A general limit lifting theorem for 2-dimensional monad ...ct2017/slides/szyld_m.pdf · KU creates...
Transcript of A general limit lifting theorem for 2-dimensional monad ...ct2017/slides/szyld_m.pdf · KU creates...
Limit lifting results Unifying morphisms Unifying limits Our result References
A general limit lifting theorem for 2-dimensionalmonad theory
(but don’t let the long title scare you!)
Martin SzyldUniversity of Buenos Aires - CONICET, Argentina
CT 2017 @ UBC, Vancouver, Canada
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 (KT−→ K , id
i⇒ T , T 2 m⇒ T )
T -Alg
U
AF //
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, `) indicates (strict, pseudo, lax) algebra morphisms
Previous results
1 (from the V-enriched case)U−→ creates all limits.
2 T -AlgpU−→ K creates lax and pseudolimits [BKP,89].
3 T -Alg`U−→ K creates oplax limits [Lack,05].
Note: All these limits are weighted, and the projections of the limitare always strict morphisms.
We will present a theorem which unifies and generalizes these results.
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 (KT−→ K , id
i⇒ T , T 2 m⇒ T )
T -Alg
U
AF //
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, `) indicates (strict, pseudo, lax) algebra morphisms
Previous results
1 (from the V-enriched case) T -AlgU−→ K creates all limits.
2 T -AlgpU−→ K creates lax and pseudolimits [BKP,89].
3 T -Alg`U−→ K creates oplax limits [Lack,05].
Note: All these limits are weighted, and the projections of the limitare always strict morphisms.
We will present a theorem which unifies and generalizes these results.
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, idi⇒ T , T 2 m⇒ T )
T -Alg
U
AF //
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, `) indicates (strict, pseudo, lax) algebra morphisms
Previous results
1 (from the V-enriched case) T -AlgU−→ K creates all limits.
2 T -AlgpU−→ K creates lax and pseudolimits [BKP,89].
3 T -Alg`U−→ K creates oplax limits [Lack,05].
Note: All these limits are weighted, and the projections of the limitare always strict morphisms.
We will present a theorem which unifies and generalizes these results.
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, idi⇒ 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, `) indicates (strict, pseudo, lax) algebra morphisms
Previous results
1 (from the V-enriched case) T -AlgU−→ K creates all limits.
2 T -AlgpU−→ K creates lax and pseudolimits [BKP,89].
3 T -Alg`U−→ K creates oplax limits [Lack,05].
Note: All these limits are weighted, and the projections of the limitare always strict morphisms.
We will present a theorem which unifies and generalizes these results.
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, idi⇒ 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, `) indicates (strict, pseudo, lax) algebra morphisms
Previous results
1 (from the V-enriched case) T -AlgsU−→ K creates all limits.
2 T -AlgpU−→ K creates lax and pseudolimits [BKP,89].
3 T -Alg`U−→ K creates oplax limits [Lack,05].
Note: All these limits are weighted, and the projections of the limitare always strict morphisms.
We will present a theorem which unifies and generalizes these results.
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, idi⇒ 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, `) indicates (strict, pseudo, lax) algebra morphisms
Previous results
1 (from the V-enriched case) T -AlgsU−→ K creates all limits.
2 T -AlgpU−→ K creates lax and pseudolimits [BKP,89].
3 T -Alg`U−→ K creates oplax limits [Lack,05].
Note: All these limits are weighted, and the projections of the limitare always strict morphisms.
We will present a theorem which unifies and generalizes these results.
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, idi⇒ 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, `) indicates (strict, pseudo, lax) algebra morphisms
Previous results
1 (from the V-enriched case) T -AlgsU−→ K creates all limits.
2 T -AlgpU−→ K creates lax and pseudolimits [BKP,89].
3 T -Alg`U−→ K creates oplax limits [Lack,05].
Note: All these limits are weighted, and the projections of the limitare always strict morphisms.
We will present a theorem which unifies and generalizes these results.
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, idi⇒ 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, `) indicates (strict, pseudo, lax) algebra morphisms
Previous results
1 (from the V-enriched case) T -AlgsU−→ K creates all limits.
2 T -AlgpU−→ K creates lax and pseudolimits [BKP,89].
3 T -Alg`U−→ K creates oplax limits [Lack,05].
Note: All these limits are weighted, and the projections of the limitare always strict morphisms.
We will present a theorem which unifies and generalizes these results.
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, idi⇒ 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, `) indicates (strict, pseudo, lax) algebra morphisms
Previous results
1 (from the V-enriched case) T -AlgsU−→ K creates all limits.
2 T -AlgpU−→ K creates lax and pseudolimits [BKP,89].
3 T -Alg`U−→ K creates oplax limits [Lack,05].
Note: All these limits are weighted, and the projections of the limitare always strict morphisms.
