[PPT] - Kan-injectivity and KZ-monads Lurdes Sousa IPV / CMUC July 10, PowerPoint Presentation

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Kan-injectivity and KZ-monads

Lurdes Sousa

IPV / CMUC

July 10, 2018

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 1 / 52

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[A. Kock, Monads for which structures are adjoints to units, 1995]: KZ

monads (lax idempotent monads) in 2-cats

Kock-Z¨

berlein

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 2 / 52

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[A. Kock, Monads for which structures are adjoints to units, 1995]: KZ

monads (lax idempotent monads) in 2-cats

Kock-Z¨

berlein

[M. Escard´

, Properly injective spaces and function spaces, 1998]:

Often, in order-enriched categories, injective objects = Eilenberg-Moore algebras of a KZ-monad = Kan-injective objects

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 2 / 52

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[A. Kock, Monads for which structures are adjoints to units, 1995]: KZ

monads (lax idempotent monads) in 2-cats

Kock-Z¨

berlein

[M. Escard´

, Properly injective spaces and function spaces, 1998]:

Often, in order-enriched categories, injective objects = Eilenberg-Moore algebras of a KZ-monad = Kan-injective objects [M. Carvalho, L.S., 2011] : Kan-injectivity/KZ-monads enjoys many features resembling Orthogonality/Idempotent monads

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 2 / 52

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M. Carvalho, L. S., Order-preserving reflectors and injectivity, TA, 2011
J. Ad´

amek, L. S., J. Velebil, Kan injectivity in order-enr. cats., MSCS, 2015

M. Carvalho, L. S., On Kan-injectivity of locales and spaces, ACS, 2017
L. S., A calculus of lax fractions, JPAA, 2017
J. Ad´

amek, L. S., KZ-monadic categories and their logic, TAC, 2017

D. Hofmann, L. S., Aspects of algebraic algebras, LMCS, 2017

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 3 / 52

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M. Carvalho, L. S., Order-preserving reflectors and injectivity, TA, 2011
J. Ad´

amek, L. S., J. Velebil, Kan injectivity in order-enr. cats., MSCS, 2015

M. Carvalho, L. S., On Kan-injectivity of locales and spaces, ACS, 2017
L. S., A calculus of lax fractions, JPAA, 2017
J. Ad´

amek, L. S., KZ-monadic categories and their logic, TAC, 2017

D. Hofmann, L. S., Aspects of algebraic algebras, LMCS, 2017
M. M. Clementino, F. Lucatelli, J. Picado: joint work in progress

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 3 / 52

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1. Kan-injectivity and KZ-monads
2. In locales and topological spaces
3. Lax fractions
4. Kan-injective subcategory problem

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 4 / 52

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Most of the time, the setting is

rder-enriched categories

A

g

≤

f

B

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 5 / 52

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1. Kan-injectivity and KZ-monads

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 6 / 52

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1. Kan-injectivity and KZ-monads

Monad T = (T, η, µ) of Kock-Z¨

berlein type: Tη ≤ ηT

(⇐ ⇒ every T-algebra (X, α) has α ⊢ ηX)

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 6 / 52

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1. Kan-injectivity and KZ-monads

Monad T = (T, η, µ) of Kock-Z¨

berlein type: Tη ≤ ηT

(⇐ ⇒ every T-algebra (X, α) has α ⊢ ηX) KZ-monadic subcategory of X= Eilenberg-Moore category of a KZ-monad over X

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 6 / 52

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1. Kan-injectivity and KZ-monads

Monad T = (T, η, µ) of Kock-Z¨

berlein type: Tη ≤ ηT

(⇐ ⇒ every T-algebra (X, α) has α ⊢ ηX) KZ-monadic subcategory of X= Eilenberg-Moore category of a KZ-monad over X Full reflective subcategory of X = Eilenberg-Moore category of an idempotent monad over X (Tη = ηT)

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 6 / 52

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1. Kan-injectivity and KZ-monads

T = (T, η, µ) idempotent: A ∈ X T iff it is orthogonal to all ηX, i.e., X(TX, A)

X(ηX ,A)

X(X, A)

is an isomorphism.

