Weak adjoint functor theorems

Weak adjoint functor theorems

Stephen Lack

Macquarie University

Kyoto, 23 December 2019

joint work with John Bourke and Luk´aˇs Vokˇr´ınek

Adjoint Functor Theorems

ur-AFT

category B with alllimits U:B → Apreserves them U has left adjoint

General AFT

categoryB with small limits U:B → Apreserves them Solution Set Condition U has left adjoint

Adjoint Functor Theorems

ur-AFT

category B with all limits U:B → Apreserves them U has left adjoint

General AFT

category B with small limits U:B → Apreserves them Solution Set Condition U has left adjoint

more Adjoint Functor Theorems

General AFT (Freyd) category B with small limits U:B → Apreserves them SSC

U has left adjoint Weak AFT (Kainen)

category B with small products U:B → Apreserves them SSC

U has weak left adjoint

Enriched AFT (Kelly)

V-categoryB with small limits U:B → Apreserves them SSC

U has left adjoint

Very (!) General AFT V-categoryB with limits U:B → Apreserves them SSC

U has left adjoint

A UFA FA

UB B

η

f Uf⁰ ∃f⁰

more Adjoint Functor Theorems

General AFT (Freyd) category B with small limits U:B → Apreserves them SSC

U has left adjoint Weak AFT (Kainen)

category B with smallproducts U:B → Apreserves them SSC

U hasweak left adjoint

Enriched AFT (Kelly)

V-categoryB with small limits U:B → Apreserves them SSC

U has left adjoint

Very (!) General AFT V-categoryB with limits U:B → Apreserves them SSC

U has left adjoint

A UFA FA

UB B

η

f Uf⁰ ∃f⁰

more Adjoint Functor Theorems

General AFT (Freyd) category B with small limits U:B → Apreserves them SSC

U has left adjoint Weak AFT (Kainen)

category B with small products U:B → Apreserves them SSC

U has weak left adjoint

Enriched AFT (Kelly)

V-categoryB with small limits U:B → Apreserves them SSC

U has left adjoint Very (!) General AFT

V-categoryB with limits U:B → Apreserves them SSC

U has left adjoint

A UFA FA

UB B

η

f Uf⁰ ∃f⁰

more Adjoint Functor Theorems

General AFT (Freyd) category B with small limits U:B → Apreserves them SSC

U has left adjoint Weak AFT (Kainen)

category B with small products U:B → Apreserves them SSC

U has weak left adjoint

Enriched AFT (Kelly)

V-categoryB with small limits U:B → Apreserves them SSC

U has left adjoint Very (!) General AFT

V-categoryB with limits U:B → Apreserves them SSC

U has left adjoint

A UFA FA

UB B

η

f Uf⁰ ∃f⁰

Enriched weakness

A UFA FA

UB B

η

f Uf⁰ ∃f⁰ B(FA,B) ^surj. A(A,UB) I enriched categories have homs C(C,D) lying inV

I (Lack-Rosicky) “Enriched Weakness” uses classE of morphisms inV to play the role of surjections

I V =Set,E ={surjections} gives unenriched weakness I E ={isomorphisms} gives “non-weak weakness”

Very (!) General AFT V-categoryB with limits U:B → Apreserves them SSC

U has E-weak left adjoint

Enriched weakness

A UFA FA

UB B

η

f Uf⁰ ∃f⁰ B(FA,B) ^E-map A(A,UB) I enriched categories have homs C(C,D) lying inV

I (Lack-Rosicky) “Enriched Weakness” uses classE of morphisms inV to play the role of surjections

I V =Set,E ={surjections} gives unenriched weakness I E ={isomorphisms} gives “non-weak weakness”

Very (!) General AFT V-categoryB with limits U:B → Apreserves them SSC

U has E-weak left adjoint

Examples

B(FA,B) ^E A(A,UB) 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 morphismp:X →Y of simplicial sets is shrinkable(dual strong deformation retract) if it is contractible insSet/Y:

I it has a section s

I with a homotopys ◦p ∼1_X

I such that induced homotopyp◦s◦p ∼p is trivial.

