L = σ-ω-oplimF a structure of algebra such that the projections are strict morphisms? We need the 2-cells θf yielding a σ-ω-opcone: TFA

T F f

  • F f

⇒ a

FA

F f

  • ⇑θf : TL

⇑T πf T πB

  • T πA
  • TFB

b

FB θf ∈ Ω if f ∈ Σ: T(Ω) ⊆ Ω

slide-31
SLIDE 31

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K

Can we give L = σ-ω-oplimF a structure of algebra such that the projections are strict morphisms? We need the 2-cells θf yielding a σ-ω-opcone: TFA

T F f

  • F f

⇒ a

FA

F f

  • ⇑θf : TL

⇑T πf T πB

  • T πA
  • TFB

b

FB θf ∈ Ω if f ∈ Σ: T(Ω) ⊆ Ω , Ω′ ⊆ Ω

slide-32
SLIDE 32

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K

Can we give L = σ-ω-oplimF a structure of algebra such that the projections are strict morphisms? We need the 2-cells θf yielding a σ-ω-opcone: TFA

T F f

  • F f

⇒ a

FA

F f

  • ⇑θf : TL

⇑T πf T πB

  • T πA
  • TFB

b

FB θf ∈ Ω if f ∈ Σ: T(Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL

ℓ

− → L.

slide-33
SLIDE 33

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (finding the hypotheses)

We consider Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). The σ-ω-limits are always taken with respect to Σ and Ω. T-AlgΩ′

U

  • A

F

  • F
  • K

Can we give L = σ-ω-oplimF a structure of algebra such that the projections are strict morphisms? We need the 2-cells θf yielding a σ-ω-opcone: TFA

T F f

  • F f

⇒ a

FA

F f

  • ⇑θf : TL

⇑T πf T πB

  • T πA
  • TFB

b

FB θf ∈ Ω if f ∈ Σ: T(Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL

ℓ

− → L. The limit L is Ω′-compatible ⇒ (TL, ℓ) is the desired lifted limit.

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SLIDE 34

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (properly stated)

Theorem: Let Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). Assume T(Ω) ⊆ Ω and Ω′ ⊆ Ω. Then T-AlgΩ′

U

− → K creates Ω′-compatible σ-ω-oplimits. the proof follows the ideas of the previous slide

slide-35
SLIDE 35

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (properly stated)

Theorem: Let Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). Assume T(Ω) ⊆ Ω and Ω′ ⊆ Ω. Then T-AlgΩ′

U

− → K creates Ω′-compatible σ-ω-oplimits. the proof follows the ideas of the previous slide We deduce the result for weighted σ-ω-limits, by showing that they can be expressed as conical σ-ω-limits.

slide-36
SLIDE 36

Limit lifting results Unifying morphisms Unifying limits Our result References

Our limit lifting theorem (properly stated)

Theorem: Let Σ ⊆ Arrows(A), Ω, Ω′ ⊆ 2-cells(K). Assume T(Ω) ⊆ Ω and Ω′ ⊆ Ω. Then T-AlgΩ′

U

− → K creates Ω′-compatible σ-ω-oplimits. the proof follows the ideas of the previous slide We deduce the result for weighted σ-ω-limits, by showing that they can be expressed as conical σ-ω-limits. The case Ω, Ω′ ∈ {Ωℓ, Ωp, Ωs} T(Ω) ⊆ Ω ✓, Ω′-compatible ✓

1 (with Ω = Ω′ = Ωs) T-Algs

U

− → K creates all (strict) limits.

2 (with Ω = Ω′ = Ωp) T-Algp

U

− → K creates σ-limits (thus in particular lax and pseudolimits).

3 (with Ω = Ω′ = Ωℓ) T-Algℓ

U

− → K creates oplax limits.

slide-37
SLIDE 37

Limit lifting results Unifying morphisms Unifying limits Our result References

Thank you for your attention!

References [BKP,89] Blackwell R., Kelly G. M., Power A.J., Two-dimensional monad theory, JPAA 59. [Gray,74] Gray J. W., Formal category theory: adjointness for 2-categories, Springer LNM 391. [Lack,05] Lack S., Limits for lax morphisms, ACS 13. A general limit lifting theorem for 2-dimensional monad theory is available as arXiv:1702.03303.