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1. Kan-injectivity and KZ-monads

T = (T, η, µ) idempotent: A ∈ X T iff it is orthogonal to all ηX, i.e., X(TX, A)

X(ηX ,A)

X(X, A)

is an isomorphism. T = (T, η, µ) KZ-monad: A ∈ X T iff X(TX, A)

X(ηX ,A)

X(X, A)

is a right adjoint retraction.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 7 / 52

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1. Kan-injectivity and KZ-monads

T = (T, η, µ) idempotent: A ∈ X T iff it is orthogonal to all ηX, i.e., X(TX, A)

X(ηX ,A)

X(X, A)

is an isomorphism. T = (T, η, µ) KZ-monad: A ∈ X T iff X(TX, A)

X(ηX ,A)

X(X, A)

is a right adjoint retraction. g is a right adjoint retraction if there is an adjunction (id, β) : f ⊣ g In order enriched categories: gf = id and fg ≤ id

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 7 / 52

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1. Kan-injectivity and KZ-monads

A is (left) Kan-injective wrt h : X → Y if X(Y , A)

X(h,A)

X(X, A)

is a right adjoint retraction.

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1. Kan-injectivity and KZ-monads

A is (left) Kan-injective wrt h : X → Y if X(Y , A)

X(h,A)

X(X, A)

is a right adjoint retraction. Equivalently: for all f : X → A, there exists a left Kan extension of f along h of the form Lanh(f ) = (f /h, id). X

f

=

h

Y

f /h=(X(h,A))∗(f )

A

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 8 / 52

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1. Kan-injectivity and KZ-monads

A is (left) Kan-injective wrt h : X → Y if X(Y , A)

X(h,A)

X(X, A)

is a right adjoint retraction. Equivalently: for all f : X → A, there exists a left Kan extension of f along h of the form Lanh(f ) = (f /h, id). X

f

=

h

Y

f /h=(X(h,A))∗(f )

A

k : A → B is (left) Kan-injective wrt h : X → Y , if A and B are so, and k preserves the left Kan extension of every f : X → A along k.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 8 / 52

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1. Kan-injectivity and KZ-monads

A is (left) Kan-injective wrt h : X → Y if X(Y , A)

X(h,A)

X(X, A)

is a right adjoint retraction. Equivalently: for all f : X → A, there exists a left Kan extension of f along h of the form Lanh(f ) = (f /h, id). X

f

=

h

Y

f /h=(X(h,A))∗(f )

A

k : A → B is (left) Kan-injective wrt h : X → Y , if A and B are so, and k preserves the left Kan extension of every f : X → A along k. Equivalently: X(Y , A)

X(Y ,k)

X(X, A)

(X(h,A))∗

X(X,k)
A

k

X(Y , B)

X(X, B)

(X(h,B))∗

B

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 8 / 52

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1. Kan-injectivity and KZ-monads

For H ⊆ Mor(X), KInj(H) := (locally full) subcategory of objects and morphisms Kan-injective wrt all h ∈ H (Left) Kan-injective subcategory

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1. Kan-injectivity and KZ-monads

For H ⊆ Mor(X), KInj(H) := (locally full) subcategory of objects and morphisms Kan-injective wrt all h ∈ H (Left) Kan-injective subcategory For T = (T, η, µ) a KZ-monad over X order-enriched, X T = KInj({ηX | X ∈ X}).

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 9 / 52

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1. Kan-injectivity and KZ-monads

A a (locally full) subcategory of X A is closed under left adjoint retractions, if, for every commutative diagram A

f

q
B

q′

X

g

Y

with q and q′ left adjoint retractions, whenever f ∈ A, then g ∈ A.