Examples

B(FA,B) ^E A(A,UB) 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 morphismp:X →Y of simplicial sets is shrinkable(dual strong deformation retract) if it is contractible insSet/Y:

I it has a section s

I with a homotopys ◦p ∼1_X

I such that induced homotopyp◦s◦p ∼p is trivial.

Mr Retract Equivalence

Examples

B(FA,B) ^E A(A,UB) 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 morphismp:X →Y of simplicial sets is shrinkable(dual strong deformation retract) if it is contractible insSet/Y:

I it has a section s

I with a homotopys ◦p ∼1_X

I such that induced homotopyp◦s◦p ∼p is trivial.

Examples

B(FA,B) ^E A(A,UB) 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 morphismp:X →Y of simplicial sets is shrinkable(dual strong deformation retract) if it is contractible insSet/Y:

I it has a section s

I with a homotopys ◦p ∼1_X

I such that induced homotopyp◦s◦p ∼p is trivial.

The setting

LetV be a monoidal model category with cofibrant unitI . . .

cofibrations I, weak equivalences W, trivial fibrationsP. Aninterval in V is a factorization

I +I i J w I

∇ C(A,B) I f

C(A,B) (f

=

g) h

X

Y p

Y 1

s X

p ' 1

of the codiagonal withi ∈ I and w ∈ W.

I A morphism A→B in aV-categoryC is a morphism I → C(A,B).

I A homotopy between morphisms A→B is a morphism J → C(A,B) (for some interval)

I A homotopy is trivial if it factorizes through w

I A morphismp is shrinkable if there exists with p◦s = 1, and homotopy s◦p ∼1 with p◦s◦p ∼p trivial.

The setting

LetV be a monoidalmodel category with cofibrant unitI . . .cofibrations I, weak equivalences W, trivial fibrationsP.

Aninterval in V is a factorization

I +I i J w I

∇ C(A,B) I f

C(A,B) (f

=

g) h

X

Y p

Y 1

s X

p ' 1

of the codiagonal withi ∈ I and w ∈ W.

I A morphism A→B in aV-categoryC is a morphism I → C(A,B).

I A homotopy between morphisms A→B is a morphism J → C(A,B) (for some interval)

I A homotopy is trivial if it factorizes through w

I A morphismp is shrinkable if there exists with p◦s = 1, and homotopy s◦p ∼1 with p◦s◦p ∼p trivial.

The setting

LetV be a monoidal model category with cofibrant unitI . . . cofibrationsI, weak equivalences W, trivial fibrationsP. Aninterval in V is a factorization

I+I i J w I

∇

C(A,B) I f

C(A,B) (f

=

g) h

X

Y p

Y 1

s X

p ' 1

of the codiagonal withi ∈ I and w ∈ W.

I A morphismA→B in aV-categoryC is a morphism I → C(A,B).

I A homotopy between morphismsA→B is a morphism J → C(A,B) (for some interval)

I A homotopy is trivial if it factorizes through w

I A morphismp is shrinkable if there exists with p◦s = 1, and homotopy s◦p ∼1 with p◦s◦p ∼p trivial.

The setting

LetV be a monoidal model category with cofibrant unitI . . . cofibrationsI, weak equivalences W, trivial fibrationsP. Aninterval in V is a factorization

I+I i J w I

∇

C(A,B)

I f

C(A,B) (f

=

g) h

X

Y p

Y 1

s X

p ' 1

of the codiagonal withi ∈ I and w ∈ W.

I A morphismA→B in aV-categoryC is a morphism I → C(A,B).

I A homotopy between morphismsA→B is a morphism J → C(A,B) (for some interval)

I A homotopy is trivial if it factorizes through w

I A morphismp is shrinkable if there exists with p◦s = 1, and homotopy s◦p ∼1 with p◦s◦p ∼p trivial.

