New model structures on simplicial sets Matt Feller University of - - PowerPoint PPT Presentation

▶
new model structures on simplicial sets
SMART_READER_LITE
LIVE PREVIEW

New model structures on simplicial sets Matt Feller University of - - PowerPoint PPT Presentation

Sep 10, 2022 232 likes •1.19k views

New model structures on simplicial sets Matt Feller University of Virginia CT 2019 Edinburgh Outline 1 Background 2 Cisinskis Theory 3 New Stuff Category Theory in Simplicial Sets 0-simplices: Category Theory in Simplicial Sets

slide-1
SLIDE 1

New model structures on simplicial sets

Matt Feller

University of Virginia

CT 2019 — Edinburgh

slide-2
SLIDE 2

Outline

1 Background 2 Cisinski’s Theory 3 New Stuff

slide-3
SLIDE 3

Category Theory in Simplicial Sets

  • 0-simplices:

‚

slide-4
SLIDE 4

Category Theory in Simplicial Sets

  • 0-simplices:

‚

  • 1-simplices:

1

slide-5
SLIDE 5

Category Theory in Simplicial Sets

  • 0-simplices:

‚

  • 1-simplices:

1

  • 2-simplices:

1 2

slide-6
SLIDE 6

Category Theory in Simplicial Sets

  • 0-simplices:

‚

  • 1-simplices:

1

  • 2-simplices:

1 2

ã Ñ

1 2

slide-7
SLIDE 7

Category Theory in Simplicial Sets

  • 0-simplices:

‚

  • 1-simplices:

1

  • 2-simplices:

1 2

ã Ñ

1 2

  • 3-simplices:

1 3 2

slide-8
SLIDE 8

Category Theory in Simplicial Sets

  • 0-simplices:

‚

  • 1-simplices:

1

  • 2-simplices:

1 2

ã Ñ

1 2

  • 3-simplices:

1 3 2

ã Ñ

1 3 2

slide-9
SLIDE 9

Category Theory in Simplicial Sets

  • 0-simplices:

‚

  • 1-simplices:

1

  • 2-simplices:

1 2

ã Ñ

1 2

  • 3-simplices:

1 3 2

ã Ñ

1 3 2

  • (etc.)
slide-10
SLIDE 10

Nerves of Categories

slide-11
SLIDE 11

Nerves of Categories

Definition

Given a small category C, define a simplicial set NpCq as follows:

slide-12
SLIDE 12

Nerves of Categories

Definition

Given a small category C, define a simplicial set NpCq as follows:

  • NpCq0 = objects of C
slide-13
SLIDE 13

Nerves of Categories

Definition

Given a small category C, define a simplicial set NpCq as follows:

  • NpCq0 = objects of C
  • NpCq1 = morphisms of C
slide-14
SLIDE 14

Nerves of Categories

Definition

Given a small category C, define a simplicial set NpCq as follows:

  • NpCq0 = objects of C
  • NpCq1 = morphisms of C
  • NpCq2 = pairs of composible morphisms in C
slide-15
SLIDE 15

Nerves of Categories

Definition

Given a small category C, define a simplicial set NpCq as follows:

  • NpCq0 = objects of C
  • NpCq1 = morphisms of C
  • NpCq2 = pairs of composible morphisms in C
  • NpCq3 = triples of composible morphisms in C
slide-16
SLIDE 16

Nerves of Categories

Definition

Given a small category C, define a simplicial set NpCq as follows:

  • NpCq0 = objects of C
  • NpCq1 = morphisms of C
  • NpCq2 = pairs of composible morphisms in C
  • NpCq3 = triples of composible morphisms in C

. . .

slide-17
SLIDE 17

Nerves of Categories

Definition

Given a small category C, define a simplicial set NpCq as follows:

  • NpCq0 = objects of C
  • NpCq1 = morphisms of C
  • NpCq2 = pairs of composible morphisms in C
  • NpCq3 = triples of composible morphisms in C

. . . N : Cat ã Ñ Set∆op (full/faithful)

slide-18
SLIDE 18

Nerves of Categories

Definition

Given a small category C, define a simplicial set NpCq as follows:

  • NpCq0 = objects of C
  • NpCq1 = morphisms of C
  • NpCq2 = pairs of composible morphisms in C
  • NpCq3 = triples of composible morphisms in C

. . . N : Cat ã Ñ Set∆op (full/faithful)

Examples

∆rns “ Np‚ . . . ‚q

g1 gn

slide-19
SLIDE 19

Nerves of Categories

Definition

Given a small category C, define a simplicial set NpCq as follows:

  • NpCq0 = objects of C
  • NpCq1 = morphisms of C
  • NpCq2 = pairs of composible morphisms in C
  • NpCq3 = triples of composible morphisms in C

. . . N : Cat ã Ñ Set∆op (full/faithful)

Examples

∆rns “ Np‚ . . . ‚q

g1 gn

J :“ Np‚ ‚q

g g -1

slide-20
SLIDE 20

Categories are “Strict”

The inclusion Sprns ã Ñ ∆rns

1 3 2

ã Ñ

1 3 2

is called a spine extension.

slide-21
SLIDE 21

Categories are “Strict”

The inclusion Sprns ã Ñ ∆rns

1 3 2

ã Ñ

1 3 2

is called a spine extension.

Unique Lifting = “Strict”

Categories are simplicial sets with unique spine extensions. X – NpCq (for some C) ð ñ Sprns X ∆rns

D! n ě 2

slide-22
SLIDE 22

Categories are “1-Segal Sets”

Unique lifting condition for categories is 1-dimensional: Sprns is composed of 1-simplices.

slide-23
SLIDE 23

Categories are “1-Segal Sets”

Unique lifting condition for categories is 1-dimensional: Sprns is composed of 1-simplices.

Interpretation

Categories are “1-Segal sets.”

slide-24
SLIDE 24

Categories are “1-Segal Sets”

Unique lifting condition for categories is 1-dimensional: Sprns is composed of 1-simplices.

Interpretation

Categories are “1-Segal sets.”

What are 2-Segal sets?

  • More general than categories
slide-25
SLIDE 25

Categories are “1-Segal Sets”

Unique lifting condition for categories is 1-dimensional: Sprns is composed of 1-simplices.

Interpretation

Categories are “1-Segal sets.”

What are 2-Segal sets?

  • More general than categories
  • Unique “2-dimensional spine extensions”
slide-26
SLIDE 26

Categories are “1-Segal Sets”

Unique lifting condition for categories is 1-dimensional: Sprns is composed of 1-simplices.

Interpretation

Categories are “1-Segal sets.”

What are 2-Segal sets?

  • More general than categories
  • Unique “2-dimensional spine extensions”

è still “strict”

slide-27
SLIDE 27

2-Segal Sets

Triangulations of the square:

2 3 T : 1 2 3 T 1 : 1

slide-28
SLIDE 28

2-Segal Sets

Triangulations of the square:

2 3 T : 1 2 3 T 1 : 1

1 3 2 1 3 2

slide-29
SLIDE 29

2-Segal Sets

Triangulations of the hexagon:

1 2 3 4 5 T : 1 2 3 4 5 T 1 : (etc.)

slide-30
SLIDE 30

2-Segal Sets

Intuition

Think of the inclusions T ã Ñ ∆rns as “2-dimensional spine extensions.”

slide-31
SLIDE 31

2-Segal Sets

Intuition

Think of the inclusions T ã Ñ ∆rns as “2-dimensional spine extensions.”

Definition

A 2-Segal set is a simplicial set X with a unique lifting condition: X is 2-Segal ð ñ T X ∆rns

D! n ě 3

slide-32
SLIDE 32

2-Segal Sets

Examples

  • (Nerves of) categories are 2-Segal.
slide-33
SLIDE 33

2-Segal Sets

Examples

  • (Nerves of) categories are 2-Segal. (1-Segal ñ 2-Segal.)
slide-34
SLIDE 34

2-Segal Sets

Examples

  • (Nerves of) categories are 2-Segal. (1-Segal ñ 2-Segal.)
  • Output of Waldhausen S‚ construction (from algebraic

K-theory) applied to nice enough double categories.

slide-35
SLIDE 35

2-Segal Sets

Examples

  • (Nerves of) categories are 2-Segal. (1-Segal ñ 2-Segal.)
  • Output of Waldhausen S‚ construction (from algebraic

K-theory) applied to nice enough double categories.