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1. Kan-injectivity and KZ-monads

A a (locally full) subcategory of X A is an inserter-ideal, provided that, for every inserter diagram A

g

I

i

i

⇑ B A

f

I

i=ins(f ,g)

A

f

g

B

f ∈ A = ⇒ i ∈ A.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 11 / 52

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1. Kan-injectivity and KZ-monads

Theorem ([CS, 2011], [ASV, 2015]) Given H ⊆ Mor(X), KInj(H) is:

1 Closed under weighted limits, i.e., the inclusion functor

KInj(H) ֒ → X creates weighted limits;

2 An inserter-ideal; 3 Closed under left adjoint retractions. Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 12 / 52

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1. Kan-injectivity and KZ-monads

Theorem ([CS, 2011], [ASV, 2015]) Given H ⊆ Mor(X), KInj(H) is:

1 Closed under weighted limits, i.e., the inclusion functor

KInj(H) ֒ → X creates weighted limits;

2 An inserter-ideal; 3 Closed under left adjoint retractions.

Corollary Every KZ-monadic subcategory enjoys properties 1, 2 and 3 above.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 12 / 52

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1. Kan-injectivity and KZ-monads

Theorem ([ASV, 2015]) Let X have inserters. A reflection of X in a subcategory A is of Kock-Z¨

berlein type (i.e. it induces a KZ-monad), iff A is an

inserter-ideal of X.

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1. Kan-injectivity and KZ-monads

Theorem ([CS, 2011]) Let A be a (locally full) subcategory of X. The inclusion functor E : A ֒ → X is a right adjoint which induces a KZ-monad over X, iff for every X ∈ X, there is an arrow ηX : X → X with X ∈ A such that: (i) A ⊆ KInj({ηX | X ∈ X}) and, for every f : X → A with A in A f /ηX ∈ A. (ii) ηX is dense, i.e., ηX/ηX = idX, X ∈ X.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 14 / 52

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1. Kan-injectivity and KZ-monads

Theorem ([CS, 2011]) Let A be a (locally full) subcategory of X. The inclusion functor E : A ֒ → X is a right adjoint which induces a KZ-monad over X, iff for every X ∈ X, there is an arrow ηX : X → X with X ∈ A such that: (i) A ⊆ KInj({ηX | X ∈ X}) and, for every f : X → A with A in A f /ηX ∈ A. (ii) ηX is dense, i.e., ηX/ηX = idX, X ∈ X. In the setting of 2-categories: [F. Marmolejo, R. Wood, Kan extensions and lax idempotent pseudomonads, TAC, 2012]

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 14 / 52

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1. Kan-injectivity and KZ-monads

Theorem ([CS, 2011]) Let A be a (locally full) subcategory of X. The inclusion functor E : A ֒ → X is a right adjoint which induces a KZ-monad over X, iff for every X ∈ X, there is an arrow ηX : X → X with X ∈ A such that: (i) A ⊆ KInj({ηX | X ∈ X}) and, for every f : X → A with A in A f /ηX ∈ A. (ii) ηX is dense, i.e., ηX/ηX = idX, X ∈ X. Furthermore, under the above conditions, A is a KZ-monadic subcategory of X iff it is closed under left adjoint retractions.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 14 / 52

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1. Kan-injectivity and KZ-monads

Eilenberg-Moore category = closure under left adjoint retractions

f the Kleisli category

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1. Kan-injectivity and KZ-monads

T = filter monad on Top0 Top0T

Idemp(Top0T) Lim(Top0T) Top0T Top0

= = = = = =

Idemp. split
compl. of Kleisli

Closure under

w. lims. of Kleisli

Closure of Kleisli under left adjoint retractions

= = = = = =

Algebraic lattices whose compacts form a coframe Algebraic lattices Continuous lattices

[HS, 2017]

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 16 / 52

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1. Kan-injectivity and KZ-monads

A subcategory of X AKInj := {h ∈ Mor(X) | A Kan-injective wrt h} Galois connection: H ✤

KInj(H)

AKInj A

✤

Category Theory 2018, Azores, 8-14 July

Kan-injectivity and KZ-monads 17 / 52

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1. Kan-injectivity and KZ-monads

A subcategory of X AKInj := {h ∈ Mor(X) | A Kan-injective wrt h} Galois connection: H ✤