The setting

LetV be a monoidal model category with cofibrant unitI . . . cofibrationsI, weak equivalences W, trivial fibrationsP. Aninterval in V is a factorization

I+I i J w I

∇

C(A,B)

I f

C(A,B) (f

=

g) h

X

Y p

Y 1

s X

p ' 1

of the codiagonal withi ∈ I and w ∈ W.

I A morphismA→B in aV-categoryC is a morphism I → C(A,B).

I A homotopy between morphismsA→B is a morphism J → C(A,B) (for some interval)

I A homotopy is trivial if it factorizes through w

I A morphismp is shrinkable if there exists with p◦s = 1, and homotopy s◦p ∼1 with p◦s◦p ∼p trivial.

The setting

LetV be a monoidal model category with cofibrant unitI . . . cofibrationsI, weak equivalences W, trivial fibrationsP. Aninterval in V is a factorization

I+I i J w I

∇

C(A,B)

I f

C(A,B)

(f=g) h

X

Y p

Y 1

s X

p ' 1

of the codiagonal withi ∈ I and w ∈ W.

I A morphismA→B in aV-categoryC is a morphism I → C(A,B).

I A homotopy between morphismsA→B is a morphism J → C(A,B) (for some interval)

I A homotopy is trivialif it factorizes through w

I A morphismp is shrinkable if there exists with p◦s = 1, and homotopy s◦p ∼1 with p◦s◦p ∼p trivial.

The setting

LetV be a monoidal model category with cofibrant unitI . . . cofibrationsI, weak equivalences W, trivial fibrationsP. Aninterval in V is a factorization

I+I i J w I

∇

C(A,B)

I f

C(A,B) (f

=

g) h

X

Y p

Y 1

s X

p ' 1

of the codiagonal withi ∈ I and w ∈ W.

I A morphismA→B in aV-categoryC is a morphism I → C(A,B).

I A homotopy between morphismsA→B is a morphism J → C(A,B) (for some interval)

I A homotopy is trivial if it factorizes through w

I A morphismp is shrinkableif there exists with p◦s = 1, and homotopy s◦p ∼1 with p◦s◦p ∼p trivial.

The setting

LetV be a monoidal model category with cofibrant unitI . . . cofibrationsI, weak equivalences W, trivial fibrationsP. Aninterval in V is a factorization

I+I i J w I

∇

C(A,B)

I f

C(A,B) (f

=

g) h

X

Y p

Y 1

s

X p '

1

of the codiagonal withi ∈ I and w ∈ W.

I A morphismA→B in aV-categoryC is a morphism I → C(A,B).

I A homotopy between morphismsA→B is a morphism J → C(A,B) (for some interval)

I A homotopy is trivial if it factorizes through w

I A morphismp is shrinkableif there exists with p◦s = 1, and homotopy s◦p ∼1 with p◦s◦p ∼p trivial.

The setting

LetV be a monoidal model category with cofibrant unitI . . . cofibrationsI, weak equivalences W, trivial fibrationsP. Aninterval in V is a factorization

I+I i J w I

∇

C(A,B)

I f

C(A,B) (f

=

g) h

X

Y p

Y 1

s X

p ' 1

of the codiagonal withi ∈ I and w ∈ W.

I A morphismA→B in aV-categoryC is a morphism I → C(A,B).

I A homotopy between morphismsA→B is a morphism J → C(A,B) (for some interval)

I A homotopy is trivial if it factorizes through w

I A morphismp is shrinkableif there exists with p◦s = 1, and homotopy s◦p ∼1 with p◦s◦p ∼p trivial.

