Model Categories and Stable Homotopy Theory Model Categories and Stable Homotopy Theory

Model categories and stable homotopy theory Model categories and stable homotopy theory

Mike Hill UCLA Mike Hill Mike Hopkins Harvard University Mike Hopkins Doug Ravenel University of Rochester Doug Ravenel Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations International Workshop on Algebraic Topology Southern University of Science and Technology Shenzhen, China June 8, 2018

1.1 in stable homotopy theory and (time permitting) in equivariant stable homotopy theory.

A spectrum X was originally deﬁned to be a sequence of pointed spaces or simplicial sets {X0, X1, X2,... } with structure X maps n :ΣXn → Xn+1. A map of spectra f : X → Y is a collection of pointed maps fn : Xn → Yn compatible with the structure maps.

There are two different notions of weak equivalence in the category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. • f : X → Y is a stable equivalence if . . .

Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.2 (time permitting) in equivariant stable homotopy theory.

There are two different notions of weak equivalence in the category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. • f : X → Y is a stable equivalence if . . .

Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories in stable homotopy theory and Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.2 in equivariant stable homotopy theory.

There are two different notions of weak equivalence in the category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. • f : X → Y is a stable equivalence if . . .

Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories in stable homotopy theory and (time permitting) Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.2 A spectrum X was originally deﬁned to be a sequence of pointed spaces or simplicial sets {X0, X1, X2,... } with structure X maps n :ΣXn → Xn+1. A map of spectra f : X → Y is a collection of pointed maps fn : Xn → Yn compatible with the structure maps.

There are two different notions of weak equivalence in the category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. • f : X → Y is a stable equivalence if . . .

Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories in stable homotopy theory and (time permitting) in Mike Hill equivariant stable homotopy theory. Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.2 with structure X maps n :ΣXn → Xn+1. A map of spectra f : X → Y is a collection of pointed maps fn : Xn → Yn compatible with the structure maps.

There are two different notions of weak equivalence in the category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. • f : X → Y is a stable equivalence if . . .

Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories in stable homotopy theory and (time permitting) in Mike Hill equivariant stable homotopy theory. Mike Hopkins Doug Ravenel

A spectrum X was originally deﬁned to be a sequence of Introduction Quillen model pointed spaces or simplicial sets {X0, X1, X2,... } categories Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.2 A map of spectra f : X → Y is a collection of pointed maps fn : Xn → Yn compatible with the structure maps.

There are two different notions of weak equivalence in the category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. • f : X → Y is a stable equivalence if . . .

Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories in stable homotopy theory and (time permitting) in Mike Hill equivariant stable homotopy theory. Mike Hopkins Doug Ravenel

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.2 There are two different notions of weak equivalence in the category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. • f : X → Y is a stable equivalence if . . .

Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories in stable homotopy theory and (time permitting) in Mike Hill equivariant stable homotopy theory. Mike Hopkins Doug Ravenel

A spectrum X was originally defined to be a sequence of Introduction Quillen model pointed spaces or simplicial sets {X0, X1, X2,... } with structure categories X Cofibrant generation maps n :ΣXn → Xn+1. A map of spectra f : X → Y is a Bousfield localization collection of pointed maps fn : Xn → Yn compatible with the Enriched category structure maps. theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.2 • f : X → Y is a strict equivalence if each map fn is a weak equivalence. • f : X → Y is a stable equivalence if . . .

Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories in stable homotopy theory and (time permitting) in Mike Hill equivariant stable homotopy theory. Mike Hopkins Doug Ravenel

Spectra as enriched There are two different notions of weak equivalence in the functors Deﬁning the smash category of spectra Sp: product of spectra Generalizations

1.2 • f : X → Y is a stable equivalence if . . .

Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories in stable homotopy theory and (time permitting) in Mike Hill equivariant stable homotopy theory. Mike Hopkins Doug Ravenel

Spectra as enriched There are two different notions of weak equivalence in the functors Deﬁning the smash category of spectra Sp: product of spectra Generalizations • f : X → Y is a strict equivalence if each map fn is a weak equivalence.

1.2 Model categories Introduction and stable homotopy theory

The purpose of this talk is to introduce the use of Quillen model categories in stable homotopy theory and (time permitting) in Mike Hill equivariant stable homotopy theory. Mike Hopkins Doug Ravenel

1.2 There are at least two different ways to ﬁnish the deﬁnition of stable equivalence:

(i) Deﬁne stable homotopy groups of spectra and require π∗f to be an isomorphism.

(ii) Deﬁne a functor Λ: Sp → Sp where (ΛX)n is the homotopy colimit (meaning the mapping telescope) of

2 Xn → ΩXn+1 → Ω Xn+2 → ...

and then require Λf to be a strict equivalence. Classically these two deﬁnitions are equivalent, but in certain variants of the deﬁnition of spectra themselves, they are different. They differ in the category SpΣ of symmetric spectra of Hovey-Shipley-Smith, which we will introduce below.

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.3 (i) Deﬁne stable homotopy groups of spectra and require π∗f to be an isomorphism.

(ii) Deﬁne a functor Λ: Sp → Sp where (ΛX)n is the homotopy colimit (meaning the mapping telescope) of

2 Xn → ΩXn+1 → Ω Xn+2 → ...

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.3 and require π∗f to be an isomorphism.

(ii) Deﬁne a functor Λ: Sp → Sp where (ΛX)n is the homotopy colimit (meaning the mapping telescope) of

2 Xn → ΩXn+1 → Ω Xn+2 → ...

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.3 (ii) Deﬁne a functor Λ: Sp → Sp where (ΛX)n is the homotopy colimit (meaning the mapping telescope) of

2 Xn → ΩXn+1 → Ω Xn+2 → ...

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.3 where (ΛX)n is the homotopy colimit (meaning the mapping telescope) of

2 Xn → ΩXn+1 → Ω Xn+2 → ...

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

(ii) Deﬁne a functor Λ: Sp → Sp Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.3 (meaning the mapping telescope) of

2 Xn → ΩXn+1 → Ω Xn+2 → ...

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

(ii) Deﬁne a functor Λ: Sp → Sp where (ΛX)n is the Spectra as enriched homotopy colimit functors Deﬁning the smash product of spectra

Generalizations

1.3 of

2 Xn → ΩXn+1 → Ω Xn+2 → ...

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

(ii) Deﬁne a functor Λ: Sp → Sp where (ΛX)n is the Spectra as enriched homotopy colimit (meaning the mapping telescope) functors Deﬁning the smash product of spectra

Generalizations

1.3 and then require Λf to be a strict equivalence. Classically these two deﬁnitions are equivalent, but in certain variants of the deﬁnition of spectra themselves, they are different. They differ in the category SpΣ of symmetric spectra of Hovey-Shipley-Smith, which we will introduce below.

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

(ii) Deﬁne a functor Λ: Sp → Sp where (ΛX)n is the Spectra as enriched homotopy colimit (meaning the mapping telescope) of functors Deﬁning the smash product of spectra 2 Xn → ΩXn+1 → Ω Xn+2 → ... Generalizations

1.3 Classically these two deﬁnitions are equivalent, but in certain variants of the deﬁnition of spectra themselves, they are different. They differ in the category SpΣ of symmetric spectra of Hovey-Shipley-Smith, which we will introduce below.

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

and then require Λf to be a strict equivalence.

1.3 but in certain variants of the deﬁnition of spectra themselves, they are different. They differ in the category SpΣ of symmetric spectra of Hovey-Shipley-Smith, which we will introduce below.

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

and then require Λf to be a strict equivalence. Classically these two deﬁnitions are equivalent,

1.3 they are different. They differ in the category SpΣ of symmetric spectra of Hovey-Shipley-Smith, which we will introduce below.

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

and then require Λf to be a strict equivalence. Classically these two deﬁnitions are equivalent, but in certain variants of the deﬁnition of spectra themselves,

1.3 They differ in the category SpΣ of symmetric spectra of Hovey-Shipley-Smith, which we will introduce below.

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

and then require Λf to be a strict equivalence. Classically these two deﬁnitions are equivalent, but in certain variants of the deﬁnition of spectra themselves, they are different.

1.3 which we will introduce below.

Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

of Hovey-Shipley-Smith, 1.3 Model categories Introduction (continued) and stable homotopy There are two different notions of weak equivalence in the theory category of spectra Sp:

• f : X → Y is a strict equivalence if each map fn is a weak equivalence. Mike Hill Mike Hopkins • f : X → Y is a stable equivalence if . . . Doug Ravenel Introduction

There are at least two different ways to finish the definition of Quillen model stable equivalence: categories Cofibrant generation

(i) Deﬁne stable homotopy groups of spectra and require π∗f Bousﬁeld localization to be an isomorphism. Enriched category theory

of Hovey-Shipley-Smith, which we will introduce below. 1.3 • Quillen model categories • Fibrant and cofibrant replacement • Cofibrant generation • Bousfield localization • Enriched category theory We will see that the passage from strict equivalence to stable equivalence is a form of Bousfield localization.

Model categories Iintroduction (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model Dan Quillen Dan Kan Pete Max Kelly categories

1940-2011 1928-2013 Bousﬁeld 1930-2007 Coﬁbrant generation

Bousﬁeld localization

In order to understand this better we need to discuss Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.4 • Fibrant and cofibrant replacement • Cofibrant generation • Bousfield localization • Enriched category theory We will see that the passage from strict equivalence to stable equivalence is a form of Bousfield localization.

Model categories Iintroduction (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model Dan Quillen Dan Kan Pete Max Kelly categories

1940-2011 1928-2013 Bousﬁeld 1930-2007 Coﬁbrant generation

Bousﬁeld localization

In order to understand this better we need to discuss Enriched category theory • Quillen model categories Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.4 • Cofibrant generation • Bousfield localization • Enriched category theory We will see that the passage from strict equivalence to stable equivalence is a form of Bousfield localization.

Model categories Iintroduction (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model Dan Quillen Dan Kan Pete Max Kelly categories

1940-2011 1928-2013 Bousﬁeld 1930-2007 Coﬁbrant generation

Bousﬁeld localization

In order to understand this better we need to discuss Enriched category theory • Quillen model categories Spectra as enriched functors

• Fibrant and coﬁbrant replacement Deﬁning the smash product of spectra

Generalizations

1.4 • Bousﬁeld localization • Enriched category theory We will see that the passage from strict equivalence to stable equivalence is a form of Bousﬁeld localization.

Model categories Iintroduction (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model Dan Quillen Dan Kan Pete Max Kelly categories

1940-2011 1928-2013 Bousﬁeld 1930-2007 Coﬁbrant generation

Bousﬁeld localization

In order to understand this better we need to discuss Enriched category theory • Quillen model categories Spectra as enriched functors

• Fibrant and cofibrant replacement Defining the smash product of spectra • Cofibrant generation Generalizations

1.4 • Enriched category theory We will see that the passage from strict equivalence to stable equivalence is a form of Bousﬁeld localization.

Model categories Iintroduction (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model Dan Quillen Dan Kan Pete Max Kelly categories

1940-2011 1928-2013 Bousﬁeld 1930-2007 Coﬁbrant generation

Bousﬁeld localization

In order to understand this better we need to discuss Enriched category theory • Quillen model categories Spectra as enriched functors

• Fibrant and cofibrant replacement Defining the smash product of spectra • Cofibrant generation Generalizations • Bousfield localization

1.4 We will see that the passage from strict equivalence to stable equivalence is a form of Bousﬁeld localization.

Model categories Iintroduction (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model Dan Quillen Dan Kan Pete Max Kelly categories

1940-2011 1928-2013 Bousﬁeld 1930-2007 Coﬁbrant generation

Bousﬁeld localization

In order to understand this better we need to discuss Enriched category theory • Quillen model categories Spectra as enriched functors

• Fibrant and cofibrant replacement Defining the smash product of spectra • Cofibrant generation Generalizations • Bousfield localization • Enriched category theory

1.4 Model categories Iintroduction (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model Dan Quillen Dan Kan Pete Max Kelly categories

1940-2011 1928-2013 Bousﬁeld 1930-2007 Coﬁbrant generation

Bousﬁeld localization

In order to understand this better we need to discuss Enriched category theory • Quillen model categories Spectra as enriched functors

• Fibrant and cofibrant replacement Defining the smash product of spectra • Cofibrant generation Generalizations • Bousfield localization • Enriched category theory We will see that the passage from strict equivalence to stable equivalence is a form of Bousfield localization.

1.4 each of which includes all isomorphisms, satisfying the following five axioms: MC1 Bicompleteness axiom. M has all small limits and colimits. These include products, coproducts, pullbacks and pushouts. This implies that M has initial and terminal objects, denoted by ∅ and ∗. g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

three classes of morphisms: weak equivalences, ﬁbrations and Mike Hill Mike Hopkins coﬁbrations, Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 satisfying the following five axioms: MC1 Bicompleteness axiom. M has all small limits and colimits. These include products, coproducts, pullbacks and pushouts. This implies that M has initial and terminal objects, denoted by ∅ and ∗. g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

three classes of morphisms: weak equivalences, ﬁbrations and Mike Hill Mike Hopkins coﬁbrations, each of which includes all isomorphisms, Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 MC1 Bicompleteness axiom. M has all small limits and colimits. These include products, coproducts, pullbacks and pushouts. This implies that M has initial and terminal objects, denoted by ∅ and ∗. g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

three classes of morphisms: weak equivalences, fibrations and Mike Hill Mike Hopkins cofibrations, each of which includes all isomorphisms, Doug Ravenel satisfying the following five axioms: Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 These include products, coproducts, pullbacks and pushouts. This implies that M has initial and terminal objects, denoted by ∅ and ∗. g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 coproducts, pullbacks and pushouts. This implies that M has initial and terminal objects, denoted by ∅ and ∗. g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 pullbacks and pushouts. This implies that M has initial and terminal objects, denoted by ∅ and ∗. g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 and pushouts. This implies that M has initial and terminal objects, denoted by ∅ and ∗. g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 This implies that M has initial and terminal objects, denoted by ∅ and ∗. g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

and pushouts. Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 denoted by ∅ and ∗. g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

and pushouts. This implies that M has initial and terminal Bousﬁeld localization objects, Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 g MC2 2-out-of-3 axiom. Let X −→f Y −→ Z be morphisms in M. Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

and pushouts. This implies that M has initial and terminal Bousﬁeld localization objects, denoted by ∅ and ∗. Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 Then if any two of f , g and gf are weak equivalences, so is the third. MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

and pushouts. This implies that M has initial and terminal Bousﬁeld localization objects, denoted by ∅ and ∗. Enriched category theory f g MC2 2-out-of-3 axiom. Let X −→ Y −→ Z be morphisms in M. Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.5 MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration is again a weak equivalence, fibration or cofibration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

Then if any two of f , g and gf are weak equivalences, so is Deﬁning the smash the third. product of spectra Generalizations

1.5 is again a weak equivalence, ﬁbration or coﬁbration.

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

Then if any two of f , g and gf are weak equivalences, so is Defining the smash the third. product of spectra Generalizations MC3 Retract axiom. A retract of a weak equivalence, fibration or cofibration

1.5 We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

1.5 if it is also a weak equivalence.

Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

We say that a ﬁbration or coﬁbration is trivial (or acyclic)

1.5 Model categories Quillen model categories and stable homotopy theory Deﬁnition A Quillen model category M is a category equipped with

colimits. These include products, coproducts, pullbacks Coﬁbrant generation

We say that a ﬁbration or coﬁbration is trivial (or acyclic) if it is also a weak equivalence. 1.5 MC5 Factorization axiom. Any morphism f : X → Y can be functorially factored in two ways as

Model categories Quillen model categories (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Definition Quillen model categories MC4 Lifting axiom. Given a commutative diagram Cofibrant generation Bousfield localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.6 MC5 Factorization axiom. Any morphism f : X → Y can be functorially factored in two ways as

Model categories Quillen model categories (continued) and stable homotopy theory

Mike Hill Mike Hopkins Deﬁnition Doug Ravenel

MC4 Lifting axiom. Given a commutative diagram Introduction Quillen model categories f A / X Coﬁbrant generation Bousﬁeld localization i p Enriched category B Y , theory g / Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.6 MC5 Factorization axiom. Any morphism f : X → Y can be functorially factored in two ways as

Model categories Quillen model categories (continued) and stable homotopy theory

Deﬁnition Mike Hill Mike Hopkins Doug Ravenel MC4 Lifting axiom. Given a commutative diagram Introduction

Quillen model f categories A 7/ X h Cofibrant generation cofibration i p trivial fibration Bousfield localization B / Y , Enriched category g theory

Spectra as enriched a morphism h (called a lifting) exists for i and p as functors indicated. Deﬁning the smash product of spectra

Generalizations

1.6 MC5 Factorization axiom. Any morphism f : X → Y can be functorially factored in two ways as

Model categories Quillen model categories (continued) and stable homotopy theory

Deﬁnition Mike Hill Mike Hopkins Doug Ravenel MC4 Lifting axiom. Given a commutative diagram Introduction

Quillen model f categories A 7/ X h Cofibrant generation trivial cofibration i p fibration Bousfield localization B / Y , Enriched category g theory

Spectra as enriched a morphism h (called a lifting) exists for i and p as functors indicated. Deﬁning the smash product of spectra

Generalizations

1.6 MC5 Factorization axiom. Any morphism f : X → Y can be functorially factored in two ways as

Model categories Quillen model categories (continued) and stable homotopy theory

Deﬁnition Mike Hill Mike Hopkins Doug Ravenel MC4 Lifting axiom. Given a commutative diagram Introduction

f Quillen model A / X categories 7 Cofibrant generation cofibration i h p trivial fibration trivial cofibration fibration Bousfield localization B / Y , Enriched category g theory

Spectra as enriched a morphism h (called a lifting) exists for i and p as functors indicated. Deﬁning the smash product of spectra

Generalizations

1.6 Model categories Quillen model categories (continued) and stable homotopy theory

Deﬁnition

MC4 Lifting axiom. Given a commutative diagram Mike Hill Mike Hopkins f Doug Ravenel A / X 7 Introduction cofibration i h p trivial fibration trivial cofibration fibration Quillen model B / Y , categories g Cofibrant generation a morphism h (called a lifting) exists for i and p as Bousfield localization Enriched category indicated. theory Spectra as enriched MC5 Factorization axiom. Any morphism f : X → Y can be functors functorially factored in two ways as Defining the smash product of spectra

Generalizations

f X / Y

1.6 Model categories Quillen model categories (continued) and stable homotopy theory Deﬁnition

MC4 Lifting axiom. Given a commutative diagram Mike Hill Mike Hopkins f Doug Ravenel A 7/ X cofibration i h p trivial fibration Introduction trivial cofibration fibration Quillen model B / Y , categories g Cofibrant generation a morphism h (called a lifting) exists for i and p as Bousfield localization Enriched category indicated. theory Spectra as enriched MC5 Factorization axiom. Any morphism f : X → Y can be functors functorially factored in two ways as Defining the smash product of spectra

Generalizations 6 ? coﬁbration = α(f ) β(f ) = trivial ﬁbration f X /( Y

1.6 Model categories Quillen model categories (continued) and stable homotopy theory Deﬁnition

Spectra as enriched MC5 Factorization axiom. Any morphism f : X → Y can be functors

functorially factored in two ways as Deﬁning the smash product of spectra

Generalizations 6 ? cofibration = α(f ) β(f ) = trivial fibration f ( X 6/ Y trivial cofibration = γ(f ) δ(f ) = fibration ( ?

1.6 Model categories Quillen model categories (continued) and stable homotopy theory Deﬁnition

MC4 Lifting axiom. Given a commutative diagram

Mike Hill f Mike Hopkins A 7/ X Doug Ravenel cofibration i h p trivial fibration trivial cofibration fibration Introduction B Y , Quillen model g / categories Cofibrant generation

a morphism h (called a lifting) exists for i and p as Bousﬁeld localization

indicated. Enriched category theory

MC5 Factorization axiom. Any morphism f : X → Y can be Spectra as enriched functorially factored in two ways as functors Defining the smash product of spectra 6 ? Generalizations cofibration = α(f ) β(f ) = trivial fibration f ( X 6/ Y trivial cofibration = γ(f ) δ(f ) = fibration ( ? This last axiom is the hardest one to verify in practice. 1.6 Weak equivalences are maps inducing isomorphisms of homotopy groups. Fibrations are Serre fibrations, that is maps p : X → Y with the right lifting property

n f I 6/ X h jn p for each n ≥ 0. In+1 Y , g /

n−1 n Coﬁbrations are maps (such as in : S → D for n ≥ 0) having the left lifting property with respect to all trivial Serre ﬁbrations.

Similar deﬁnitions can be made for T , the category of pointed topological spaces and basepoint preserving maps.

Model categories Two classical examples and stable homotopy theory

Let T op denote the category of (compactly generated weak Hausdorff) topological spaces. Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.7 Fibrations are Serre ﬁbrations, that is maps p : X → Y with the right lifting property

n f I 6/ X h jn p for each n ≥ 0. In+1 Y , g /

n−1 n Coﬁbrations are maps (such as in : S → D for n ≥ 0) having the left lifting property with respect to all trivial Serre ﬁbrations.

Similar deﬁnitions can be made for T , the category of pointed topological spaces and basepoint preserving maps.

Model categories Two classical examples and stable homotopy theory

Let T op denote the category of (compactly generated weak Hausdorff) topological spaces. Weak equivalences are maps Mike Hill inducing isomorphisms of homotopy groups. Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.7 that is maps p : X → Y with the right lifting property

n f I 6/ X h jn p for each n ≥ 0. In+1 Y , g /

n−1 n Coﬁbrations are maps (such as in : S → D for n ≥ 0) having the left lifting property with respect to all trivial Serre ﬁbrations.

Similar deﬁnitions can be made for T , the category of pointed topological spaces and basepoint preserving maps.

Model categories Two classical examples and stable homotopy theory

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.7 n−1 n Coﬁbrations are maps (such as in : S → D for n ≥ 0) having the left lifting property with respect to all trivial Serre ﬁbrations.

Similar deﬁnitions can be made for T , the category of pointed topological spaces and basepoint preserving maps.

Model categories Two classical examples and stable homotopy theory

Let T op denote the category of (compactly generated weak Hausdorff) topological spaces. Weak equivalences are maps Mike Hill inducing isomorphisms of homotopy groups. Fibrations are Mike Hopkins Serre fibrations, that is maps p : X → Y with the right lifting Doug Ravenel property Introduction Quillen model categories n f I 6/ X Cofibrant generation h Bousfield localization jn p for each n ≥ 0. Enriched category In+1 Y , theory g / Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.7 Similar deﬁnitions can be made for T , the category of pointed topological spaces and basepoint preserving maps.

Model categories Two classical examples and stable homotopy theory

n−1 n Deﬁning the smash Coﬁbrations are maps (such as in : S → D for n ≥ 0) product of spectra

having the left lifting property with respect to all trivial Serre Generalizations ﬁbrations.

1.7 the category of pointed topological spaces and basepoint preserving maps.

Model categories Two classical examples and stable homotopy theory

n−1 n Deﬁning the smash Coﬁbrations are maps (such as in : S → D for n ≥ 0) product of spectra

having the left lifting property with respect to all trivial Serre Generalizations ﬁbrations.

Similar deﬁnitions can be made for T ,

1.7 Model categories Two classical examples and stable homotopy theory

n−1 n Deﬁning the smash Coﬁbrations are maps (such as in : S → D for n ≥ 0) product of spectra

having the left lifting property with respect to all trivial Serre Generalizations ﬁbrations.

Similar deﬁnitions can be made for T , the category of pointed topological spaces and basepoint preserving maps.

1.7 Definition An object X is cofibrant if the unique map ∅ → X is a cofibration. It X is fibrant if the unique map X → ∗ is a fibration.

All objects in T and T op are ﬁbrant. The coﬁbrant objects are the CW-complexes.

By MC5, for any object X, the unique maps ∅ → X and X → ∗ have factorizations

∅ → QX → X and X → RX → ∗ where QX is a coﬁbrant object weakly equivalent to X, and RX is a ﬁbrant object weakly equivalent to X.