KInj(H)

AKInj A

✤

In case A is an Eilenberg-Moore category of a KZ-monad T

AKInj = {f | f is a T-embedding

}

Tf is a left adjoint section

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 17 / 52

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2. In Loc and Top0

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 18 / 52

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2. In Loc and Top0

[D. Scott, LN, 1972]: In Top0, continuous lattices = spaces injective wrt embeddings [P. Johnstone, JPAA, 1981]: In Loc, stably locally compact locales = retracts of coherent locales = locales injective wrt flat embeddings

M. Escard´
, in 1990’s:

Several examples of injective objs. = EM-algebras of a KZ-monad

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 18 / 52

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2. In Loc and Top0

Loc = Frmop Locale = frame = complete lattice L with (

A) ∧ b =
a∈A

(a ∧ b) Localic map = infima-preserving map f : L → M with f ∗ : M → L preserving finite meets

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 19 / 52

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2. In Loc and Top0

Loc = Frmop Locale = frame = complete lattice L with (

A) ∧ b =
a∈A

(a ∧ b) Localic map = infima-preserving map f : L → M with f ∗ : M → L preserving finite meets Embeddings = one-to-one localic maps

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 19 / 52

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2. In Loc and Top0

Loc = Frmop Locale = frame = complete lattice L with (

A) ∧ b =
a∈A

(a ∧ b) Localic map = infima-preserving map f : L → M with f ∗ : M → L preserving finite meets Embeddings = one-to-one localic maps f : L → M is n-flat, if f (

i∈I

xi) =

i∈I

f (xi), for |I| ≤ n.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 19 / 52

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2. In Loc and Top0

Loc = Frmop Locale = frame = complete lattice L with (

A) ∧ b =
a∈A

(a ∧ b) Localic map = infima-preserving map f : L → M with f ∗ : M → L preserving finite meets Embeddings = one-to-one localic maps f : L → M is n-flat, if f (

i∈I

xi) =

i∈I

f (xi), for |I| ≤ n. (0-flat =) 1-flat = dense (f (0) = 0) (0-flat =) 2-flat = flat

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 19 / 52

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2. In Loc and Top0

For every cardinal n, Fn = free frame generated by the set n Fn = ({downsets of ({finite subsets of X}, ⊇) , ⊆)

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 20 / 52

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2. In Loc and Top0

F1 =

1
s

Fn =

s
1

↓{i}

. . .

✟ ✟ ✟

↓{i+1}

❍❍ ❍

↓{j}

. . . . . . . . .

↓{i} = {A ⊆n |A fin.,i ∈ A}

Dn = F0

f0

Fn

fn

F1

with fn(1) = 1, fn(s) = s, and fn(x) = 0 otherwise

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 21 / 52

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2. In Loc and Top0

F1 =

1
s

Fn =

s
1

↓{i}

. . .

✟ ✟ ✟

↓{i+1}

❍❍ ❍

↓{j}

. . . . . . . . .

↓{i} = {A ⊆n |A fin.,i ∈ A}

Dn = F0

f0

Fn

fn

F1

with fn(1) = 1, fn(s) = s, and fn(x) = 0 otherwise Theorem ([CS, 2017])

Embeddings = F KInj

1

n-flat embeddings = DKInj

n

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 21 / 52

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2. In Loc and Top0

D =

n∈Card

Dn is a subcategory of Loc made of spatial locales. Corollary Loc is the Kan-injective hull of a subcategory made of spatial locales: Loc = KInj

DKInj

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 22 / 52

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2. In Loc and Top0

D =

n∈Card

Dn is a subcategory of Loc made of spatial locales. Corollary Loc is the Kan-injective hull of a subcategory made of spatial locales: Loc = KInj

DKInj

Proof. DKInj =

n∈Card

DKInj

n

= {{f ∈ Loc | f∗ ∈ Loc and f∗f = id} = {f ∈ Loc | f is a left adjoint section in Loc}

H

Thus, KInj(H) = Loc.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 22 / 52

