The homotopy relation in a category with weak equivalences Martin - - PowerPoint PPT Presentation

▶
the homotopy relation in a category with weak equivalences
SMART_READER_LITE
LIVE PREVIEW

The homotopy relation in a category with weak equivalences Martin - - PowerPoint PPT Presentation

Mar 05, 2023 355 likes •856 views

Motivation The relation W and Whitehead Constructing W The case of model categories The homotopy relation in a category with weak equivalences Martin Szyld University of Buenos Aires - CONICET, Argentina CT 2018 @ UA c, Ponta Delgada,

slide-1
SLIDE 1

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The homotopy relation in a category with weak equivalences

Martin Szyld University of Buenos Aires - CONICET, Argentina CT 2018 @ UA¸ c, Ponta Delgada, Portugal

slide-2
SLIDE 2

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Model (bi)categories:

a structure (C, F, coF, W), with C a (bi)category, and F coF W families of arrows of C · · · · ·

∼

· satisfying some axioms.

slide-3
SLIDE 3

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Model (bi)categories:

a structure (C, F, coF, W), with C a (bi)category, and F coF W families of arrows of C · · · · ·

∼

· satisfying some axioms. A taste of the axioms: · ·

  • ·
  • ·

·

  • ∼

and ·

  • ∼

· ·

  • ·
  • ·

∼

  • ·

·

  • r

·

  • ∼

·

  • ·

·

  • ·
  • ·
  • ·
  • ·
slide-4
SLIDE 4

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Model (bi)categories:

a structure (C, F, coF, W), with C a (bi)category, and F coF W families of arrows of C · · · · ·

∼

· satisfying some axioms. A taste of the axioms: · ·

  • ·
  • ∼

=

· ·

  • ∼

and ·

  • ∼

∼ =

· ·

  • ·
  • ·

∼

  • ·

·

  • r

·

  • ∼

·

  • ·

·

  • ·
  • ·
  • ·
  • ·
slide-5
SLIDE 5

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Model (bi)categories:

a structure (C, F, coF, W), with C a (bi)category, and F coF W families of arrows of C · · · · ·

∼

· satisfying some axioms. A taste of the axioms: · ·

  • ·
  • ∼

=

· ·

  • ∼

and ·

  • ∼

∼ =

· ·

  • ·
  • ∼

=

·

∼

  • ·

·

  • r

·

  • ∼

= ∼

·

  • ·

·

  • ·
  • ·
  • ·
  • ·
slide-6
SLIDE 6

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Model (bi)categories:

a structure (C, F, coF, W), with C a (bi)category, and F coF W families of arrows of C · · · · ·

∼

· satisfying some axioms. A taste of the axioms: · ·

  • ·
  • ∼

=

· ·

  • ∼

and ·

  • ∼

∼ =

· ·

  • ·
  • ∼

=

·

∼

  • ·

·

  • r

·

  • ∼

= ∼

·

  • ·

·

  • ·
  • ∼

=

·

  • ·
  • ·

∼ =

slide-7
SLIDE 7

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Model (bi)categories:

a structure (C, F, coF, W), with C a (bi)category, and F coF W families of arrows of C · · · · ·

∼

· satisfying some axioms. A taste of the axioms: · ·

  • ·
  • ∼

=

· ·

  • ∼

and ·

  • ∼

∼ =

· ·

  • ·
  • ∼

=

·

∼

  • ·

·

  • r

·

  • ∼

= ∼

·

  • ·

·

  • ·
  • ∼

=

·

  • ·
  • ·

∼ =

Ho(C) = C[W−1] admits a construction “quotienting by homotopy”.

slide-8
SLIDE 8

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Our original problem: homotopy in a model bicategory

We1 seek a construction of the homotopy bicategory Ho(C):

  • Objects and arrows are those of Cfc ( 0

X 1 ).

  • 2-cells: classes [H] of “homotopies” by an eq. relation.