We will present a theorem which unifies and generalizes these results.
Limit lifting results Unifying morphisms Unifying limits Our result References
Ω-morphisms of T -algebras
A lax morphism Af−→ B between T -algebras has a structural 2-cell
TATf//
a
⇓f
TB
b
Af// B
1 lax (`) 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 Ω` = 2-cells(K), Ωp = invertible 2-cells,Ωs = identities, we recover the three cases above.
Limit lifting results Unifying morphisms Unifying limits Our result References
Ω-morphisms of T -algebras
A lax morphism Af−→ B between T -algebras has a structural 2-cell
TATf//
a
⇓f
TB
b
Af// B
1 lax (`) 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 Ω` = 2-cells(K), Ωp = invertible 2-cells,Ωs = identities, we recover the three cases above.
Limit lifting results Unifying morphisms Unifying limits Our result References
Ω-morphisms of T -algebras
A lax morphism Af−→ B between T -algebras has a structural 2-cell
TATf//
a
⇓f
TB
b
Af// B
1 lax (`) 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 Ω` = 2-cells(K), Ωp = invertible 2-cells,Ωs = identities, we recover the three cases above.
Limit lifting results Unifying morphisms Unifying limits Our result References
Ω-morphisms of T -algebras
A lax morphism Af−→ B between T -algebras has a structural 2-cell
TATf//
a
⇓f
TB
b
Af// B
1 lax (`) 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 Ω` = 2-cells(K), Ωp = invertible 2-cells,Ωs = identities, we recover the three cases above.
Limit lifting results Unifying morphisms Unifying limits Our result References
Ω-morphisms of T -algebras
A lax morphism Af−→ B between T -algebras has a structural 2-cell
TATf//
a
⇓f
TB
b
Af// B
1 lax (`) 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 Ω` = 2-cells(K), Ωp = invertible 2-cells,Ωs = identities, we recover the three cases above.
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: AF //θ⇓G// B, θ is a lax natural
transformation
FAθA //
Ff
⇓θf
GA
Gf
FBθB
// 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 A4E
//θ⇓F// B,
i.e.
FA
Ff
E
θA 77
θB''
⇓θf
FB
such that θf is in Ω when f is in Σ.
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: AF //θ⇓G// B, θ is a lax natural
transformation
FAθA //
Ff
⇓θf
GA
Gf
FBθB
// 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 A4E
//θ⇓F// B,
i.e.
FA
Ff
E
θA 77
θB''
⇓θf
FB
such that θf is in Ω when f is in Σ.
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: AF //θ⇓G// B, θ is a lax natural
transformation
FAθA //
Ff
⇓θf
GA
Gf
FBθB
// 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 A4E
//θ⇓F// B,
i.e.
FA
Ff
E
θA 77
θB''
⇓θf
FB
such that θf is in Ω when f is in Σ.
Limit lifting results Unifying morphisms Unifying limits Our result References
• σ-ω-limit: is the universal σ-ω-cone, in the sense that the followingis an isomorphism
B(E,L)π∗−→ σ-ω-Cones(E,F )
On objects: ϕ oo // θ
FA
Ff
E
θA //
θB//
L
πA99
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 withparticular choices of Ω (and Σ).
Limit lifting results Unifying morphisms Unifying limits Our result References
• σ-ω-limit: is the universal σ-ω-cone, in the sense that the followingis an isomorphism
B(E,L)π∗−→ σ-ω-Cones(E,F )
On objects: ϕ oo // θ
FA
Ff
E ⇓θf
θA //
θB//
L ⇓πf
πA99
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 withparticular choices of Ω (and Σ).
Limit lifting results Unifying morphisms Unifying limits Our result References
• σ-ω-limit: is the universal σ-ω-cone, in the sense that the followingis an isomorphism
B(E,L)π∗−→ σ-ω-Cones(E,F )
On objects: ϕ oo // θ
FA
Ff
E∃ !ϕ
⇓θf//
θA //
θB//
L ⇓πf
πA99
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 withparticular choices of Ω (and Σ).
Limit lifting results Unifying morphisms Unifying limits Our result References
• σ-ω-limit: is the universal σ-ω-cone, in the sense that the followingis an isomorphism
B(E,L)π∗−→ σ-ω-Cones(E,F )
On objects: ϕ oo // θ
FA
Ff
E∃ !ϕ
⇑θf//
θA //
θB//
L ⇑πf
πA99
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 withparticular choices of Ω (and Σ).
Limit lifting results Unifying morphisms Unifying limits Our result References
• σ-ω-limit: is the universal σ-ω-cone, in the sense that the followingis an isomorphism
B(E,L)π∗−→ σ-ω-Cones(E,F )
On objects: ϕ oo // θ
FA
Ff
E∃ !ϕ
⇑θf//
θA //
θB//
L ⇑πf
πA99
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 withparticular choices of Ω (and Σ).