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2. In Loc and Top0

L ∈ Loc Given n, GnL := {U ⊆ L | U =↓ U, U closed under

I

, |I| ≤ n} with ⊆ Gn : Loc → Loc gives rise to the functor part of a KZ-monad.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 23 / 52

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2. In Loc and Top0

L ∈ Loc Given n, GnL := {U ⊆ L | U =↓ U, U closed under

I

, |I| ≤ n} with ⊆ Gn : Loc → Loc gives rise to the functor part of a KZ-monad. a ≪n b, if, ∀U ∈ GnL, b ≤ U ⇒ a ∈ U

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 23 / 52

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2. In Loc and Top0

L ∈ Loc Given n, GnL := {U ⊆ L | U =↓ U, U closed under

I

, |I| ≤ n} with ⊆ Gn : Loc → Loc gives rise to the functor part of a KZ-monad. a ≪n b, if, ∀U ∈ GnL, b ≤ U ⇒ a ∈ U L is stably locally n-compact if

∀a ∈ L, a =
x≪na

x

∀x, a, b, (x ≪n a, x ≪n b) ⇒ x ≪n a ∧ b
1≪n1

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 23 / 52

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2. In Loc and Top0

SLCompn = category of stably locally n-compact locales and localic maps f such that f ∗ preserves ≪n Theorem ([CS, 2017]) For every n, SLCompn is a KZ-monadic subcategory, and it is the Kan-injective hull of Dn, i.e., SLCompn = KInj

DKInj

n

.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 24 / 52

SLIDE 49

2. In Loc and Top0

Top0

Lc

Loc

X

✤

Ω(X)

f

✤

(f −1)∗

Lc ⊣ Sp : Loc → Top0

rder-enriched adjunction

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 25 / 52

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2. In Loc and Top0

Top0

Lc

Loc

X

✤

Ω(X)

f

✤

(f −1)∗

Lc ⊣ Sp : Loc → Top0

rder-enriched adjunction

X

f

Y

is an embedding in Top0 iff Lc(f ) is an embedding in Loc is dense in Top0 iff Lc(f ) is dense in Loc is n-flat in Top0 iff Lc(f ) is n-flat in Loc

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 25 / 52

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2. In Loc and Top0

Lemma Let F ⊣ G : A → X be an order-enriched adjunction. Then, given h in X and an object A (resp., a morphism f ) in A, A (resp., f ) is Kan-injective wrt Fh

GA (resp., Gf ) is Kan-injective wrt h.
Proof. Immediate from the natural isomorphism

A(FX, A) ∼ = X(X, GA).

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 26 / 52

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2. In Loc and Top0

Corollary ([CS, 2017]) In Top0:

Embeddings are precisely the morphisms wrt which the

Sierpi´ nski space is Kan-injective.

n-flat embeddings are precisely the morphisms wrt which

Sp[Dn] is Kan-injective.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 27 / 52

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2. In Loc and Top0

In Top0: A AKInj KInj

AKInj

(KZ-monadic) 2 = Sierpi´ nski embeddings continuous lattices & maps pres. all and ↑ 1

2

dense embeddings Scott conts. lats. & maps pres. (= ∅) and ↑ 1

2

flat

embeddings stably locally compact spaces & convenient maps espaco

❅

❅

Category Theory 2018, Azores, 8-14 July

Kan-injectivity and KZ-monads 28 / 52

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3. Lax fractions

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 29 / 52

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3. Lax fractions

We need Kan-injectivity w.r.t. squares

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 29 / 52

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3. Lax fractions

full reflective subcategory: the Kleisli category of the idemp. monad T is a category of fractions of {h | Th is an iso}

aaaaaaaaaaaaaaaaaaaaaaaaaaaa= AOrth

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 30 / 52

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3. Lax fractions

full reflective subcategory: the Kleisli category of the idemp. monad T is a category of fractions of {h | Th is an iso}

aaaaaaaaaaaaaaaaaaaaaaaaaaaa= AOrth

KZ-monadic subcategory: AKInj = {h | Th is a left adjoint section}

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 30 / 52

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3. Lax fractions

full reflective subcategory: the Kleisli category of the idemp. monad T is a category of fractions of {h | Th is an iso}