Model categories Some deﬁnitions and stable homotopy theory

Recall that we denote the initial and terminal objects of M by and ∗. When they are the same, we say that M is pointed. ∅ Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.8 All objects in T and T op are ﬁbrant. The coﬁbrant objects are the CW-complexes.

By MC5, for any object X, the unique maps ∅ → X and X → ∗ have factorizations

∅ → QX → X and X → RX → ∗ where QX is a coﬁbrant object weakly equivalent to X, and RX is a ﬁbrant object weakly equivalent to X.

Model categories Some deﬁnitions and stable homotopy theory

Recall that we denote the initial and terminal objects of M by and ∗. When they are the same, we say that M is pointed. ∅ Mike Hill Mike Hopkins Definition Doug Ravenel An object X is cofibrant if the unique map ∅ → X is a Introduction Quillen model cofibration. It X is fibrant if the unique map X → ∗ is a fibration. categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.8 The coﬁbrant objects are the CW-complexes.

By MC5, for any object X, the unique maps ∅ → X and X → ∗ have factorizations

∅ → QX → X and X → RX → ∗ where QX is a coﬁbrant object weakly equivalent to X, and RX is a ﬁbrant object weakly equivalent to X.

Model categories Some deﬁnitions and stable homotopy theory

Coﬁbrant generation

All objects in T and T op are ﬁbrant. Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.8 By MC5, for any object X, the unique maps ∅ → X and X → ∗ have factorizations

∅ → QX → X and X → RX → ∗ where QX is a coﬁbrant object weakly equivalent to X, and RX is a ﬁbrant object weakly equivalent to X.

Model categories Some deﬁnitions and stable homotopy theory

Coﬁbrant generation

All objects in T and T op are fibrant. The cofibrant objects are Bousfield localization the CW-complexes. Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.8 where QX is a coﬁbrant object weakly equivalent to X, and RX is a ﬁbrant object weakly equivalent to X.

Model categories Some deﬁnitions and stable homotopy theory

Coﬁbrant generation

All objects in T and T op are fibrant. The cofibrant objects are Bousfield localization the CW-complexes. Enriched category theory

Spectra as enriched By MC5, for any object X, the unique maps ∅ → X and X → ∗ functors have factorizations Deﬁning the smash product of spectra

Generalizations ∅ → QX → X and X → RX → ∗

1.8 and RX is a ﬁbrant object weakly equivalent to X.

Model categories Some deﬁnitions and stable homotopy theory

Coﬁbrant generation

All objects in T and T op are fibrant. The cofibrant objects are Bousfield localization the CW-complexes. Enriched category theory

Spectra as enriched By MC5, for any object X, the unique maps ∅ → X and X → ∗ functors have factorizations Deﬁning the smash product of spectra

Generalizations ∅ → QX → X and X → RX → ∗ where QX is a coﬁbrant object weakly equivalent to X,

1.8 Model categories Some deﬁnitions and stable homotopy theory

Coﬁbrant generation

All objects in T and T op are fibrant. The cofibrant objects are Bousfield localization the CW-complexes. Enriched category theory

Spectra as enriched By MC5, for any object X, the unique maps ∅ → X and X → ∗ functors have factorizations Deﬁning the smash product of spectra

Generalizations ∅ → QX → X and X → RX → ∗ where QX is a coﬁbrant object weakly equivalent to X, and RX is a ﬁbrant object weakly equivalent to X.

1.8 These maps to and from X are called cofibrant and fibrant approximations. The objects QX and RX are called cofibrant and fibrant replacements of X.

Model categories Some deﬁnitions (continued) and stable homotopy theory

Mike Hill Mike Hopkins By MC5, for any object X, the unique maps ∅ → X and X → ∗ Doug Ravenel have factorizations Introduction

Quillen model ∅ → QX → X and X → RX → ∗ categories Cofibrant generation where QX is a cofibrant object weakly equivalent to X, Bousfield localization Enriched category and RX is a fibrant object weakly equivalent to X. theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.9 The objects QX and RX are called coﬁbrant and ﬁbrant replacements of X.

Model categories Some deﬁnitions (continued) and stable homotopy theory

Mike Hill Mike Hopkins By MC5, for any object X, the unique maps ∅ → X and X → ∗ Doug Ravenel have factorizations Introduction

Spectra as enriched These maps to and from X are called cofibrant and fibrant functors Defining the smash approximations. product of spectra

Generalizations

1.9 Model categories Some deﬁnitions (continued) and stable homotopy theory

Mike Hill Mike Hopkins By MC5, for any object X, the unique maps ∅ → X and X → ∗ Doug Ravenel have factorizations Introduction

Spectra as enriched These maps to and from X are called cofibrant and fibrant functors Defining the smash approximations. The objects QX and RX are called cofibrant product of spectra and fibrant replacements of X. Generalizations

1.9 It is known that every (trivial) coﬁbration in T op can be derived from the ones in (J ) I by iterating certain elementary constructions. A map is a (trivial) ﬁbration iff it has the right lifting property with respect to each map in (I) J . This condition is easier to verify than the previous one.

Similarly in T (the category of pointed spaces), let

n n−1 n o I+ = in+ : S+ → D+, n ≥ 0

and n n+1 J+ = jn+ : I+ → I+ , n ≥ 0 ,

where X+ denotes the space X with a disjoint base point.

Model categories Coﬁbrant generation and stable homotopy theory Example In T op, let

Mike Hill n−1 n n n+1 Mike Hopkins I = in : S → D , n ≥ 0 and J = jn : I → I , n ≥ 0 . Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.10 A map is a (trivial) ﬁbration iff it has the right lifting property with respect to each map in (I) J . This condition is easier to verify than the previous one.

Similarly in T (the category of pointed spaces), let

n n−1 n o I+ = in+ : S+ → D+, n ≥ 0

and n n+1 J+ = jn+ : I+ → I+ , n ≥ 0 ,

where X+ denotes the space X with a disjoint base point.

Model categories Coﬁbrant generation and stable homotopy theory Example In T op, let

Mike Hill n−1 n n n+1 Mike Hopkins I = in : S → D , n ≥ 0 and J = jn : I → I , n ≥ 0 . Doug Ravenel

It is known that every (trivial) cofibration in T op can be derived Introduction Quillen model from the ones in (J ) I by iterating certain elementary categories constructions. Cofibrant generation Bousfield localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.10 This condition is easier to verify than the previous one.

Similarly in T (the category of pointed spaces), let

n n−1 n o I+ = in+ : S+ → D+, n ≥ 0

and n n+1 J+ = jn+ : I+ → I+ , n ≥ 0 ,

where X+ denotes the space X with a disjoint base point.

Model categories Coﬁbrant generation and stable homotopy theory Example In T op, let

Mike Hill n−1 n n n+1 Mike Hopkins I = in : S → D , n ≥ 0 and J = jn : I → I , n ≥ 0 . Doug Ravenel

It is known that every (trivial) cofibration in T op can be derived Introduction Quillen model from the ones in (J ) I by iterating certain elementary categories constructions. A map is a (trivial) fibration iff it has the right Cofibrant generation lifting property with respect to each map in (I) J . Bousfield localization Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.10 Similarly in T (the category of pointed spaces), let

n n−1 n o I+ = in+ : S+ → D+, n ≥ 0

and n n+1 J+ = jn+ : I+ → I+ , n ≥ 0 ,

where X+ denotes the space X with a disjoint base point.

Model categories Coﬁbrant generation and stable homotopy theory Example In T op, let

Mike Hill n−1 n n n+1 Mike Hopkins I = in : S → D , n ≥ 0 and J = jn : I → I , n ≥ 0 . Doug Ravenel

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.10 n n−1 n o I+ = in+ : S+ → D+, n ≥ 0

and n n+1 J+ = jn+ : I+ → I+ , n ≥ 0 ,

where X+ denotes the space X with a disjoint base point.

