SLIDE 1
Weak adjoint functor theorems Stephen Lack Macquarie University - - PowerPoint PPT Presentation
Weak adjoint functor theorems Stephen Lack Macquarie University - - PowerPoint PPT Presentation
Weak adjoint functor theorems Stephen Lack Macquarie University Kyoto, 23 December 2019 joint work with John Bourke and Luk a s Vok r nek Adjoint Functor Theorems ur-AFT category B with all limits U : B A preserves them U has
SLIDE 2
SLIDE 3
Adjoint Functor Theorems
ur-AFT
category B with all limits U : B → A preserves them U has left adjoint
General AFT
category B with small limits U : B → A preserves them Solution Set Condition U has left adjoint
SLIDE 4
more Adjoint Functor Theorems
General AFT (Freyd)
category B with small limits U : B → A preserves them SSC U has left adjoint
Weak AFT (Kainen)
category B with small products U : B → A preserves them SSC U has weak left adjoint
Enriched AFT (Kelly)
V-category B with small limits U : B → A preserves them SSC U has left adjoint
SLIDE 5
more Adjoint Functor Theorems
General AFT (Freyd)
category B with small limits U : B → A preserves them SSC U has left adjoint
Weak AFT (Kainen)
category B with small products U : B → A preserves them SSC U has weak left adjoint
Enriched AFT (Kelly)
V-category B with small limits U : B → A preserves them SSC U has left adjoint A UFA FA UB B
η f ∃f ′ Uf ′
SLIDE 6
more Adjoint Functor Theorems
General AFT (Freyd)
category B with small limits U : B → A preserves them SSC U has left adjoint
Weak AFT (Kainen)
category B with small products U : B → A preserves them SSC U has weak left adjoint
Enriched AFT (Kelly)
V-category B with small limits U : B → A preserves them SSC U has left adjoint
Very (!) General AFT
V-category B with limits U : B → A preserves them SSC U has left adjoint A UFA FA UB B
η f ∃f ′ Uf ′
SLIDE 7
more Adjoint Functor Theorems
General AFT (Freyd)
category B with small limits U : B → A preserves them SSC U has left adjoint
Weak AFT (Kainen)
category B with small products U : B → A preserves them SSC U has weak left adjoint
Enriched AFT (Kelly)
V-category B with small limits U : B → A preserves them SSC U has left adjoint
Very (!) General AFT
V-category B with limits U : B → A preserves them SSC U has left adjoint A UFA FA UB B
η f ∃f ′ Uf ′
SLIDE 8
Enriched weakness
A UFA FA UB B
η f ∃f ′ Uf ′
B(FA, B) A(A, UB)
surj.
◮ enriched categories have homs C(C, D) lying in V ◮ (Lack-Rosicky) “Enriched Weakness” uses class E of morphisms in V to play the role of surjections ◮ V = Set, E = {surjections} gives unenriched weakness ◮ E = {isomorphisms} gives “non-weak weakness”
SLIDE 9
Enriched weakness
A UFA FA UB B
η f ∃f ′ Uf ′
B(FA, B) A(A, UB)
E-map
◮ enriched categories have homs C(C, D) lying in V ◮ (Lack-Rosicky) “Enriched Weakness” uses class E of morphisms in V to play the role of surjections ◮ V = Set, E = {surjections} gives unenriched weakness ◮ E = {isomorphisms} gives “non-weak weakness”
Very (!) General AFT
V-category B with limits U : B → A preserves them SSC U has E-weak left adjoint
SLIDE 10
Examples
B(FA, B) A(A, UB)
E
V E E-weak left adjoint Set isos left adjoint Set surjections weak left adjoint V isos (enriched) left adjoint Cat equivalences left biadjoint
SLIDE 11
Examples
B(FA, B) A(A, UB)
E
V E E-weak left adjoint Set isos left adjoint Set surjections weak left adjoint V isos (enriched) left adjoint Cat equivalences left biadjoint Cat retract equivalences (. . . ) left biadjoint
SLIDE 12
Mr Retract Equivalence
SLIDE 13
Examples
B(FA, B) A(A, UB)
E
V E E-weak left adjoint Set isos left adjoint Set surjections weak left adjoint V isos (enriched) left adjoint Cat equivalences left biadjoint Cat retract equivalences (. . . ) left biadjoint
SLIDE 14
Examples
B(FA, B) A(A, UB)
E
V E E-weak left adjoint Set isos left adjoint Set surjections weak left adjoint V isos (enriched) left adjoint Cat equivalences left biadjoint Cat retract equivalences (. . . ) left biadjoint sSet shrinkable morphisms
Definition
A morphism p : X → Y of simplicial sets is shrinkable (dual strong deformation retract) if it is contractible in sSet/Y : ◮ it has a section s ◮ with a homotopy s ◦ p ∼ 1X ◮ such that induced homotopy p ◦ s ◦ p ∼ p is trivial.