aaaaaaaaaaaaaaaaaaaaaaaaaaaa= AOrth

KZ-monadic subcategory: AKInj = {h | Th is a left adjoint section} fraction: h−1 · f

f

h−1
h
lax fraction: h∗ · f

f

h∗
h
Category Theory 2018, Azores, 8-14 July

Kan-injectivity and KZ-monads 30 / 52

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3. Lax fractions

AOrth closed under colimits in X →

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 31 / 52

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3. Lax fractions

AOrth closed under colimits in X →

Applications:

AOrth admits a calculus of

fractions

an affirmative answer to the
Orthog. Subcat. Problem

[Gabriel, Ulmer, 1971] [Kelly, 1980] . . .

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 31 / 52

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3. Lax fractions

AOrth closed under colimits in X →

Applications:

AOrth admits a calculus of

fractions

an affirmative answer to the
Orthog. Subcat. Problem

[Gabriel, Ulmer, 1971] [Kelly, 1980] . . .

What about AKInj ?

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 31 / 52

SLIDE 62

3. Lax fractions

S =

h1
u
v
h2

is a square in X. It represents the morphism (u, v) : h1 → h2 in X →.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 32 / 52

SLIDE 63

3. Lax fractions

S =

h1
u
v
h2

is a square in X. It represents the morphism (u, v) : h1 → h2 in X →. A is Kan-injective wrt S if it is Kan-injective wrt h1 and h2 and, for every f , (fu)/h1 = (f /h2)v:

h1
u
v
(fu)/h1
h2
f
f /h2
A

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 32 / 52

SLIDE 64

3. Lax fractions

S =

h1
u
v
h2

is a square in X. It represents the morphism (u, v) : h1 → h2 in X →. A is Kan-injective wrt S if it is Kan-injective wrt h1 and h2 and, for every f , (fu)/h1 = (f /h2)v:

h1
u
v
(fu)/h1
h2
f
f /h2
A

k : A → B is Kan-injective wrt S if it is Kan-injective wrt h1 and h2.

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SLIDE 65

3. Lax fractions

AKInj = subcategory of X → of morphisms and squares wrt which A is Kan-injective

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SLIDE 66

3. Lax fractions

AKInj = subcategory of X → of morphisms and squares wrt which A is Kan-injective Theorem Let X have weighted colimits. AKInj is closed under weighted colimits in X →. And it is a coinserter-ideal.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 33 / 52

SLIDE 67

3. Lax fractions

A morphism h as a square: S(h) =

h
h

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 34 / 52

SLIDE 68

3. Lax fractions

A split square is a square

u
h1

(h1)∗

v
h2

(h2)∗

with h1 and h2 left adjoint sections and (h2)∗v = u(h1)∗.

A square S is a split square iff KInj(S) = X.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 35 / 52

SLIDE 69

3. Lax fractions

Let H be a class of squares of X. A category of lax fractions for H is a functor F : X → X[H∗] such that:

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 36 / 52

SLIDE 70

3. Lax fractions

Let H be a class of squares of X. A category of lax fractions for H is a functor F : X → X[H∗] such that:

1 For every square

h1
u
v
h2

in H, its image by F

Fh1

Fu

Fv
Fh2

is a split square.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 36 / 52

SLIDE 71

3. Lax fractions

Let H be a class of squares of X. A category of lax fractions for H is a functor F : X → X[H∗] such that:

1 For every square

h1
u
v
h2

in H, its image by F

Fh1

Fu

Fv
Fh2

is a split square.