  • Lots of other examples from combinatorics.
slide-36
SLIDE 36

2-Segal Sets

Examples

  • (Nerves of) categories are 2-Segal. (1-Segal ñ 2-Segal.)
  • Output of Waldhausen S‚ construction (from algebraic

K-theory) applied to nice enough double categories.

  • Lots of other examples from combinatorics.

Another Perspective

2-Segal sets are equivalent to multivalued categories, where composition is not always unique or defined, but is associative.

slide-37
SLIDE 37

Homotopical/8/Non-Strict Versions

Categories have a homotopical analogue in simplicial sets.

slide-38
SLIDE 38

Homotopical/8/Non-Strict Versions

Categories have a homotopical analogue in simplicial sets.

Definition

A quasi-category is a simplicial set X with a (non-unique) lifting condition: X is a quasi-category ð ñ Λirns X ∆rns

D 0 ă i ă n

slide-39
SLIDE 39

Homotopical/8/Non-Strict Versions

Categories have a homotopical analogue in simplicial sets.

Definition

A quasi-category is a simplicial set X with a (non-unique) lifting condition: X is a quasi-category ð ñ Λirns X ∆rns

D 0 ă i ă n

  • Composition is always defined, but only “unique up to

homotopy.”

slide-40
SLIDE 40

Homotopical/8/Non-Strict Versions

Categories have a homotopical analogue in simplicial sets.

Definition

A quasi-category is a simplicial set X with a (non-unique) lifting condition: X is a quasi-category ð ñ Λirns X ∆rns

D 0 ă i ă n

  • Composition is always defined, but only “unique up to

homotopy.”

  • Special Case: If all morphisms are invertible, we have a Kan

complex—also defined by a non-unique lifting condition.

slide-41
SLIDE 41

Question

Strict Homotopical Category

slide-42
SLIDE 42

Question

Strict Homotopical Category Quasi-category

slide-43
SLIDE 43

Question

Strict Homotopical Groupoid Category Quasi-category

slide-44
SLIDE 44

Question

Strict Homotopical Groupoid Kan Complex Category Quasi-category

slide-45
SLIDE 45

Question

Strict Homotopical Groupoid Kan Complex Category Quasi-category 2-Segal Set

slide-46
SLIDE 46

Question

Strict Homotopical Groupoid Kan Complex Category Quasi-category 2-Segal Set ???

slide-47
SLIDE 47

Model Structure = “Homotopy Theory”

We can endow a category with a “homotopy theory” by putting a model structure on it.

slide-48
SLIDE 48

Model Structure = “Homotopy Theory”

We can endow a category with a “homotopy theory” by putting a model structure on it. Every model structure comes with a class of well-behaved objects, called the fibrant objects, defined by a lifting condition.

slide-49
SLIDE 49

Model Structure = “Homotopy Theory”

We can endow a category with a “homotopy theory” by putting a model structure on it. Every model structure comes with a class of well-behaved objects, called the fibrant objects, defined by a lifting condition.

Examples

  • Classical model structure on Set∆op:
slide-50
SLIDE 50

Model Structure = “Homotopy Theory”

We can endow a category with a “homotopy theory” by putting a model structure on it. Every model structure comes with a class of well-behaved objects, called the fibrant objects, defined by a lifting condition.

Examples

  • Classical model structure on Set∆op:

è equivalent to homotopy theory of topological spaces

slide-51
SLIDE 51

Model Structure = “Homotopy Theory”

We can endow a category with a “homotopy theory” by putting a model structure on it. Every model structure comes with a class of well-behaved objects, called the fibrant objects, defined by a lifting condition.

Examples

  • Classical model structure on Set∆op:

è equivalent to homotopy theory of topological spaces è fibrant objects: Kan complexes

slide-52
SLIDE 52

Model Structure = “Homotopy Theory”

We can endow a category with a “homotopy theory” by putting a model structure on it. Every model structure comes with a class of well-behaved objects, called the fibrant objects, defined by a lifting condition.