SLIDE 15
The setting
Let V be a monoidal model category with cofibrant unit I . . .
SLIDE 16
The setting
Let V be a monoidal model category with cofibrant unit I . . . cofibrations I, weak equivalences W, trivial fibrations P.
SLIDE 17
The setting
Let V be a monoidal model category with cofibrant unit I . . . cofibrations I, weak equivalences W, trivial fibrations P. An interval in V is a factorization I + I J I i w ∇
- f the codiagonal with i ∈ I and w ∈ W.
◮ A morphism A → B in a V-category C is a morphism I → C(A, B). ◮ A homotopy between morphisms A → B is a morphism J → C(A, B) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is shrinkable if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.
SLIDE 18
The setting
Let V be a monoidal model category with cofibrant unit I . . . cofibrations I, weak equivalences W, trivial fibrations P. An interval in V is a factorization I + I J I i w C(A, B) I f
- f the codiagonal with i ∈ I and w ∈ W.
◮ A morphism A → B in a V-category C is a morphism I → C(A, B). ◮ A homotopy between morphisms A → B is a morphism J → C(A, B) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is shrinkable if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.
SLIDE 19
The setting
Let V be a monoidal model category with cofibrant unit I . . . cofibrations I, weak equivalences W, trivial fibrations P. An interval in V is a factorization I + I J I i w C(A, B) C(A, B) (f g) h
- f the codiagonal with i ∈ I and w ∈ W.
◮ A morphism A → B in a V-category C is a morphism I → C(A, B). ◮ A homotopy between morphisms A → B is a morphism J → C(A, B) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is shrinkable if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.
SLIDE 20
The setting
Let V be a monoidal model category with cofibrant unit I . . . cofibrations I, weak equivalences W, trivial fibrations P. An interval in V is a factorization I + I J I i w C(A, B) C(A, B) (f =g) h
- f the codiagonal with i ∈ I and w ∈ W.
◮ A morphism A → B in a V-category C is a morphism I → C(A, B). ◮ A homotopy between morphisms A → B is a morphism J → C(A, B) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is shrinkable if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.
SLIDE 21
The setting
Let V be a monoidal model category with cofibrant unit I . . . cofibrations I, weak equivalences W, trivial fibrations P. An interval in V is a factorization I + I J I i w C(A, B) C(A, B) (f g) h X Y p
- f the codiagonal with i ∈ I and w ∈ W.
◮ A morphism A → B in a V-category C is a morphism I → C(A, B). ◮ A homotopy between morphisms A → B is a morphism J → C(A, B) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is shrinkable if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.
SLIDE 22
The setting
Let V be a monoidal model category with cofibrant unit I . . . cofibrations I, weak equivalences W, trivial fibrations P. An interval in V is a factorization I + I J I i w C(A, B) C(A, B) (f g) h X Y p Y 1 s
- f the codiagonal with i ∈ I and w ∈ W.
◮ A morphism A → B in a V-category C is a morphism I → C(A, B). ◮ A homotopy between morphisms A → B is a morphism J → C(A, B) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is shrinkable if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.
SLIDE 23
The setting
Let V be a monoidal model category with cofibrant unit I . . . cofibrations I, weak equivalences W, trivial fibrations P. An interval in V is a factorization I + I J I i w C(A, B) C(A, B) (f g) h X Y p Y 1 s X ≃ p 1
- f the codiagonal with i ∈ I and w ∈ W.
◮ A morphism A → B in a V-category C is a morphism I → C(A, B). ◮ A homotopy between morphisms A → B is a morphism J → C(A, B) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is shrinkable if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.
