Lecture 2: Homotopical Algebra Nicola Gambino School of Mathematics - - PowerPoint PPT Presentation

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Lecture 2: Homotopical Algebra

Nicola Gambino

School of Mathematics University of Leeds

Young Set Theory Copenhagen June 13th, 2016

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Homotopical algebra

Motivation

◮ Axiomatic development of homotopy theory ◮ Addressing size issues in localizations

Key notion

◮ Model category

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Outline

Part I: Model categories Part II: Groupoids Part III: Simplicial sets

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Part I: Model categories

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Lifting problems

Fix a category C.

• Definition. Let p : B → A and i : X → Y .

◮ We say p has the right lifting property w.r.t. i if for every diagram

X

• i
• B

p

• Y

A

there exists a diagonal filler X

• i
• B

p

• Y
• A

Notation: i ⋔ p

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Examples

◮ In Set, if i injective and p surjective then i ⋔ p

X

• i
• B

p

• Y
• A

◮ In Top, a map p : B → A is a Hurewicz fibration if iX ⋔ p

X × {0}

• iX
• B

p

• X × I
• A

for all X. The case X = {∗} is a path-lifting property.

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Lifting problems: special cases

• 1. If A = 1, then we have an ‘extension property’ (cf. injective objects):

X

• i
• B

Y

• 2. If X = 0, then we have a ‘lifting property’ (cf. projective objects):

B

p

• Y
• A
• Note. General case is a combination of these:

X

• i
• B

p

• Y
• A

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Weak factorisation systems

• Definition. A weak factorisation system on C is a pair (L, R) of classes of

maps such that:

• 1. L = {i | (∀p ∈ R) i ⋔ p}
• 2. R = {p | (∀i ∈ L) i ⋔ p}
• 3. Every f : X → Y admits a factorisation

X

f

• i
• Y

B

p

• with i ∈ L and p ∈ R.

Example

◮ ( Inj , Surj ) is a weak factorisation system on Set.

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Model structures

• Definition. A Model structure on C consists of three classes of maps,
• Weq , Fib , Cof
• ,

such that

• 1. Weq satisfies 3-for-2, i.e. for all

X

h

• f
• Z

Y

g

• if two out of f , g, h are in Weq, then so is the third.
• 2. (Weq ∩ Cof, Fib) is a weak factorisation system.
• 3. (Cof, Weq ∩ Fib) is a weak factorisation system

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Model structures (II)

Examples

• 1. Any category C admits a model structure where

Weq = { isomorphisms } , Fib = { all maps } , Cof = { all maps }

• 2. The category Top admits a model structure where

Weq = { homotopy equivalences } , Fib = { Hurewicz fibrations }

• 3. The category Top admits a model structure where

Weq = { weak homotopy equivalences } , Fib = { Serre fibrations }

• Terminology. An object X ∈ C is said to be

◮ fibrant if X → 1 is in Fib ◮ cofibrant if 0 → X is in Cof.

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Model structures: factorisations

Remark 1. Every f : X → Y admits two factorisations X

f

• i
• Y

B

p

• 1. i ∈ Weq ∩ Cof , p ∈ Fib
• 2. i ∈ Cof , p ∈ Weq ∩ Fib
• Example. The ‘path object’ factorisation

A

∆A

• r
• A × A

P

(s,t)

• with r ∈ Weq ∩ Cof and (s, t) ∈ Fib.

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Model structures: lifting problems

Remark 2. We have diagonal fillers X

• i
• B

p

• Y
• A

in two cases:

• 1. i ∈ Weq ∩ Cof , p ∈ Fib
• 2. i ∈ Cof , p ∈ Weq ∩ Fib
• Example. We have diagonal fillers for

A

• r
• E

p

• P

(s,t)

• A × A

for all p ∈ Fib.