Model categories Coﬁbrant generation and stable homotopy theory Example In T op, let

Mike Hill n−1 n n n+1 Mike Hopkins I = in : S → D , n ≥ 0 and J = jn : I → I , n ≥ 0 . Doug Ravenel

Spectra as enriched functors Similarly in T (the category of pointed spaces), let Deﬁning the smash product of spectra

Generalizations

1.10 and n n+1 J+ = jn+ : I+ → I+ , n ≥ 0 ,

where X+ denotes the space X with a disjoint base point.

Model categories Coﬁbrant generation and stable homotopy theory Example In T op, let

Mike Hill n−1 n n n+1 Mike Hopkins I = in : S → D , n ≥ 0 and J = jn : I → I , n ≥ 0 . Doug Ravenel

Spectra as enriched functors Similarly in T (the category of pointed spaces), let Deﬁning the smash product of spectra

n n−1 n o Generalizations I+ = in+ : S+ → D+, n ≥ 0

1.10 where X+ denotes the space X with a disjoint base point.

Model categories Coﬁbrant generation and stable homotopy theory Example In T op, let

Mike Hill n−1 n n n+1 Mike Hopkins I = in : S → D , n ≥ 0 and J = jn : I → I , n ≥ 0 . Doug Ravenel

Spectra as enriched functors Similarly in T (the category of pointed spaces), let Deﬁning the smash product of spectra

n n−1 n o Generalizations I+ = in+ : S+ → D+, n ≥ 0

and n n+1 J+ = jn+ : I+ → I+ , n ≥ 0 ,

1.10 Model categories Coﬁbrant generation and stable homotopy theory Example In T op, let

Mike Hill n−1 n n n+1 Mike Hopkins I = in : S → D , n ≥ 0 and J = jn : I → I , n ≥ 0 . Doug Ravenel

Spectra as enriched functors Similarly in T (the category of pointed spaces), let Deﬁning the smash product of spectra

n n−1 n o Generalizations I+ = in+ : S+ → D+, n ≥ 0

and n n+1 J+ = jn+ : I+ → I+ , n ≥ 0 ,

where X+ denotes the space X with a disjoint base point.

1.10 In practice, specifying the generating sets I and J , and deﬁning weak equivalences is the most convenient way to describe a model category.

The Kan Recognition Theorem gives four necessary and sufﬁcient conditions for morphism sets I and J to be generating sets as above, assuming that weak equivalences have already been deﬁned. They are too technical for this talk.

Model categories Coﬁbrant generation (continued) and stable homotopy theory

Definition Mike Hill Mike Hopkins A cofibrantly generated model category M is one with Doug Ravenel morphism sets I and J having similar properties to the ones in Introduction Quillen model T op. I (J ) is a generating set of (trivial) cofibrations. categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.11 The Kan Recognition Theorem gives four necessary and sufﬁcient conditions for morphism sets I and J to be generating sets as above, assuming that weak equivalences have already been deﬁned. They are too technical for this talk.

Model categories Coﬁbrant generation (continued) and stable homotopy theory

Coﬁbrant generation

In practice, specifying the generating sets I and J , and Bousﬁeld localization

deﬁning weak equivalences is the most convenient way to Enriched category describe a model category. theory Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.11 assuming that weak equivalences have already been deﬁned. They are too technical for this talk.

Model categories Coﬁbrant generation (continued) and stable homotopy theory

Coﬁbrant generation

In practice, specifying the generating sets I and J , and Bousﬁeld localization

deﬁning weak equivalences is the most convenient way to Enriched category describe a model category. theory Spectra as enriched functors The Kan Recognition Theorem gives four necessary and Deﬁning the smash product of spectra

sufﬁcient conditions for morphism sets I and J to be Generalizations generating sets as above,

1.11 They are too technical for this talk.

Model categories Coﬁbrant generation (continued) and stable homotopy theory

Coﬁbrant generation

In practice, specifying the generating sets I and J , and Bousﬁeld localization

sufﬁcient conditions for morphism sets I and J to be Generalizations generating sets as above, assuming that weak equivalences have already been deﬁned.

1.11 Model categories Coﬁbrant generation (continued) and stable homotopy theory

Coﬁbrant generation

In practice, specifying the generating sets I and J , and Bousﬁeld localization

sufﬁcient conditions for morphism sets I and J to be Generalizations generating sets as above, assuming that weak equivalences have already been deﬁned. They are too technical for this talk.

1.11 Around 1975 Pete Bousﬁeld had a brilliant idea.

Suppose we have a model category M, and we wish to change the model structure (without altering the underlying category) as follows. • Enlarge the class of weak equivalences in some way. • Keep the same class of cofibrations as before. • Define fibrations in terms of right lifting properties with respect to the newly defined trivial cofibrations. The class of trivial fibrations remains unaltered. Since there are more weak equivalences, there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

The hardest part of this is showing that the new classes of weak equivalences and ﬁbrations, along with the original class of coﬁbrations, satisfy the second Factorization Axiom MC5. The proof involves some delicate set theory.

Model categories Bousﬁeld localization and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.12 Suppose we have a model category M, and we wish to change the model structure (without altering the underlying category) as follows. • Enlarge the class of weak equivalences in some way. • Keep the same class of cofibrations as before. • Define fibrations in terms of right lifting properties with respect to the newly defined trivial cofibrations. The class of trivial fibrations remains unaltered. Since there are more weak equivalences, there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.12 (without altering the underlying category) as follows. • Enlarge the class of weak equivalences in some way. • Keep the same class of cofibrations as before. • Define fibrations in terms of right lifting properties with respect to the newly defined trivial cofibrations. The class of trivial fibrations remains unaltered. Since there are more weak equivalences, there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Mike Hill Suppose we have a model category M, and we wish to Mike Hopkins change the model structure Doug Ravenel Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.12 as follows. • Enlarge the class of weak equivalences in some way. • Keep the same class of cofibrations as before. • Define fibrations in terms of right lifting properties with respect to the newly defined trivial cofibrations. The class of trivial fibrations remains unaltered. Since there are more weak equivalences, there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Mike Hill Suppose we have a model category M, and we wish to Mike Hopkins change the model structure (without altering the underlying Doug Ravenel category) Introduction Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.12 • Enlarge the class of weak equivalences in some way. • Keep the same class of cofibrations as before. • Define fibrations in terms of right lifting properties with respect to the newly defined trivial cofibrations. The class of trivial fibrations remains unaltered. Since there are more weak equivalences, there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.12 • Keep the same class of cofibrations as before. • Define fibrations in terms of right lifting properties with respect to the newly defined trivial cofibrations. The class of trivial fibrations remains unaltered. Since there are more weak equivalences, there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.12 • Define fibrations in terms of right lifting properties with respect to the newly defined trivial cofibrations. The class of trivial fibrations remains unaltered. Since there are more weak equivalences, there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.12 The class of trivial fibrations remains unaltered. Since there are more weak equivalences, there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Mike Hill Suppose we have a model category M, and we wish to Mike Hopkins change the model structure (without altering the underlying Doug Ravenel category) as follows. Introduction Quillen model • Enlarge the class of weak equivalences in some way. categories • Keep the same class of cofibrations as before. Cofibrant generation • Define fibrations in terms of right lifting properties with Bousfield localization Enriched category respect to the newly defined trivial cofibrations. theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.12 Since there are more weak equivalences, there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Deﬁning the smash product of spectra

Generalizations

1.12 there are more trivial cofibrations. Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Since there are more weak equivalences, Deﬁning the smash product of spectra

Generalizations

1.12 Hence there are fewer fibrations and fewer fibrant objects. This could make the fibrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Since there are more weak equivalences, there are more trivial Deﬁning the smash coﬁbrations. product of spectra Generalizations

1.12 and fewer ﬁbrant objects. This could make the ﬁbrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Since there are more weak equivalences, there are more trivial Defining the smash cofibrations. Hence there are fewer fibrations product of spectra Generalizations

1.12 This could make the ﬁbrant replacement functor much more interesting.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Since there are more weak equivalences, there are more trivial Defining the smash cofibrations. Hence there are fewer fibrations and fewer fibrant product of spectra objects. Generalizations

1.12 The hardest part of this is showing that the new classes of weak equivalences and ﬁbrations, along with the original class of coﬁbrations, satisfy the second Factorization Axiom MC5. The proof involves some delicate set theory.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

Since there are more weak equivalences, there are more trivial Defining the smash cofibrations. Hence there are fewer fibrations and fewer fibrant product of spectra objects. This could make the fibrant replacement functor much Generalizations more interesting.