Examples

  • Classical model structure on Set∆op:

è equivalent to homotopy theory of topological spaces è fibrant objects: Kan complexes

  • Joyal model structure on Set∆op:
slide-53
SLIDE 53

Model Structure = “Homotopy Theory”

We can endow a category with a “homotopy theory” by putting a model structure on it. Every model structure comes with a class of well-behaved objects, called the fibrant objects, defined by a lifting condition.

Examples

  • Classical model structure on Set∆op:

è equivalent to homotopy theory of topological spaces è fibrant objects: Kan complexes

  • Joyal model structure on Set∆op:

è fibrant objects: quasi-categories

slide-54
SLIDE 54

Finding New Model Structure

Model structures are very finicky. Most lifting conditions will not give us a model structure.

slide-55
SLIDE 55

Finding New Model Structure

Model structures are very finicky. Most lifting conditions will not give us a model structure.

Idea

Look for a model structure first, then decide if the fibrant objects have the properties we want.

slide-56
SLIDE 56

Finding New Model Structure

Model structures are very finicky. Most lifting conditions will not give us a model structure.

Idea

Look for a model structure first, then decide if the fibrant objects have the properties we want. How do we find new model structures?

slide-57
SLIDE 57

Cisinski’s Theory

Classical/Joyal model structures share some properties:

slide-58
SLIDE 58

Cisinski’s Theory

Classical/Joyal model structures share some properties:

  • Cofibrations = Monomorphisms
slide-59
SLIDE 59

Cisinski’s Theory

Classical/Joyal model structures share some properties:

  • Cofibrations = Monomorphisms
  • Fibrant objects defined by lifting against a set. (The model

structures are cofibrantly generated.)

slide-60
SLIDE 60

Cisinski’s Theory

Classical/Joyal model structures share some properties:

  • Cofibrations = Monomorphisms
  • Fibrant objects defined by lifting against a set. (The model

structures are cofibrantly generated.) Cisinski gives us a way to find model structures with these properties.

slide-61
SLIDE 61

Pushout-Product

Pushout-Product

Given A ã Ñ B and C ã Ñ D, the induced map A ˆ C A ˆ D B ˆ C pB ˆ Cq Y pA ˆ Dq B ˆ D is their pushout-product, denoted pA ã Ñ Bq lpC ã Ñ Dq.

slide-62
SLIDE 62

Pushout-Product

Example

The pushout product of 0 ã Ñ ∆r1s and B∆r1s ã Ñ ∆r1s is p∆r1s ˆ B∆r1sq Y p0 ˆ ∆r1sq ∆r1s ˆ ∆r1s which looks like ã Ñ

slide-63
SLIDE 63

Cisinski’s Theory

Bdry :“ tB∆rns ã Ñ ∆rnsuně0 J “ Np‚ ‚q BJ “ 0 Y 1

slide-64
SLIDE 64

Cisinski’s Theory

Bdry :“ tB∆rns ã Ñ ∆rnsuně0 J “ Np‚ ‚q BJ “ 0 Y 1 S a set of AJpSq :“ Bdry lp0 ã Ñ Jq monomorphisms Y S Y S lpBJ ã Ñ Jq Y pS lpBJ ã Ñ Jqq lpBJ ã Ñ Jq . . .

slide-65
SLIDE 65

Cisinski’s Theory

Bdry :“ tB∆rns ã Ñ ∆rnsuně0 J “ Np‚ ‚q BJ “ 0 Y 1 S a set of AJpSq :“ Bdry lp0 ã Ñ Jq monomorphisms Y S Y S lpBJ ã Ñ Jq Y pS lpBJ ã Ñ Jqq lpBJ ã Ñ Jq . . .

Theorem (Cisinski)

  • For any set S of monomorphisms, there is a model structure

whose fibrant objects are those with lifts against AJpSq.

slide-66
SLIDE 66

Cisinski’s Theory

Bdry :“ tB∆rns ã Ñ ∆rnsuně0 J “ Np‚ ‚q BJ “ 0 Y 1 S a set of AJpSq :“ Bdry lp0 ã Ñ Jq monomorphisms Y S Y S lpBJ ã Ñ Jq Y pS lpBJ ã Ñ Jqq lpBJ ã Ñ Jq . . .