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: I trivial fibration⇒ shrinkable

( ifX,Y cofibrant ) I shrinkable⇒ weak equiv

(if X fibrant or cofibrant)

Very (!) General AFT V-categoryB with limits U:B → Apreserves them SSC

U hasE-weak left adjoint for E ={shrinkables}

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:

I trivial fibration⇒ shrinkable ( ifX,Y cofibrant )

I shrinkable⇒ weak equiv (if X fibrant or cofibrant)

Very (!) General AFT V-categoryB with limits U:B → Apreserves them SSC

U hasE-weak left adjoint for E ={shrinkables}

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:

I trivial fibration⇒ shrinkable ( ifX,Y cofibrant )

I shrinkable⇒ weak equiv (if X fibrant or cofibrant)

Very (!) General AFT V-categoryB with limits U:B → Apreserves them SSC

U hasE-weak left adjoint for E ={shrinkables}

The limits in question

Limitof a functorS:D → B is defined by natural isos B(B,limS)∼= [D,Set](∆1,B(B,S)).

Limit ofV-functorS:D → B weighted byG:D → V defined by B(B,lim

G S)∼= [D,V](G,B(B,S)).

The power ofA∈ B byX ∈ V defined by B(B,A^X)∼=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

∃

AV-category Bhas enough cofibrant limits if for any weight G there is a cofibrantG⁰ with a pointwise trivial fibration G⁰ →G for whichB hasG⁰-weighted limits.

The limits in question

Limit of a functorS:D → B is defined by natural isos B(B,limS)∼= [D,Set](∆1,B(B,S)).

LimitofV-functorS:D → B weighted byG:D → V defined by B(B,lim

G S)∼= [D,V](G,B(B,S)).

The power ofA∈ B byX ∈ V defined by B(B,A^X)∼=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

∃

AV-category Bhas enough cofibrant limits if for any weight G there is a cofibrantG⁰ with a pointwise trivial fibration G⁰ →G for whichB hasG⁰-weighted limits.

The limits in question

Limit of a functorS:D → B is defined by natural isos B(B,limS)∼= [D,Set](∆1,B(B,S)).

Limit ofV-functorS:D → B weighted byG:D → V defined by B(B,lim

G S)∼= [D,V](G,B(B,S)).

Thepower ofA∈ B byX ∈ V defined by B(B,A^X)∼=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

∃

AV-category Bhas enough cofibrant limits if for any weight G there is a cofibrantG⁰ with a pointwise trivial fibration G⁰ →G for whichB hasG⁰-weighted limits.

The limits in question

Limit of a functorS:D → B is defined by natural isos B(B,limS)∼= [D,Set](∆1,B(B,S)).

Limit ofV-functorS:D → B weighted byG:D → V defined by B(B,lim

G S)∼= [D,V](G,B(B,S)).

The power ofA∈ B byX ∈ V defined by B(B,A^X)∼=V(X,B(B,A)) A weight G:D → V iscofibrant

if it is projective with respect to the pointwise trivial fibrations:

G K

H

KD HD p pD ∈ P

∃

AV-category Bhas enough cofibrant limits if for any weight G there is a cofibrantG⁰ with a pointwise trivial fibration G⁰ →G for whichB hasG⁰-weighted limits.

The limits in question

Limit of a functorS:D → B is defined by natural isos B(B,limS)∼= [D,Set](∆1,B(B,S)).

Limit ofV-functorS:D → B weighted byG:D → V defined by B(B,lim

G S)∼= [D,V](G,B(B,S)).

The power ofA∈ B byX ∈ V defined by B(B,A^X)∼=V(X,B(B,A)) A weight G:D → V iscofibrant

if it is projective with respect to

the pointwise trivial fibrations: G K H

KD HD p pD ∈ P

∃

AV-category Bhas enough cofibrant limits if for any weight G there is a cofibrantG⁰ with a pointwise trivial fibration G⁰ →G for whichB hasG⁰-weighted limits.

The limits in question

Limit of a functorS:D → B is defined by natural isos B(B,limS)∼= [D,Set](∆1,B(B,S)).

Limit ofV-functorS:D → B weighted byG:D → V defined by B(B,lim

G S)∼= [D,V](G,B(B,S)).