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-limF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-cone:
TFAa // FA
TL
TπA77
L
TFB FB
θf ∈ Ω if f ∈ Σ: T (Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-limF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-cone:
TFAa // FA
TL
TπA77
`// L
πA77
TFB FB
θf ∈ Ω if f ∈ Σ: T (Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-limF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-cone:
TFAa // FA
Ff
TL
TπB''
TπA77
`
⇓θf// L ⇓πf
πA77
πB ''TFB
b// FB
θf ∈ Ω if f ∈ Σ: T (Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-limF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-cone:
TFAa // FA
Ff
TL ⇓θf
TπB''
TπA77
L
TFBb// FB
θf ∈ Ω if f ∈ Σ:
T (Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-limF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-cone:
TFA
TFf
Ff⇒
a // FA
Ff
⇓θf : TL ⇓Tπf
TπB''
TπA77
L
TFBb// FB
θf ∈ Ω if f ∈ Σ:
T (Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-oplimF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-opcone:
TFA
TFf
Ff⇒
a // FA
Ff
⇑θf : TL ⇑Tπf
TπB''
TπA77
L
TFBb// FB
θf ∈ Ω if f ∈ Σ:
T (Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-oplimF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-opcone:
TFA
TFf
Ff⇒
a // FA
Ff
⇑θf : TL ⇑Tπf
TπB''
TπA77
L
TFBb// FB
θf ∈ Ω if f ∈ Σ: T (Ω) ⊆ Ω
, Ω′ ⊆ Ω ⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-oplimF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-opcone:
TFA
TFf
Ff⇒
a // FA
Ff
⇑θf : TL ⇑Tπf
TπB''
TπA77
L
TFBb// FB
θf ∈ Ω if f ∈ Σ: T (Ω) ⊆ Ω , Ω′ ⊆ Ω
⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-oplimF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-opcone:
TFA
TFf
Ff⇒
a // FA
Ff
⇑θf : TL ⇑Tπf
TπB''
TπA77
L
TFBb// FB
θf ∈ Ω if f ∈ Σ: T (Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 arealways taken with respect to Σ and Ω.
T -AlgΩ′
U
A F //
F
;;
K
Can we give L = σ-ω-oplimF a structure of algebrasuch that the projections are strict morphisms?
We need the 2-cells θf yielding a σ-ω-opcone:
TFA
TFf
Ff⇒
a // FA
Ff
⇑θf : TL ⇑Tπf
TπB''
TπA77
L
TFBb// FB
θf ∈ Ω if f ∈ Σ: T (Ω) ⊆ Ω , Ω′ ⊆ Ω ⇒ TL`−→ L.
The limit L is Ω′-compatible ⇒ (TL, `) is the desired lifted limit.
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 theycan be expressed as conical σ-ω-limits.
The case Ω,Ω′ ∈ Ω`,Ωp,ΩsT (Ω) ⊆ Ω 3, Ω′-compatible 3
1 (with Ω = Ω′ = Ωs) T -AlgsU−→ K creates all (strict) limits.
2 (with Ω = Ω′ = Ωp) T -AlgpU−→ K creates σ-limits (thus in
particular lax and pseudolimits).
3 (with Ω = Ω′ = Ω`) T -Alg`U−→ K creates oplax limits.
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 theycan be expressed as conical σ-ω-limits.
The case Ω,Ω′ ∈ Ω`,Ωp,ΩsT (Ω) ⊆ Ω 3, Ω′-compatible 3
1 (with Ω = Ω′ = Ωs) T -AlgsU−→ K creates all (strict) limits.
2 (with Ω = Ω′ = Ωp) T -AlgpU−→ K creates σ-limits (thus in
particular lax and pseudolimits).
3 (with Ω = Ω′ = Ω`) T -Alg`U−→ K creates oplax limits.
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 theycan be expressed as conical σ-ω-limits.
The case Ω,Ω′ ∈ Ω`,Ωp,ΩsT (Ω) ⊆ Ω 3, Ω′-compatible 3
1 (with Ω = Ω′ = Ωs) T -AlgsU−→ K creates all (strict) limits.
2 (with Ω = Ω′ = Ωp) T -AlgpU−→ K creates σ-limits (thus in
particular lax and pseudolimits).
3 (with Ω = Ω′ = Ω`) T -Alg`U−→ K creates oplax limits.
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-dimensionalmonad theory, JPAA 59.
[Gray,74] Gray J. W., Formal category theory: adjointness for2-categories, Springer LNM 391.
[Lack,05] Lack S., Limits for lax morphisms, ACS 13.
A general limit lifting theorem for 2-dimensional monad theory isavailable as arXiv:1702.03303.