Theorem (Cisinski)

  • For any set S of monomorphisms, there is a model structure

whose fibrant objects are those with lifts against AJpSq.

  • When the fibrant objects of a given model structure all lift

against S, they also lift against AJpSq.

slide-67
SLIDE 67

Joyal Model Structure

Example

If S is the set of spine extensions, we get the Joyal model structure.

slide-68
SLIDE 68

Joyal Model Structure

Example

If S is the set of spine extensions, we get the Joyal model structure.

Idea

If we didn’t know what a quasi-category was, we could let S be the spine extensions, and Cisinski’s theory would tell us what a quasi-category should be.

slide-69
SLIDE 69

Are We Done?

Now let S “ tT ã Ñ ∆rnsu, the “two dimensional spine extensions.”

slide-70
SLIDE 70

Are We Done?

Now let S “ tT ã Ñ ∆rnsu, the “two dimensional spine extensions.” Shouldn’t AJpSq answer our question?

slide-71
SLIDE 71

Are We Done?

Now let S “ tT ã Ñ ∆rnsu, the “two dimensional spine extensions.” Shouldn’t AJpSq answer our question? Yes, but. . .

slide-72
SLIDE 72

Are We Done?

Now let S “ tT ã Ñ ∆rnsu, the “two dimensional spine extensions.” Shouldn’t AJpSq answer our question? Yes, but. . .

Analogy

Group presentations often don’t tell us that much about a group.

slide-73
SLIDE 73

Are We Done?

Now let S “ tT ã Ñ ∆rnsu, the “two dimensional spine extensions.” Shouldn’t AJpSq answer our question? Yes, but. . .

Analogy

Group presentations often don’t tell us that much about a group. Similarly, even with the description from Cisinski’s theory, there is still a lot we don’t know about our model structure.

slide-74
SLIDE 74

Minimal Model Structure

Bdry :“ tB∆rns ã Ñ ∆rnsuně0 J “ Np‚ ‚q AJp∅q :“ Bdry lp0 ã Ñ Jq

Theorem (Cisinski)

There is a model structure whose fibrant objects are those with lifts against AJp∅q; the minimal model structure on Set∆op.

slide-75
SLIDE 75

Minimal Model Structure

Bdry :“ tB∆rns ã Ñ ∆rnsuně0 J “ Np‚ ‚q AJp∅q :“ Bdry lp0 ã Ñ Jq

Theorem (Cisinski)

There is a model structure whose fibrant objects are those with lifts against AJp∅q; the minimal model structure on Set∆op. Ex: n “ 1 ã Ñ

slide-76
SLIDE 76

Horns and Iso-Horns

Horns

  • Simplex =

Np‚ ‚ . . . ‚ ‚q

  • Face = delete one vertex
  • Horn = union of all faces but one

n “ 2 Horn Extension

ã Ñ

slide-77
SLIDE 77

Horns and Iso-Horns

Horns

  • Simplex =

Np‚ ‚ . . . ‚ ‚q

  • Face = delete one vertex
  • Horn = union of all faces but one

n “ 2 Horn Extension

ã Ñ

Iso-Horns

  • Isoplex =

Np‚ . . . ‚ ‚ . . . ‚q

  • Face = delete one vertex
  • Horn = union of all faces but one,

the one opposite a vertex of the isomorphism

n “ 2 Iso-Horn Ext’n

ã Ñ

slide-78
SLIDE 78

Theorem

Theorem (F.)

The fibrant objects in the minimal model structure are those with lifts against IsoHorn.

slide-79
SLIDE 79

Theorem

Theorem (F.)

The fibrant objects in the minimal model structure are those with lifts against IsoHorn. In fact, AJp∅q “ IsoHorn.

slide-80
SLIDE 80

Theorem

Theorem (F.)

The fibrant objects in the minimal model structure are those with lifts against IsoHorn. In fact, AJp∅q “ IsoHorn.

30-Second Sketch of Proof

slide-81
SLIDE 81

Theorem

Theorem (F.)