1together with E. Descotte and E. Dubuc.

slide-9
SLIDE 9

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Our original problem: homotopy in a model bicategory

We1 seek a construction of the homotopy bicategory Ho(C):

  • Objects and arrows are those of Cfc ( 0

X 1 ).

  • 2-cells: classes [H] of “homotopies” by an eq. relation.

Simultaneous requirements

  • Vertical composition
  • Horizontal composition
  • compatible with the eq. relation
  • (Non invertible) 2-cell → homotopy

1together with E. Descotte and E. Dubuc.

slide-10
SLIDE 10

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Our original problem: homotopy in a model bicategory

We1 seek a construction of the homotopy bicategory Ho(C):

  • Objects and arrows are those of Cfc ( 0

X 1 ).

  • 2-cells: classes [H] of “homotopies” by an eq. relation.

Simultaneous requirements

  • Vertical composition
  • Horizontal composition
  • compatible with the eq. relation
  • (Non invertible) 2-cell → homotopy

Considering Quillen’s notion an obstacle f

ℓ

∼ g if and only if there is a diagram in which σ is a weak equivalence (and A ∐ A

∂0+∂1

− − − − → A × I is a cofibration) A

f

  • g
  • id
  • ∂0
  • ∂1
  • B

A A × I

σ

  • ∼

h

  • 1together with E. Descotte and E. Dubuc.
slide-11
SLIDE 11

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Our original problem: homotopy in a model bicategory

We1 seek a construction of the homotopy bicategory Ho(C):

  • Objects and arrows are those of Cfc ( 0

X 1 ).

  • 2-cells: classes [H] of “homotopies” by an eq. relation.

Simultaneous requirements

  • Vertical composition
  • Horizontal composition
  • compatible with the eq. relation
  • (Non invertible) 2-cell → homotopy

Considering Quillen’s notion an obstacle f

ℓ

∼ g ⇒ jf

ℓ

∼ jg ✓ A

f

  • g
  • id
  • ∂0
  • ∂1
  • B

A A × I

σ

  • ∼

h

  • 1together with E. Descotte and E. Dubuc.
slide-12
SLIDE 12

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Our original problem: homotopy in a model bicategory

We1 seek a construction of the homotopy bicategory Ho(C):

  • Objects and arrows are those of Cfc ( 0

X 1 ).

  • 2-cells: classes [H] of “homotopies” by an eq. relation.

Simultaneous requirements

  • Vertical composition
  • Horizontal composition
  • compatible with the eq. relation
  • (Non invertible) 2-cell → homotopy

Considering Quillen’s notion an obstacle f

ℓ

∼ g ⇒ jf

ℓ

∼ jg ✓ f

ℓ

∼ g ⇒ fj

ℓ

∼ gj : A

f

  • g
  • id
  • ∂0
  • ∂1
  • B

A A × I

σ

  • ∼

h

  • A′

j

  • 1together with E. Descotte and E. Dubuc.
slide-13
SLIDE 13

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Our original problem: homotopy in a model bicategory

We1 seek a construction of the homotopy bicategory Ho(C):

  • Objects and arrows are those of Cfc ( 0

X 1 ).

  • 2-cells: classes [H] of “homotopies” by an eq. relation.

Simultaneous requirements

  • Vertical composition
  • Horizontal composition
  • compatible with the eq. relation
  • (Non invertible) 2-cell → homotopy

Considering Quillen’s notion an obstacle f

ℓ

∼ g ⇒ jf

ℓ

∼ jg ✓ f

ℓ

∼ g ⇒ fj

ℓ

∼ gj : f

ℓ

∼ g ⇒ f

r

∼ g ⇒ fj

r

∼ gj ⇒ fj

ℓ

∼ gj A

f

  • g
  • h

B BI

∂0

  • ∂1
  • B

∼ σ

  • id
  • A′

j

  • 1together with E. Descotte and E. Dubuc.
slide-14
SLIDE 14

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Homotopy in a category with weak equivalences