SLIDE 24
Examples (of V)
V I W shrinkable morphisms Set all isos isos V all isos isos Set mono all surj Cat inj obj equiv retract equivalences sSet mono wk hty equiv (Kan) shrinkable sSet mono wk cat equiv (Joyal) shrinkable 2-Cat biequivalences surj, full biequivalences
SLIDE 25
Examples (of V)
V I W shrinkable morphisms Set all isos isos V all isos isos Set mono all surj Cat inj obj equiv retract equivalences sSet mono wk hty equiv (Kan) shrinkable sSet mono wk cat equiv (Joyal) shrinkable 2-Cat biequivalences surj, full biequivalences In general, for f : X → Y in V: ◮ trivial fibration ⇒ shrinkable ( if X, Y cofibrant ) ◮ shrinkable ⇒ weak equiv (if X fibrant or cofibrant)
SLIDE 26
Examples (of V)
V I W shrinkable morphisms Set all isos isos V all isos isos Set mono all surj Cat inj obj equiv retract equivalences sSet mono wk hty equiv (Kan) shrinkable sSet mono wk cat equiv (Joyal) shrinkable 2-Cat biequivalences surj, full biequivalences In general, for f : X → Y in V: ◮ trivial fibration ⇒ shrinkable ( if X, Y cofibrant ) ◮ shrinkable ⇒ weak equiv (if X fibrant or cofibrant)
Very (!) General AFT
V-category B with limits U : B → A preserves them SSC U has E-weak left adjoint for E = {shrinkables}
SLIDE 27
The limits in question
Limit of a functor S : D → B is defined by natural isos B(B, lim S) ∼ = [D, Set](∆1, B(B, S)). Limit of V-functor S : D → B weighted by G : D → V defined by B(B, lim
G S) ∼
= [D, V](G, B(B, S)). The power of A ∈ B by X ∈ V defined by B(B, AX) ∼ = V(X, B(B, A)) A weight G : D → V is cofibrant if it is projective with respect to the pointwise trivial fibrations: A V-category B has enough cofibrant limits if for any weight G there is a cofibrant G ′ with a pointwise trivial fibration G ′ → G for which B has G ′-weighted limits.
SLIDE 28
The limits in question
Limit of a functor S : D → B is defined by natural isos B(B, lim S) ∼ = [D, Set](∆1, B(B, S)). Limit of V-functor S : D → B weighted by G : D → V defined by B(B, lim
G S) ∼
= [D, V](G, B(B, S)). The power of A ∈ B by X ∈ V defined by B(B, AX) ∼ = V(X, B(B, A)) A weight G : D → V is cofibrant if it is projective with respect to the pointwise trivial fibrations: A V-category B has enough cofibrant limits if for any weight G there is a cofibrant G ′ with a pointwise trivial fibration G ′ → G for which B has G ′-weighted limits.
SLIDE 29
The limits in question
Limit of a functor S : D → B is defined by natural isos B(B, lim S) ∼ = [D, Set](∆1, B(B, S)). Limit of V-functor S : D → B weighted by G : D → V defined by B(B, lim
G S) ∼
= [D, V](G, B(B, S)). The power of A ∈ B by X ∈ V defined by B(B, AX) ∼ = V(X, B(B, A)) A weight G : D → V is cofibrant if it is projective with respect to the pointwise trivial fibrations: A V-category B has enough cofibrant limits if for any weight G there is a cofibrant G ′ with a pointwise trivial fibration G ′ → G for which B has G ′-weighted limits.
SLIDE 30
The limits in question
Limit of a functor S : D → B is defined by natural isos B(B, lim S) ∼ = [D, Set](∆1, B(B, S)). Limit of V-functor S : D → B weighted by G : D → V defined by B(B, lim
G S) ∼
= [D, V](G, B(B, S)). The power of A ∈ B by X ∈ V defined by B(B, AX) ∼ = V(X, B(B, A)) A weight G : D → V is cofibrant if it is projective with respect to the pointwise trivial fibrations: A V-category B has enough cofibrant limits if for any weight G there is a cofibrant G ′ with a pointwise trivial fibration G ′ → G for which B has G ′-weighted limits.
SLIDE 31
The limits in question
Limit of a functor S : D → B is defined by natural isos B(B, lim S) ∼ = [D, Set](∆1, B(B, S)). Limit of V-functor S : D → B weighted by G : D → V defined by B(B, lim
G S) ∼
= [D, V](G, B(B, S)). The power of A ∈ B by X ∈ V defined by B(B, AX) ∼ = V(X, B(B, A)) A weight G : D → V is cofibrant if it is projective with respect to the pointwise trivial fibrations: G K H KD HD p pD ∈ P A V-category B has enough cofibrant limits if for any weight G there is a cofibrant G ′ with a pointwise trivial fibration G ′ → G for which B has G ′-weighted limits.
SLIDE 32
The limits in question
Limit of a functor S : D → B is defined by natural isos B(B, lim S) ∼ = [D, Set](∆1, B(B, S)). Limit of V-functor S : D → B weighted by G : D → V defined by B(B, lim
G S) ∼
= [D, V](G, B(B, S)). The power of A ∈ B by X ∈ V defined by B(B, AX) ∼ = V(X, B(B, A)) A weight G : D → V is cofibrant if it is projective with respect to the pointwise trivial fibrations: G K H KD HD p pD ∈ P A V-category B has enough cofibrant limits if for any weight G there is a cofibrant G ′ with a pointwise trivial fibration G ′ → G for which B has G ′-weighted limits.