The power ofA∈ B byX ∈ V defined by B(B,A^X)∼=V(X,B(B,A)) A weight G:D → V iscofibrant

if it is projective with respect to

the pointwise trivial fibrations: G K H

KD HD p pD ∈ P

∃

AV-category Bhas enough cofibrant limits if for any weight G there is a cofibrantG⁰ with a pointwise trivial fibration G⁰ →G for whichB hasG⁰-weighted limits.

The limits in question

Limit of a functorS:D → B is defined by natural isos B(B,limS)∼= [D,Set](∆1,B(B,S)).

Limit ofV-functorS:D → B weighted byG:D → V defined by B(B,lim

G S)∼= [D,V](G,B(B,S)).

The power ofA∈ B byX ∈ V defined by B(B,A^X)∼=V(X,B(B,A)) A weight G:D → V iscofibrant

if it is projective with respect to

the pointwise trivial fibrations: G K H

KD HD p pD ∈ P

∃

AV-category Bhas enough cofibrant limits if for any weight G there is a cofibrantG⁰ with a pointwise trivial fibration G⁰ →G for whichB hasG⁰-weighted limits.

The limits in question

Limit of a functorS:D → B is defined by natural isos B(B,limS)∼= [D,Set](∆1,B(B,S)).

Limit ofV-functorS:D → B weighted byG:D → V defined by B(B,lim

G S)∼= [D,V](G,B(B,S)).

The power ofA∈ B byX ∈ V defined by B(B,A^X)∼=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

∃

AV-category Bhasenough cofibrant limits if for any weightG there is a cofibrantG⁰ with a pointwise trivial fibration G⁰ →G for whichB hasG⁰-weighted limits.

The VGAFT

LetV be a monoidal model category with cofibrant unit, andE the shrinkable morphisms.

V(ery) G(eneral) AFT

V-categoryB with all powers and enough cofibrant limits U:B → Apreserves them

SSC

U has anE-weak left adjoint

N(ot) Q(uite) S(o) G(eneral) AFT

V-categoryB with all powers and enough cofibrant limits U:B → Apreserves them

B₀ andA₀ are accessible,U₀ accessible functor (unenriched) U has anE-weak left adjoint

The VGAFT

LetV be a monoidal model category with cofibrant unit, andE the shrinkable morphisms.

V(ery) G(eneral) AFT

V-categoryB with all powers and enough cofibrant limits U:B → Apreserves them

SSC

U has anE-weak left adjoint N(ot) Q(uite) S(o) G(eneral) AFT

V-categoryB with all powers and enough cofibrant limits U:B → Apreserves them

B₀ andA₀ are accessible,U₀ accessible functor (unenriched) U has anE-weak left adjoint

Applications (to 2-categories)

This involves the caseV=Cat.

A 2-category will have enough cofibrant limits if it has PIE limits:

that is, if it has I products I inserters I equifiers.

Theorem

LetB 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.

Mr PIE

Applications (to 2-categories)

This involves the caseV=Cat.

A 2-category will have enough cofibrant limits if it has PIE limits:

that is, if it has I products I inserters I equifiers.

Theorem

LetB 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.

Applications (to 2-categories)

This involves the caseV=Cat.

A 2-category will have enough cofibrant limits if it has PIE limits:

that is, if it has I products I inserters I equifiers.

Theorem

LetB 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.

Application (to Riehl-Verity ∞-cosmoi)

An∞-cosmosis 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.

Application (to Riehl-Verity ∞-cosmoi)

An∞-cosmos is asSet-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.

Application (to Riehl-Verity ∞-cosmoi)

An∞-cosmos is asSet-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.

Acosmological functoris an enriched functor between ∞-cosmoi which preserves this structure.

Corollary

Any cosomological functor satisfying the SSC has a weak left adjoint.

Application (to Riehl-Verity ∞-cosmoi)

An∞-cosmos is asSet-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.

Application (to Riehl-Verity ∞-cosmoi)

An∞-cosmos is asSet-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.