Quote from [DHKS] book Many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious.

slide-15
SLIDE 15

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Homotopy in a category with weak equivalences

Quote from [DHKS] book Many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious. Section 1: model categories, Section 2: categories with weak equivalences (C, W). Section 1: Ho(Cfc) = Cfc/∼, with ∼ =

ℓ

∼ =

r

∼ “long and technical”

slide-16
SLIDE 16

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Homotopy in a category with weak equivalences

Quote from [DHKS] book Many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious. Section 1: model categories, Section 2: categories with weak equivalences (C, W). Section 1: Ho(Cfc) = Cfc/∼, with ∼ =

ℓ

∼ =

r

∼ “long and technical” Considering ∼W for (C, W) simplifies and clarifies this argument

slide-17
SLIDE 17

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Homotopy in a category with weak equivalences

Quote from [DHKS] book Many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious. Section 1: model categories, Section 2: categories with weak equivalences (C, W). Section 1: Ho(Cfc) = Cfc/∼, with ∼ =

ℓ

∼ =

r

∼ “long and technical” Considering ∼W for (C, W) simplifies and clarifies this argument

1 Condition for (C, W) under which Ho(C) = C/∼W

slide-18
SLIDE 18

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Homotopy in a category with weak equivalences

Quote from [DHKS] book Many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious. Section 1: model categories, Section 2: categories with weak equivalences (C, W). Section 1: Ho(Cfc) = Cfc/∼, with ∼ =

ℓ

∼ =

r

∼ “long and technical” Considering ∼W for (C, W) simplifies and clarifies this argument

1 Condition for (C, W) under which Ho(C) = C/∼W 2 Explicit construction of ∼W, similar to

ℓ

∼

slide-19
SLIDE 19

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Homotopy in a category with weak equivalences

Quote from [DHKS] book Many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious. Section 1: model categories, Section 2: categories with weak equivalences (C, W). Section 1: Ho(Cfc) = Cfc/∼, with ∼ =

ℓ

∼ =

r

∼ “long and technical” Considering ∼W for (C, W) simplifies and clarifies this argument

1 Condition for (C, W) under which Ho(C) = C/∼W 2 Explicit construction of ∼W, similar to

ℓ

∼

3 For C model: (Cfc, W) satisfies this condition, and ∼W =

ℓ

∼

slide-20
SLIDE 20

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

R = (RAB), RAB relation in C(A, B). C/R = C/∼, where ∼ is the least congruence that contains R. Ho(C)

ϕ

  • C

γ

  • π
  • C/∼

ψ

  • If C/∼ = Ho(C), then ∼ has to be ∼W:

f ∼W g if and only if γf = γg. The relation ∼W depends only on W.

slide-21
SLIDE 21

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

R = (RAB), RAB relation in C(A, B). C/R = C/∼, where ∼ is the least congruence that contains R. Ho(C)

ϕ

  • C

γ

  • π
  • C/∼

ψ

  • If C/∼ = Ho(C), then ∼ has to be ∼W:

f ∼W g if and only if γf = γg. The relation ∼W depends only on W. {R}

ω

  • π
  • ⊥
  • C ↓ Cat
  • ⊤
  • γ

{W}

∼(−)

  • C/R = Ho(C) if and only if

1 W ⊆ ωR and R ⊆ ∼W Fix W. Then C/∼W = Ho(C) if and only if 2 W ⊆ ω ∼W.

slide-22
SLIDE 22

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition

ωR is the family of R-equivalences (arrows that admit an R-inverse). 2 W ⊆ ω ∼W: any w.e. is a homotopical equivalence. We say that such a (C, W) is Whitehead.

slide-23
SLIDE 23

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition

ωR is the family of R-equivalences (arrows that admit an R-inverse). 2 W ⊆ ω ∼W: any w.e. is a homotopical equivalence. We say that such a (C, W) is Whitehead. An arrow splits if it is a retraction or a section ( ·

r

·

s

  • , rs = id)