1.12 along with the original class of coﬁbrations, satisfy the second Factorization Axiom MC5. The proof involves some delicate set theory.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

The hardest part of this is showing that the new classes of weak equivalences and ﬁbrations,

1.12 satisfy the second Factorization Axiom MC5. The proof involves some delicate set theory.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

The hardest part of this is showing that the new classes of weak equivalences and ﬁbrations, along with the original class of coﬁbrations,

1.12 The proof involves some delicate set theory.

Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

The hardest part of this is showing that the new classes of weak equivalences and ﬁbrations, along with the original class of coﬁbrations, satisfy the second Factorization Axiom MC5.

1.12 Model categories Bousﬁeld localization and stable homotopy theory

Around 1975 Pete Bousﬁeld had a brilliant idea.

The hardest part of this is showing that the new classes of weak equivalences and ﬁbrations, along with the original class of coﬁbrations, satisfy the second Factorization Axiom MC5.

The proof involves some delicate set theory. 1.12 Let T op be the category of topological spaces with its usual model structure.

1 Choose an integer n > 0. Define a map f to be a weak equivalence if πk f is an isomorphism for k ≤ n. Then the fibrant objects are the spaces with no homotopy above dimension n. The fibrant replacement functor is the nth Postnikov section. It was originally constructed by attaching cells to kill all homotopy above dimension n. 2 Choose a prime p. Define a map to be a weak equivalence if it induces an isomorphism in mod p homology. On simply connected spaces, the fibrant replacement functor is p-adic completion.

3 Choose a generalized homology theory h∗. Define a map f to be a weak equivalence if h∗f is an isomorphism. The resulting fibrant replacement functor is Bousfield localization with respect to h∗. One can do the same with the category of spectra, once we have the right model structure on it.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.13 1 Choose an integer n > 0. Define a map f to be a weak equivalence if πk f is an isomorphism for k ≤ n. Then the fibrant objects are the spaces with no homotopy above dimension n. The fibrant replacement functor is the nth Postnikov section. It was originally constructed by attaching cells to kill all homotopy above dimension n. 2 Choose a prime p. Define a map to be a weak equivalence if it induces an isomorphism in mod p homology. On simply connected spaces, the fibrant replacement functor is p-adic completion.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.13 Define a map f to be a weak equivalence if πk f is an isomorphism for k ≤ n. Then the fibrant objects are the spaces with no homotopy above dimension n. The fibrant replacement functor is the nth Postnikov section. It was originally constructed by attaching cells to kill all homotopy above dimension n. 2 Choose a prime p. Define a map to be a weak equivalence if it induces an isomorphism in mod p homology. On simply connected spaces, the fibrant replacement functor is p-adic completion.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

1 Choose an integer n > 0. Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.13 Then the fibrant objects are the spaces with no homotopy above dimension n. The fibrant replacement functor is the nth Postnikov section. It was originally constructed by attaching cells to kill all homotopy above dimension n. 2 Choose a prime p. Define a map to be a weak equivalence if it induces an isomorphism in mod p homology. On simply connected spaces, the fibrant replacement functor is p-adic completion.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

1 Choose an integer n > 0. Deﬁne a map f to be a weak Mike Hill Mike Hopkins equivalence if πk f is an isomorphism for k ≤ n. Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.13 The fibrant replacement functor is the nth Postnikov section. It was originally constructed by attaching cells to kill all homotopy above dimension n. 2 Choose a prime p. Define a map to be a weak equivalence if it induces an isomorphism in mod p homology. On simply connected spaces, the fibrant replacement functor is p-adic completion.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

1 Choose an integer n > 0. Deﬁne a map f to be a weak Mike Hill Mike Hopkins equivalence if πk f is an isomorphism for k ≤ n. Then the Doug Ravenel ﬁbrant objects are the spaces with no homotopy above Introduction

dimension n. Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.13 It was originally constructed by attaching cells to kill all homotopy above dimension n. 2 Choose a prime p. Deﬁne a map to be a weak equivalence if it induces an isomorphism in mod p homology. On simply connected spaces, the ﬁbrant replacement functor is p-adic completion.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

dimension n. The ﬁbrant replacement functor is the nth Quillen model Postnikov section. categories Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.13 2 Choose a prime p. Deﬁne a map to be a weak equivalence if it induces an isomorphism in mod p homology. On simply connected spaces, the ﬁbrant replacement functor is p-adic completion.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

dimension n. The fibrant replacement functor is the nth Quillen model Postnikov section. It was originally constructed by categories attaching cells to kill all homotopy above dimension n. Cofibrant generation Bousfield localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.13 Deﬁne a map to be a weak equivalence if it induces an isomorphism in mod p homology. On simply connected spaces, the ﬁbrant replacement functor is p-adic completion.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

2 Choose a prime p. Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.13 On simply connected spaces, the ﬁbrant replacement functor is p-adic completion.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

2 Choose a prime p. Deﬁne a map to be a weak Enriched category equivalence if it induces an isomorphism in mod p theory Spectra as enriched homology. functors

Deﬁning the smash product of spectra

Generalizations

1.13 3 Choose a generalized homology theory h∗. Define a map f to be a weak equivalence if h∗f is an isomorphism. The resulting fibrant replacement functor is Bousfield localization with respect to h∗. One can do the same with the category of spectra, once we have the right model structure on it.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

2 Choose a prime p. Define a map to be a weak Enriched category equivalence if it induces an isomorphism in mod p theory Spectra as enriched homology. On simply connected spaces, the fibrant functors replacement functor is p-adic completion. Defining the smash product of spectra

Generalizations

1.13 Define a map f to be a weak equivalence if h∗f is an isomorphism. The resulting fibrant replacement functor is Bousfield localization with respect to h∗. One can do the same with the category of spectra, once we have the right model structure on it.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

3 Choose a generalized homology theory h∗. Generalizations

1.13 The resulting ﬁbrant replacement functor is Bousﬁeld localization with respect to h∗. One can do the same with the category of spectra, once we have the right model structure on it.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

3 Choose a generalized homology theory h∗. Deﬁne a map f Generalizations to be a weak equivalence if h∗f is an isomorphism.

1.13 One can do the same with the category of spectra, once we have the right model structure on it.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

1.13 once we have the right model structure on it.

Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

3 Choose a generalized homology theory h∗. Define a map f Generalizations to be a weak equivalence if h∗f is an isomorphism. The resulting fibrant replacement functor is Bousfield localization with respect to h∗. One can do the same with the category of spectra,

1.13 Model categories Three examples of Bousﬁeld localization and stable homotopy theory Let T op be the category of topological spaces with its usual model structure.

1.13 A symmetric monoidal structure on a category V0 is a functor

⊗ V0 × V0 −→V0

sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is required to have suitable properties including • a natural isomorphism t : X ⊗ Y → Y ⊗ X and • a unit object 1 such that 1 ⊗ X is naturally isomorphic to X.

We denote this by V = (V0, ⊗, 1).

Familiar examples include (Set, ×, ∗), (T op, ×, ∗), (T , ∧, S0), where T is the category of pointed topological spaces, and (Set∆, ×, ∗), where Set∆ is the category of simplicial sets.

Model categories Enriched category theory and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.14 sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is required to have suitable properties including • a natural isomorphism t : X ⊗ Y → Y ⊗ X and • a unit object 1 such that 1 ⊗ X is naturally isomorphic to X.

We denote this by V = (V0, ⊗, 1).

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.14 • a natural isomorphism t : X ⊗ Y → Y ⊗ X and • a unit object 1 such that 1 ⊗ X is naturally isomorphic to X.

We denote this by V = (V0, ⊗, 1).

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

required to have suitable properties including Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.14 • a unit object 1 such that 1 ⊗ X is naturally isomorphic to X.

We denote this by V = (V0, ⊗, 1).

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

required to have suitable properties including Coﬁbrant generation • a natural isomorphism t : X ⊗ Y → Y ⊗ X and Bousﬁeld localization Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.14 We denote this by V = (V0, ⊗, 1).