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Part II: Groupoids

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Example: groupoids

The category Gpd

◮ objects: groupoids, i.e. categories in which every arrow is invertible ◮ maps: functors

Examples

• 1. Sets and bijections.
• 2. A group G is a groupoid with one object, ∗, and Map(∗, ∗) = G.
• 3. Every topological space has a fundamental groupoid, Π1(X) of points

and homotopy classes of maps.

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Isofibrations

• Definition. A functor p : B → A between groupoids is a isofibration if it has

the following path lifting property: B

p

• b0

∃β

❴

• ∃b1

A a0

∀α

a1

• Note. p : B → A is isofibration iff

{0}

b

• i0
• B

p

• J

a

A

has a diagonal filler, where J =

• 1

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The model structure on groupoids

• Theorem. The category Gpd admits a model structure

◮ Weq = equivalence of categories ◮ Fib = isofibrations ◮ Cof = functors injective on objects

• Note. The (Weq ∩ Cof, Fib)-factorisation of f : A → B is given by

A

f

• i
• B

{(x, y, β) | β : fx → y}

p

• In particular

A

∆A

• r
• A × A

AJ

(s,t)

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Part III: Simplicial sets

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Simplicial sets

The simplicial category ∆ has

◮ objects: [n] = {0 < . . . < n}, non-empty finite linear orders ◮ morphisms: order-preserving functions

• Definition. A simplicial set is a functor

A : ∆op → Set [n] → An The category SSet

◮ objects: simplicial sets ◮ maps: natural transformations

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Simplicial sets as spaces

• Idea. We think of a simplicial set as a set of instructions to construct a space:

◮ For n ≥ 0, define the topological standard n-simplex

|∆n| = {(x0, . . . , xn) ∈ Rn+1 | xi ≥ 0 , x0 + . . . + xn = 1}

◮ For A ∈ SSet, define its geometric realization

R(A) =

[n]∈∆

An × |∆n|

• /≃

This gives a functor R : SSet → Top.

• Note. For [n] ∈ ∆, there is ∆n ∈ SSet such that

R(∆n) ∼ = |∆n| This is called the (simplicial) standard n-simplex.

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Examples: nerve of a groupoid

Given a groupoid G, its nerve is the simplicial set NG : ∆op → Set defined by (NG)n = set of strings of n-composable arrows in G = { x0

f1

x1

f1

x2 . . .

fn

xn }

Note.

◮ NG captures objects and maps of G, not composition. ◮ This gives a functor N : Gpd → SSet.

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The category of simplicial sets

SSet is a presheaf category ⇒

◮ it has all small limits and colimits ◮ it is locally cartesian closed: all slices are cartesian closed.

Equivalently: B

f

• SSet/B

Πf ⊣

• Σf

⊣

• A

SSet/A

∆f

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Kan fibrations

• Definition. A map p : B → A is a Kan fibration if every diagram

Λn

k hn

k

• B

p

• ∆n

A

has a diagonal filler. Here, Λn

k is obtained by removing from ∆n its interior and

the interior of the face opposite the k-th vertex, and hn

k the inclusion.

Examples. Λ2

1 h1

B

p

• ∆1

A

Λ1

h1

B

p

• ∆2

A

• Note. p : B → A is an isofibration in Gpd ⇒ Np : NB → NA Kan fibration.

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The model structure on simplicial sets

• Theorem. The category SSet admits a model structure where

◮ Weq = weak homotopy equivalences ◮ Fib = Kan fibrations ◮ Cof = monomorphisms

• Note. The fibrant objects are the Kan complexes:

Λn

k hn

k

• B

∆n

• e.g.

Λ2

1 h1

B

∆1

• Λ1

h1

B

∆2

• Note. G groupoid ⇒ NG Kan complex (using the composition and inverses)

Kan complexes can be seen as weak ∞-groupoids.

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Summary

Part I: Model structures Part II: Groupoids Part III: Simplicial sets Tomorrow: the type theory T has models in groupoids and simplicial sets.