(C, W) is split-generated if any w.e. is a composition of split w.e.

slide-24
SLIDE 24

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition

ωR is the family of R-equivalences (arrows that admit an R-inverse). 2 W ⊆ ω ∼W: any w.e. is a homotopical equivalence. We say that such a (C, W) is Whitehead. An arrow splits if it is a retraction or a section ( ·

r

·

s

  • , rs = id)

(C, W) is split-generated if any w.e. is a composition of split w.e. Toy examples

1

·

∼ f ·

slide-25
SLIDE 25

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition

ωR is the family of R-equivalences (arrows that admit an R-inverse). 2 W ⊆ ω ∼W: any w.e. is a homotopical equivalence. We say that such a (C, W) is Whitehead. An arrow splits if it is a retraction or a section ( ·

r

·

s

  • , rs = id)

(C, W) is split-generated if any w.e. is a composition of split w.e. Toy examples

1

·

∼ f · is not Whitehead.

slide-26
SLIDE 26

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition

ωR is the family of R-equivalences (arrows that admit an R-inverse). 2 W ⊆ ω ∼W: any w.e. is a homotopical equivalence. We say that such a (C, W) is Whitehead. An arrow splits if it is a retraction or a section ( ·

r

·

s

  • , rs = id)

(C, W) is split-generated if any w.e. is a composition of split w.e. Toy examples

1

·

∼ f · is not Whitehead.

2

·

a

  • ∼ f ·

g

  • b
  • , gf = a, fg = b, (a2 = a, b2 = b)
slide-27
SLIDE 27

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition

ωR is the family of R-equivalences (arrows that admit an R-inverse). 2 W ⊆ ω ∼W: any w.e. is a homotopical equivalence. We say that such a (C, W) is Whitehead. An arrow splits if it is a retraction or a section ( ·

r

·

s

  • , rs = id)

(C, W) is split-generated if any w.e. is a composition of split w.e. Toy examples

1

·

∼ f · is not Whitehead.

2

·

a

  • ∼ f ·

g

  • b
  • , gf = a, fg = b, (a2 = a, b2 = b) is

Whitehead and not split-generated.

slide-28
SLIDE 28

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition in model categories

Prop: Split-generated ⇒ Whitehead. Proof: Because split w.e. are homotopical equivalences:

slide-29
SLIDE 29

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition in model categories

Prop: Split-generated ⇒ Whitehead. Proof: Because split w.e. are homotopical equivalences: rs = id ⇒ γ(rs) = γ(id), i.e. rs ∼W id.

slide-30
SLIDE 30

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition in model categories

Prop: Split-generated ⇒ Whitehead. Proof: Because split w.e. are homotopical equivalences: rs = id ⇒ γ(rs) = γ(id), i.e. rs ∼W id. rsr = r ⇒ γ(r)γ(sr) = γ(r) ⇒ γ(sr) = γ(id), i.e. sr ∼W id.

slide-31
SLIDE 31

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition in model categories

Prop: Split-generated ⇒ Whitehead. Proof: Because split w.e. are homotopical equivalences: rs = id ⇒ γ(rs) = γ(id), i.e. rs ∼W id. rsr = r ⇒ γ(r)γ(sr) = γ(r) ⇒ γ(sr) = γ(id), i.e. sr ∼W id. When C is a model category (Cfc, W) is split-generated (any w.e. is a section followed by a retraction, both w.e.)

slide-32
SLIDE 32

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition in model categories

Prop: Split-generated ⇒ Whitehead. Proof: Because split w.e. are homotopical equivalences: rs = id ⇒ γ(rs) = γ(id), i.e. rs ∼W id. rsr = r ⇒ γ(r)γ(sr) = γ(r) ⇒ γ(sr) = γ(id), i.e. sr ∼W id. When C is a model category (Cfc, W) is split-generated (any w.e. is a section followed by a retraction, both w.e.) It follows Cfc/∼W = Ho(Cfc).