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

Deﬁning the smash product of spectra

Generalizations

1.14 Familiar examples include (Set, ×, ∗), (T op, ×, ∗), (T , ∧, S0), where T is the category of pointed topological spaces, and (Set∆, ×, ∗), where Set∆ is the category of simplicial sets.

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

Generalizations

1.14 (T op, ×, ∗), (T , ∧, S0), where T is the category of pointed topological spaces, and (Set∆, ×, ∗), where Set∆ is the category of simplicial sets.

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

required to have suitable properties including Cofibrant generation • a natural isomorphism t : X ⊗ Y → Y ⊗ X and Bousfield localization Enriched category • a unit object 1 such that 1 ⊗ X is naturally isomorphic to X. theory Spectra as enriched functors We denote this by V = (V0, ⊗, 1). Defining the smash product of spectra Familiar examples include (Set, ×, ∗), Generalizations

1.14 (T , ∧, S0), where T is the category of pointed topological spaces, and (Set∆, ×, ∗), where Set∆ is the category of simplicial sets.

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

1.14 where T is the category of pointed topological spaces, and (Set∆, ×, ∗), where Set∆ is the category of simplicial sets.

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

required to have suitable properties including Cofibrant generation • a natural isomorphism t : X ⊗ Y → Y ⊗ X and Bousfield localization Enriched category • a unit object 1 such that 1 ⊗ X is naturally isomorphic to X. theory Spectra as enriched functors We denote this by V = (V0, ⊗, 1). Defining the smash product of spectra 0 Familiar examples include (Set, ×, ∗), (T op, ×, ∗), (T , ∧, S ), Generalizations

1.14 and (Set∆, ×, ∗), where Set∆ is the category of simplicial sets.

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

required to have suitable properties including Cofibrant generation • a natural isomorphism t : X ⊗ Y → Y ⊗ X and Bousfield localization Enriched category • a unit object 1 such that 1 ⊗ X is naturally isomorphic to X. theory Spectra as enriched functors We denote this by V = (V0, ⊗, 1). Defining the smash product of spectra 0 Familiar examples include (Set, ×, ∗), (T op, ×, ∗), (T , ∧, S ), Generalizations where T is the category of pointed topological spaces,

1.14 where Set∆ is the category of simplicial sets.

Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

required to have suitable properties including Cofibrant generation • a natural isomorphism t : X ⊗ Y → Y ⊗ X and Bousfield localization Enriched category • a unit object 1 such that 1 ⊗ X is naturally isomorphic to X. theory Spectra as enriched functors We denote this by V = (V0, ⊗, 1). Defining the smash product of spectra 0 Familiar examples include (Set, ×, ∗), (T op, ×, ∗), (T , ∧, S ), Generalizations where T is the category of pointed topological spaces, and (Set∆, ×, ∗),

1.14 Model categories Enriched category theory and stable homotopy theory

A symmetric monoidal structure on a category V0 is a functor Mike Hill Mike Hopkins ⊗ Doug Ravenel V0 × V0 −→V0 Introduction

Quillen model sending a pair of objects (X, Y ) to a third object X ⊗ Y . It is categories

required to have suitable properties including Cofibrant generation • a natural isomorphism t : X ⊗ Y → Y ⊗ X and Bousfield localization Enriched category • a unit object 1 such that 1 ⊗ X is naturally isomorphic to X. theory Spectra as enriched functors We denote this by V = (V0, ⊗, 1). Defining the smash product of spectra 0 Familiar examples include (Set, ×, ∗), (T op, ×, ∗), (T , ∧, S ), Generalizations where T is the category of pointed topological spaces, and (Set∆, ×, ∗), where Set∆ is the category of simplicial sets.

1.14 Let V = (V0, ⊗, 1) be a symmetric monoidal category as above. Deﬁnition A V-category (or a category enriched over V) C consists of • a collection of objects, • for each pair of objects (X, Y ), a morphism object C(X, Y ) in V0 (instead of a set of morphisms X → Y ), • for each triple of objects (X, Y , Z ), a composition morphism in V0

cX,Y ,Z : C(Y , Z) ⊗ C(X, Y ) → C(X, Z)

(replacing the usual composition) and

• for each object X, an identity morphism in V0 1 → C(X, X), replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

Model categories Enriched category theory (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.15 (or a category enriched over V) C consists of • a collection of objects, • for each pair of objects (X, Y ), a morphism object C(X, Y ) in V0 (instead of a set of morphisms X → Y ), • for each triple of objects (X, Y , Z ), a composition morphism in V0

cX,Y ,Z : C(Y , Z) ⊗ C(X, Y ) → C(X, Z)

(replacing the usual composition) and

• for each object X, an identity morphism in V0 1 → C(X, X), replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

Model categories Enriched category theory (continued) and stable homotopy theory Let V = (V0, ⊗, 1) be a symmetric monoidal category as above. Deﬁnition A V-category Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.15 consists of • a collection of objects, • for each pair of objects (X, Y ), a morphism object C(X, Y ) in V0 (instead of a set of morphisms X → Y ), • for each triple of objects (X, Y , Z ), a composition morphism in V0

cX,Y ,Z : C(Y , Z) ⊗ C(X, Y ) → C(X, Z)

(replacing the usual composition) and

• for each object X, an identity morphism in V0 1 → C(X, X), replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

Model categories Enriched category theory (continued) and stable homotopy theory Let V = (V0, ⊗, 1) be a symmetric monoidal category as above. Deﬁnition A V-category (or a category enriched over V) C Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.15 • a collection of objects, • for each pair of objects (X, Y ), a morphism object C(X, Y ) in V0 (instead of a set of morphisms X → Y ), • for each triple of objects (X, Y , Z ), a composition morphism in V0

cX,Y ,Z : C(Y , Z) ⊗ C(X, Y ) → C(X, Z)

(replacing the usual composition) and

• for each object X, an identity morphism in V0 1 → C(X, X), replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.15 • for each pair of objects (X, Y ), a morphism object C(X, Y ) in V0 (instead of a set of morphisms X → Y ), • for each triple of objects (X, Y , Z ), a composition morphism in V0

cX,Y ,Z : C(Y , Z) ⊗ C(X, Y ) → C(X, Z)

(replacing the usual composition) and

• for each object X, an identity morphism in V0 1 → C(X, X), replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.15 (instead of a set of morphisms X → Y ), • for each triple of objects (X, Y , Z ), a composition morphism in V0

cX,Y ,Z : C(Y , Z) ⊗ C(X, Y ) → C(X, Z)

(replacing the usual composition) and

• for each object X, an identity morphism in V0 1 → C(X, X), replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

• for each pair of objects (X, Y ), a morphism object C(X, Y ) Introduction in V0 Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.15 • for each triple of objects (X, Y , Z ), a composition morphism in V0

cX,Y ,Z : C(Y , Z) ⊗ C(X, Y ) → C(X, Z)

(replacing the usual composition) and

• for each object X, an identity morphism in V0 1 → C(X, X), replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

• for each pair of objects (X, Y ), a morphism object C(X, Y ) Introduction in V0 (instead of a set of morphisms X → Y ), Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.15 (replacing the usual composition) and

• for each object X, an identity morphism in V0 1 → C(X, X), replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

• for each pair of objects (X, Y ), a morphism object C(X, Y ) Introduction in V0 (instead of a set of morphisms X → Y ), Quillen model categories

Deﬁning the smash product of spectra

Generalizations

1.15 and

• for each object X, an identity morphism in V0 1 → C(X, X), replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

• for each pair of objects (X, Y ), a morphism object C(X, Y ) Introduction in V0 (instead of a set of morphisms X → Y ), Quillen model categories

Generalizations

1.15 replacing the usual identity morphism X → X.