2 If G : X → C is another functor under the above condition,

then there is a unique functor H : X[H∗] → C such that HF = G.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 36 / 52

SLIDE 72

3. Lax fractions

Theorem ([S, 2017]) Let A be a KZ-monadic subcategory of X. Then the corresponding Kleisli category is a category of lax fractions for H = AKInj.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 37 / 52

SLIDE 73

3. Lax fractions

Theorem ([S, 2017]) Let A be a KZ-monadic subcategory of X. Then the corresponding Kleisli category is a category of lax fractions for H = AKInj. Also in [S., 2017]: a calculus of lax fractions, via a calculus of squares

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 37 / 52

SLIDE 74

4. Kan-injective subcategory problem

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 38 / 52

SLIDE 75

4. Kan-injective subcategory problem

In locally bounded categories, Orth(H) is reflective (for each set H). Each reflection of X in Orth(H) is given by a convenient chain X = X0

X1 . . . Xi . . . Xλ [M. Kelly, 1980]

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 38 / 52

SLIDE 76

4. Kan-injective subcategory problem

In locally bounded categories, Orth(H) is reflective (for each set H). Each reflection of X in Orth(H) is given by a convenient chain X = X0

X1 . . . Xi . . . Xλ [M. Kelly, 1980]

“Rules” used in the construction of the chain

+

Some rules



          whose closure is precisely all morphisms k with Orth(H) ⊆ Orth({k})

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 38 / 52

SLIDE 77

4. Kan-injective subcategory problem

In locally bounded categories, Orth(H) is reflective (for each set H). Each reflection of X in Orth(H) is given by a convenient chain X = X0

X1 . . . Xi . . . Xλ [M. Kelly, 1980]

“Rules” used in the construction of the chain

+

Some rules



          whose closure is precisely all morphisms k with Orth(H) ⊆ Orth({k}) Logic for Orthogonality [J. Ad´ amek, M. H´ ebert, L.S., The orthog. subcat. probl. ..., 2009] [J. Ad´ amek, M. Sobral, L.S., A logic of implications ..., 2009]

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SLIDE 78

4. Kan-injective subcategory problem

Analogously, two related problems:

Kan-Injective Subcategory Problem
A Logic for Kan-injectivity

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SLIDE 79

4. Kan-injective subcategory problem

Theorem ([ASV, 2015]) In a locally bounded order-enriched category, KInj(H) is KZ-monadic, for every set H of morphisms. To obtain a complete logic for Kan-injectivity, we need squares.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 40 / 52

SLIDE 80

4. Kan-injective subcategory problem

Theorem ([ASV, 2015], [AS, 2017]) In a locally bounded order-enriched category, KInj(H) is KZ-monadic, for every set H of squares. To obtain a complete logic for Kan-injectivity, we need squares.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 40 / 52

SLIDE 81

4. Kan-injective subcategory problem

X is locally bounded, that is:

it has weighted colimits;
it has a proper f. s. (E, M), i.e., E ⊆ Epi, M ⊆ OrderMono;

(mf ≤ mg ⇒ f ≤ g)

it is E-cowellpowered;
every object X has bound, i.e.,

X(X, −) preserves λ-direced M-unions, for some λ.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 41 / 52

SLIDE 82

4. Kan-injective subcategory problem

The reflection chain

Given a set H of squares, for every X, the chain X = X0

X1 X2 . . . Xi . . . (i ∈ Ord)

is constructed as follows:

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SLIDE 83

4. Kan-injective subcategory problem

Limit step i. Xi = colim

j<i

Xj Isolated step i → i + 1i even). S = A1

a h1 B1 b

A2

h2

B2

Ar

f

hr

Br

f / /hr

Xi

Xi+1

Wide pushout of all pushouts of f ’s along hr’s of S ∈ H

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 43 / 52

SLIDE 84

4. Kan-injective subcategory problem

Limit step i. Xi = colim

j<i

Xj Isolated step i → i + 1 (i even). S = A1

a h1 B1 b

A2

h2

B2

Ar

f

hr

Br

f / /hr

Xi

Xi+1

Wide pushout of all pushouts of f ’s along hr’s of S ∈ H

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 43 / 52

SLIDE 85

4. Kan-injective subcategory problem

Limit step i. Xi = colim

j<i

Xj Isolated step i → i + 1 (i even). S = A1

a h1 B1 b

A2

h2

B2

Ar

f

hr

Br

f / /hr

Xi

Xi+1

Wide pushout of all pushouts of f ’s along hr’s (r = 1, 2) of S ∈ H

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 43 / 52

SLIDE 86

4. Kan-injective subcategory problem

Isolated step i + 1 → i + 2 . A1

a

h1

B1

b

g

≤

A2

f

h2
f /

/h2

Xi

Xi+1

Ar

f

hr

Br

g

≤

Xj
Xi

Xi+1

coins(f / /h2 · b, g) coins(xj+1,i+1 · f / /hr, g) Xi+1

Xi+2

is the wide pushout of all these coinserters for S ∈ H, and possible f ’s and g’s.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 44 / 52

SLIDE 87

4. Kan-injective subcategory problem

Isolated step i + 1 → i + 2 . A1

a

h1

B1

b

g

≤

A2

f

h2
f /

/h2

Xi

Xi+1

Ar

f

hr

Br

g

≤

Xj
Xi

Xi+1

coins(f / /h2 · b, g) coins(xj+1,i+1 · f / /hr, g) Xi+1

Xi+2

is the wide pushout of all these coinserters for S ∈ H, and possible f ’s and g’s.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 44 / 52

SLIDE 88

4. Kan-injective subcategory problem

Isolated step i + 1 → i + 2 . A1

a

h1

B1

b

g

≤

A2

f

h2
f /

/h2

Xi

Xi+1

Ar

f

hr

Br

g

≤

Xj
Xi

Xi+1

coins(f / /h2 · b, g) coins(xj+1,i+1 · f / /hr, g) Xi+1

Xi+2

is the wide pushout of all these coinserters for S ∈ H, and possible f ’s and g’s.

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SLIDE 89

4. Kan-injective subcategory problem

There is a cardinal λ, greater than the bounds of the objects appearing in the squares of H, such that X0

Xλ

is a KZ-reflection in KInj(H).

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 45 / 52

SLIDE 90

4. Kan-injective subcategory problem

Aim: System of deduction rules such that, for every set of squares H and every square S, H ⊢ S iff H | = S where H | = S means that KInj(H) ⊆ KInj({S}).

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 46 / 52

SLIDE 91

4. Kan-injective subcategory problem

Kan-Injectivity Deduction System

AXIOM

S for split squares S

COMPOSITION

S1 S2 S for a composite S, horizontal or vertical, of S1 and S2

PUSHOUT h1

h2
hr

a

for a pushout of hr, r = 1 or 2,

along an arbitrary morphism a

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SLIDE 92

4. Kan-injective subcategory problem

Kan-Injectivity Deduction System

WIDE PUSHOUT h bi (i∈I)

bih
h

bj ¯ bj

h
k
for any wide pushout

bi k

¯

bi

and j ∈ I

COINSERTER

S1 , S2, S3·S2 S4

h S1

b
g
S2
S3
b′
S4
c
for (b′b)h ≤ gh

and c = coins(b′b, g)

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 48 / 52

SLIDE 93

4. Kan-injective subcategory problem

Kan-Injectivity Deduction System

RIGHT CANCELLATION

S , S(h) , S(k) S0 for S =

S0
h
k
UPPER

CANCELLATION

Si , Si (i ∈ I) S for Si =

hi ai Si

bi
a

S

b
h
with (bi)i∈I

collectively epic

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 49 / 52

SLIDE 94

4. Kan-injective subcategory problem

Theorem In any order-enriched locally bounded category, the Kan-injectivity Deduction System is sound and complete: H | = S iff H ⊢ S

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SLIDE 95

4. Kan-injective subcategory problem

Theorem In any locally bounded order-enriched category, for every set of squares H, the class {S ∈ Square(X) | H | = S} is the smallest subcategory of X → containing H and all split squares, and closed under horizontal composition, weighted colimits, the coinserter rule, and right and upper cancellations.

Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 51 / 52

SLIDE 96

Open question Let X have weighted colimits. Do Eilenberg-Moore categories of a KZ-monad over X have weighted colimits (at least under mild conditions)?

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