slide-33
SLIDE 33

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The Whitehead condition in model categories

Prop: Split-generated ⇒ Whitehead. Proof: Because split w.e. are homotopical equivalences: rs = id ⇒ γ(rs) = γ(id), i.e. rs ∼W id. rsr = r ⇒ γ(r)γ(sr) = γ(r) ⇒ γ(sr) = γ(id), i.e. sr ∼W id. When C is a model category (Cfc, W) is split-generated (any w.e. is a section followed by a retraction, both w.e.) It follows Cfc/∼W = Ho(Cfc). Recall that ∼W is the only possible congruence such that this equality holds.

slide-34
SLIDE 34

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The congruence ∼W can be constructed from different R satisfying 1 ∼W

ℓ

∼

r

∼ Rℓ Rr W ⊆ ωR, “R inverts W” R ⊆ ∼W Whitehead Split-gen. Model

slide-35
SLIDE 35

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The congruence ∼W can be constructed from different R satisfying 1 ∼W

ℓ

∼

r

∼ Rℓ Rr W ⊆ ωR, “R inverts W” R ⊆ ∼W Whitehead Split-gen. Model fRℓ g if and only if ·

f

  • g

·

∼ w

· (wf = wg, w w.e.)

slide-36
SLIDE 36

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The congruence ∼W can be constructed from different R satisfying 1 ∼W

ℓ

∼

r

∼ Rℓ Rr W ⊆ ωR, “R inverts W” R ⊆ ∼W Whitehead Split-gen. Model fRℓ g if and only if ·

f

  • g

·

∼ w

· (wf = wg, w w.e.)

  • Rℓ ⊆ ∼W ✓
slide-37
SLIDE 37

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The congruence ∼W can be constructed from different R satisfying 1 ∼W

ℓ

∼

r

∼ Rℓ Rr W ⊆ ωR, “R inverts W” R ⊆ ∼W Whitehead Split-gen. Model fRℓ g if and only if ·

f

  • g

·

∼ w

· (wf = wg, w w.e.)

  • Rℓ ⊆ ∼W ✓
  • Rℓ inverts split w.e.

rs = id ⇒ rsRℓ id rsr = r ⇒ srRℓ id

slide-38
SLIDE 38

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

A construction of ∼W from Rℓ

First we close Rℓ by composition, then by transitivity. fRc

ℓ g :

A

f

  • g
  • a
  • d0
  • d1
  • B

C

  • A

∼ w

  • h
  • Rc

ℓ is a relaxed version of ℓ

∼ in which A

id

− → A can be any arrow a.

slide-39
SLIDE 39

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

A construction of ∼W from Rℓ

First we close Rℓ by composition, then by transitivity. fRc

ℓ g :

A

f

  • g
  • a
  • d0
  • d1
  • B

C

  • A

∼ w

  • h
  • Rc

ℓ is a relaxed version of ℓ

∼ in which A

id

− → A can be any arrow a. f ∼W g if and only if fRc

ℓ f1Rc ℓ ... fnRc ℓ g.

slide-40
SLIDE 40

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

A construction of ∼W from Rℓ

First we close Rℓ by composition, then by transitivity. fRc

ℓ g :

A

f

  • g
  • a
  • d0
  • d1
  • B

C

  • A

∼ w

  • h
  • Rc

ℓ is a relaxed version of ℓ

∼ in which A

id

− → A can be any arrow a. f ∼W g if and only if fRc

ℓ f1Rc ℓ ... fnRc ℓ g.