SLIDE 33
The limits in question
Limit of a functor S : D → B is defined by natural isos B(B, lim S) ∼ = [D, Set](∆1, B(B, S)). Limit of V-functor S : D → B weighted by G : D → V defined by B(B, lim
G S) ∼
= [D, V](G, B(B, S)). The power of A ∈ B by X ∈ V defined by B(B, AX) ∼ = V(X, B(B, A)) A weight G : D → V is cofibrant if it is projective with respect to the pointwise trivial fibrations: G K H KD HD p pD ∈ P
∃
A V-category B has enough cofibrant limits if for any weight G there is a cofibrant G ′ with a pointwise trivial fibration G ′ → G for which B has G ′-weighted limits.
SLIDE 34
The limits in question
Limit of a functor S : D → B is defined by natural isos B(B, lim S) ∼ = [D, Set](∆1, B(B, S)). Limit of V-functor S : D → B weighted by G : D → V defined by B(B, lim
G S) ∼
= [D, V](G, B(B, S)). The power of A ∈ B by X ∈ V defined by B(B, AX) ∼ = V(X, B(B, A)) A weight G : D → V is cofibrant if it is projective with respect to the pointwise trivial fibrations: G K H KD HD p pD ∈ P
∃
A V-category B has enough cofibrant limits if for any weight G there is a cofibrant G ′ with a pointwise trivial fibration G ′ → G for which B has G ′-weighted limits.
SLIDE 35
The VGAFT
Let V be a monoidal model category with cofibrant unit, and E the shrinkable morphisms.
V(ery) G(eneral) AFT
V-category B with all powers and enough cofibrant limits U : B → A preserves them SSC U has an E-weak left adjoint
SLIDE 36
The VGAFT
Let V be a monoidal model category with cofibrant unit, and E the shrinkable morphisms.
V(ery) G(eneral) AFT
V-category B with all powers and enough cofibrant limits U : B → A preserves them SSC U has an E-weak left adjoint
N(ot) Q(uite) S(o) G(eneral) AFT
V-category B with all powers and enough cofibrant limits U : B → A preserves them B0 and A0 are accessible, U0 accessible functor (unenriched) U has an E-weak left adjoint
SLIDE 37
Applications (to 2-categories)
This involves the case V = Cat. A 2-category will have enough cofibrant limits if it has PIE limits: that is, if it has ◮ products ◮ inserters ◮ equifiers.
SLIDE 38
Mr PIE
SLIDE 39
Applications (to 2-categories)
This involves the case V = Cat. A 2-category will have enough cofibrant limits if it has PIE limits: that is, if it has ◮ products ◮ inserters ◮ equifiers.
SLIDE 40
Applications (to 2-categories)
This involves the case V = Cat. A 2-category will have enough cofibrant limits if it has PIE limits: that is, if it has ◮ products ◮ inserters ◮ equifiers.
Theorem
Let B be a 2-category with PIE limits, and let U : B → A preserve
- them. If U satisfies SSC, then it has a left biadjoint.
Theorem
Any accessible 2-category with PIE limits has bicolimits.
SLIDE 41
Application (to Riehl-Verity ∞-cosmoi)
An ∞-cosmos is a sSet-category with all powers, enough cofibrant limits, and certain further structure. These are intended to be a model-independent framework in which to study the totality of (∞, 1)-categories and related structures.
Corollary
Any accessible ∞-cosmos has weak colimits. A cosmological functor is an enriched functor between ∞-cosmoi which preserves this structure.
Corollary
Any cosomological functor satisfying the SSC has a weak left adjoint.
SLIDE 42
Application (to Riehl-Verity ∞-cosmoi)
An ∞-cosmos is a sSet-category with all powers, enough cofibrant limits, and certain further structure. These are intended to be a model-independent framework in which to study the totality of (∞, 1)-categories and related structures.
Corollary
Any accessible ∞-cosmos has weak colimits. A cosmological functor is an enriched functor between ∞-cosmoi which preserves this structure.
Corollary
Any cosomological functor satisfying the SSC has a weak left adjoint.
SLIDE 43
Application (to Riehl-Verity ∞-cosmoi)
An ∞-cosmos is a sSet-category with all powers, enough cofibrant limits, and certain further structure. These are intended to be a model-independent framework in which to study the totality of (∞, 1)-categories and related structures.
Corollary
Any accessible ∞-cosmos has weak colimits. A cosmological functor is an enriched functor between ∞-cosmoi which preserves this structure.
Corollary
Any cosomological functor satisfying the SSC has a weak left adjoint.
SLIDE 44
Application (to Riehl-Verity ∞-cosmoi)
An ∞-cosmos is a sSet-category with all powers, enough cofibrant limits, and certain further structure. These are intended to be a model-independent framework in which to study the totality of (∞, 1)-categories and related structures.
Corollary
Any accessible ∞-cosmos has weak colimits. A cosmological functor is an enriched functor between ∞-cosmoi which preserves this structure.
Corollary
Any cosomological functor satisfying the SSC has a weak left adjoint.
SLIDE 45