The fibrant objects in the minimal model structure are those with lifts against IsoHorn. In fact, AJp∅q “ IsoHorn.

30-Second Sketch of Proof

  • Elements of IsoHorn are retracts of things in AJp∅q.
slide-82
SLIDE 82

Theorem

Theorem (F.)

The fibrant objects in the minimal model structure are those with lifts against IsoHorn. In fact, AJp∅q “ IsoHorn.

30-Second Sketch of Proof

  • Elements of IsoHorn are retracts of things in AJp∅q.
  • Elements of AJp∅q are built out of elements of IsoHorn (via

transfinite composition of pushouts).

slide-83
SLIDE 83

Theorem

Theorem (F.)

The fibrant objects in the minimal model structure are those with lifts against IsoHorn. In fact, AJp∅q “ IsoHorn.

30-Second Sketch of Proof

  • Elements of IsoHorn are retracts of things in AJp∅q.
  • Elements of AJp∅q are built out of elements of IsoHorn (via

transfinite composition of pushouts). è Codomains are categories, so n-simplices are equivalent to paths of arrows.

slide-84
SLIDE 84

Interpretation

In category theory: c – d ù ñ Hompc, xq – Hompd, xq for all x

slide-85
SLIDE 85

Interpretation

In category theory: c – d ù ñ Hompc, xq – Hompd, xq for all x When c and d are isomorphic, their relationship to the rest of the category is also equivalent.

slide-86
SLIDE 86

Interpretation

In category theory: c – d ù ñ Hompc, xq – Hompd, xq for all x When c and d are isomorphic, their relationship to the rest of the category is also equivalent.

Fibrant Objects in the Minimal Model Structure

Similar thing happens: if two 0-simplices are “isomorphic,” then there is a correspondence between the n simplices of which they are vertices.

slide-87
SLIDE 87

Interpretation

In category theory: c – d ù ñ Hompc, xq – Hompd, xq for all x When c and d are isomorphic, their relationship to the rest of the category is also equivalent.

Fibrant Objects in the Minimal Model Structure

Similar thing happens: if two 0-simplices are “isomorphic,” then there is a correspondence between the n simplices of which they are vertices. è x – y ù ñ x and y interact with the rest of X equivalently.

slide-88
SLIDE 88

Model Structures on Set∆op

QCat Kan M Ñ N indicates . . . . . . FibObM Ě FibObN QCatn Kann . . . . . . Homology QCat0 Kan0 QCat´1 Kan´1 Trivial

slide-89
SLIDE 89

Model Structures on Set∆op

Minimal QCat M Ñ N indicates FibObM Ě FibObN

slide-90
SLIDE 90

Model Structures on Set∆op

Minimal QCat M Ñ N indicates FibObM Ě FibObN

slide-91
SLIDE 91

Model Structures on Set∆op

Minimal Q-2-Seg? QCat M Ñ N indicates FibObM Ě FibObN

slide-92
SLIDE 92

Model Structures on Set∆op

Minimal . . . Q-3-Seg? Q-2-Seg? QCat M Ñ N indicates FibObM Ě FibObN

slide-93
SLIDE 93

Model Structures on Set∆op

Minimal Htpy? . . . Q-3-Seg? Q-2-Seg? QCat M Ñ N indicates FibObM Ě FibObN

slide-94
SLIDE 94

Model Structures on Set∆op

Minimal Htpy? . . . Q-3-Seg? Q-2-Seg? QCat

? ?

M Ñ N indicates FibObM Ě FibObN

slide-95
SLIDE 95

References

  • J. Bergner, A. Osorno, V. Ozornova, M. Rovelli,
  • C. Scheimbauer.

2-Segal sets and the Waldhausen construction. Topology and its Applications, 235, 10.1016/j.topol.2017.12.009, 2016

  • A. Campbell, E. Lanari.

On truncated quasi-categories. Preprint, arXiv:1810.11188, 2018. D.-C. Cisinski. Les pr´ efaisceaux comme mod` eles des types d’homotopie Ast´ erisque, no. 308, Soc. Math. France, 2006.

  • T. Dyckerhoff, M. Kapranov

Higher Segal Spaces I Preprint, arXiv:1212.3563, 2012.