In dimension 2 f

H

g : A

⇐ ⇒ ⇐ ⇒ f

  • g
  • a
  • d0
  • d1
  • B

C

  • A

w

  • ∼

h

  • “homotopy respect to the w.e.”

behaves better for forming the 2-cells of Ho(C).

slide-41
SLIDE 41

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The case of model categories

Prop: If fRc

ℓ g then for any cylinder object,

A

f

  • g
  • id
  • ∂0
  • ∂1
  • B

A A × I

σ

  • ∼
slide-42
SLIDE 42

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The case of model categories

Prop: If fRc

ℓ g then for any cylinder object,

A

f

  • g
  • id
  • ∂0
  • ∂1
  • B

A A × I

σ

  • ∼
  • Proof: in 2 steps. Step 1: In fRc

ℓ g we may assume w a fibration

A

f

  • g
  • a
  • d0
  • d1
  • B

C

  • A

w

  • ∼

h

  • i
  • ∼
  • A′

p

  • ∼

r

  • A

f

  • g
  • a
  • i d0
  • i d1
  • B
  • A

h

  • C
  • A′

p ∼

  • r
slide-43
SLIDE 43

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The case of model categories

Prop: If fRc

ℓ g then for any cylinder object,

A

f

  • g
  • id
  • ∂0
  • ∂1
  • B

A A × I

σ

  • ∼
  • Proof: in 2 steps. Step 1: In fRc

ℓ g we may assume w a fibration

Step 2: A

f

  • g
  • a
  • d0
  • d1
  • B

C

  • A

h

  • w

∼

  • A ∐ A

d0+d1 ∂0+∂1

  • A

w ∼

  • h

B A × I

aσ

  • C
slide-44
SLIDE 44

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The case of model categories

Prop: If fRc

ℓ g then for any cylinder object,

A

f

  • g
  • id
  • ∂0
  • ∂1
  • B

A A × I

σ

  • ∼
  • Consequences:

1 Rc

ℓ = ℓ

∼ = ∼W, in particular we recover Cfc/

ℓ

∼ = Ho(Cfc).

slide-45
SLIDE 45

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

The case of model categories

Prop: If fRc

ℓ g then for any cylinder object,

A

f

  • g
  • id
  • ∂0
  • ∂1
  • B

A A × I

σ

  • ∼
  • Consequences:

1 Rc

ℓ = ℓ

∼ = ∼W, in particular we recover Cfc/

ℓ

∼ = Ho(Cfc).

2 New proofs of

ℓ

∼ =

r

∼ and of transitivity, both follow from: f1

ℓ

∼ f2, f2

ℓ

∼ f3 ⇒ f1

r

∼ f3 A

f1

  • f2
  • ∂0
  • ∂1
  • B

A × I

h

  • ,

A

f3

  • f2
  • ∂0
  • ∂1
  • B

A × I

h′

  • A

∂0

  • f1
  • f3

B A × I

h

  • h′
  • A

∂1

  • ∼

f2

slide-46
SLIDE 46

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Further Results

Fibrant-cofibrant replacement in this context. Analysis of the saturated condition in this case. Corollary: any model category is saturated.

slide-47
SLIDE 47

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Further Results

Fibrant-cofibrant replacement in this context. Analysis of the saturated condition in this case. Corollary: any model category is saturated. References

  • [DHKS]: Dwyer, Hirschhorn, Kan, Smith, Homotopy Limit

Functors on Model Categories and Homotopical Categories.

  • Results presented in this talk: The homotopy relation in a

category with weak equivalences, arXiv.

  • 2-dimensional case: talks by Dubuc and Descotte, also in arXiv.
slide-48
SLIDE 48

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories

Further Results

Fibrant-cofibrant replacement in this context. Analysis of the saturated condition in this case. Corollary: any model category is saturated. References

  • [DHKS]: Dwyer, Hirschhorn, Kan, Smith, Homotopy Limit

Functors on Model Categories and Homotopical Categories.

  • Results presented in this talk: The homotopy relation in a

category with weak equivalences, arXiv.

  • 2-dimensional case: talks by Dubuc and Descotte, also in arXiv.

Thank you!