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

• for each pair of objects (X, Y ), a morphism object C(X, Y ) Introduction in V0 (instead of a set of morphisms X → Y ), Quillen model categories

• for each triple of objects (X, Y , Z ), a composition Cofibrant generation morphism in V0 Bousfield localization Enriched category theory cX,Y ,Z : C(Y , Z) ⊗ C(X, Y ) → C(X, Z) Spectra as enriched functors (replacing the usual composition) and Defining the smash product of spectra • for each object X, an identity morphism in V0 1 → C(X, X), Generalizations

1.15 There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

• for each pair of objects (X, Y ), a morphism object C(X, Y ) Introduction in V0 (instead of a set of morphisms X → Y ), Quillen model categories

1.15 and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

• for each pair of objects (X, Y ), a morphism object C(X, Y ) Introduction in V0 (instead of a set of morphisms X → Y ), Quillen model categories

There is an underlying ordinary category C0 with the same objects as C

1.15 Model categories Enriched category theory (continued) and stable homotopy theory Let V = (V0, ⊗, 1) be a symmetric monoidal category as above. Deﬁnition A V-category (or a category enriched over V) C consists of Mike Hill Mike Hopkins • a collection of objects, Doug Ravenel

• for each pair of objects (X, Y ), a morphism object C(X, Y ) Introduction in V0 (instead of a set of morphisms X → Y ), Quillen model categories

There is an underlying ordinary category C0 with the same objects as C and morphism sets

C0(X, Y ) = V0(1, C(X, Y )).

1.15 One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural transformations) between them.

In this language, an ordinary category is enriched over Set.

A topological category is one that is enriched over T op.

A simplicial category is one that is enriched over Set∆, the category of simplicial sets.

A symmetric monoidal category V0 is closed if it enriched over itself. This means that for each pair of objects (X, Y ) there is an internal Hom object V(X, Y ) with natural isomorphisms ∼ V0(X ⊗ Y , Z) = V0(X, V(Y , Z)).

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.16 and enriched natural transformations (V-natural transformations) between them.

In this language, an ordinary category is enriched over Set.

A topological category is one that is enriched over T op.

A simplicial category is one that is enriched over Set∆, the category of simplicial sets.

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.16 In this language, an ordinary category is enriched over Set.

A topological category is one that is enriched over T op.

A simplicial category is one that is enriched over Set∆, the category of simplicial sets.

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural

transformations) between them. Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.16 A topological category is one that is enriched over T op.

A simplicial category is one that is enriched over Set∆, the category of simplicial sets.

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural

transformations) between them. Mike Hill Mike Hopkins Doug Ravenel In this language, an ordinary category is enriched over Set. Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.16 A simplicial category is one that is enriched over Set∆, the category of simplicial sets.

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural

transformations) between them. Mike Hill Mike Hopkins Doug Ravenel In this language, an ordinary category is enriched over Set. Introduction

Quillen model A topological category is one that is enriched over T op. categories Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.16 A symmetric monoidal category V0 is closed if it enriched over itself. This means that for each pair of objects (X, Y ) there is an internal Hom object V(X, Y ) with natural isomorphisms ∼ V0(X ⊗ Y , Z) = V0(X, V(Y , Z)).

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural

transformations) between them. Mike Hill Mike Hopkins Doug Ravenel In this language, an ordinary category is enriched over Set. Introduction

Quillen model A topological category is one that is enriched over T op. categories Coﬁbrant generation

Bousﬁeld localization A simplicial category is one that is enriched over Set∆, the Enriched category category of simplicial sets. theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.16 This means that for each pair of objects (X, Y ) there is an internal Hom object V(X, Y ) with natural isomorphisms ∼ V0(X ⊗ Y , Z) = V0(X, V(Y , Z)).

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural

transformations) between them. Mike Hill Mike Hopkins Doug Ravenel In this language, an ordinary category is enriched over Set. Introduction

Quillen model A topological category is one that is enriched over T op. categories Coﬁbrant generation

Bousﬁeld localization A simplicial category is one that is enriched over Set∆, the Enriched category category of simplicial sets. theory

Spectra as enriched A symmetric monoidal category V is closed if it enriched over functors 0 Deﬁning the smash itself. product of spectra Generalizations

1.16 there is an internal Hom object V(X, Y ) with natural isomorphisms ∼ V0(X ⊗ Y , Z) = V0(X, V(Y , Z)).

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural

transformations) between them. Mike Hill Mike Hopkins Doug Ravenel In this language, an ordinary category is enriched over Set. Introduction

Quillen model A topological category is one that is enriched over T op. categories Coﬁbrant generation

Bousﬁeld localization A simplicial category is one that is enriched over Set∆, the Enriched category category of simplicial sets. theory

Spectra as enriched A symmetric monoidal category V is closed if it enriched over functors 0 Deﬁning the smash itself. This means that for each pair of objects (X, Y ) product of spectra Generalizations

1.16 with natural isomorphisms ∼ V0(X ⊗ Y , Z) = V0(X, V(Y , Z)).

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural

transformations) between them. Mike Hill Mike Hopkins Doug Ravenel In this language, an ordinary category is enriched over Set. Introduction

Quillen model A topological category is one that is enriched over T op. categories Coﬁbrant generation

Bousﬁeld localization A simplicial category is one that is enriched over Set∆, the Enriched category category of simplicial sets. theory

Spectra as enriched A symmetric monoidal category V is closed if it enriched over functors 0 Deﬁning the smash itself. This means that for each pair of objects (X, Y ) there is product of spectra an internal Hom object V(X, Y ) Generalizations

1.16 The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural

transformations) between them. Mike Hill Mike Hopkins Doug Ravenel In this language, an ordinary category is enriched over Set. Introduction

Quillen model A topological category is one that is enriched over T op. categories Coﬁbrant generation

Bousﬁeld localization A simplicial category is one that is enriched over Set∆, the Enriched category category of simplicial sets. theory

1.16 Model categories Enriched category theory (continued) and stable homotopy theory One can deﬁne enriched functors (V-functors) between V-categories and enriched natural transformations (V-natural

transformations) between them. Mike Hill Mike Hopkins Doug Ravenel In this language, an ordinary category is enriched over Set. Introduction

Quillen model A topological category is one that is enriched over T op. categories Coﬁbrant generation

Bousﬁeld localization A simplicial category is one that is enriched over Set∆, the Enriched category category of simplicial sets. theory

The symmetric monoidal categories Set, T op, T and Set∆ are each closed.

1.16 with structure maps ΣXn → Xn+1. We will redeﬁne it to be an enriched T -valued functor on a small T -category J N. This will make the structure maps built in to the functor. Maps between spectra will be enriched natural transformations. Deﬁnition The indexing category J N has natural numbers n ≥ 0 as objects with

Sn−m for n ≥ m J N(m, n) = ∗ otherwise.

For m ≤ n ≤ p, the composition morphism

p−n n−m p−m jm,n,p : S ∧ S → S

is the standard homeomorphism.

Model categories Spectra as enriched functors and stable homotopy theory Recall that a spectrum X was originally deﬁned to be a sequence of pointed spaces {Xn}

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.17 We will redeﬁne it to be an enriched T -valued functor on a small T -category J N. This will make the structure maps built in to the functor. Maps between spectra will be enriched natural transformations. Deﬁnition The indexing category J N has natural numbers n ≥ 0 as objects with

Sn−m for n ≥ m J N(m, n) = ∗ otherwise.

For m ≤ n ≤ p, the composition morphism

p−n n−m p−m jm,n,p : S ∧ S → S

is the standard homeomorphism.

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.17 This will make the structure maps built in to the functor. Maps between spectra will be enriched natural transformations. Deﬁnition The indexing category J N has natural numbers n ≥ 0 as objects with

Sn−m for n ≥ m J N(m, n) = ∗ otherwise.

For m ≤ n ≤ p, the composition morphism

p−n n−m p−m jm,n,p : S ∧ S → S

is the standard homeomorphism.

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.17 Maps between spectra will be enriched natural transformations. Deﬁnition The indexing category J N has natural numbers n ≥ 0 as objects with

Sn−m for n ≥ m J N(m, n) = ∗ otherwise.

For m ≤ n ≤ p, the composition morphism

p−n n−m p−m jm,n,p : S ∧ S → S

is the standard homeomorphism.

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.17 Deﬁnition The indexing category J N has natural numbers n ≥ 0 as objects with

Sn−m for n ≥ m J N(m, n) = ∗ otherwise.

For m ≤ n ≤ p, the composition morphism

p−n n−m p−m jm,n,p : S ∧ S → S

is the standard homeomorphism.

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.17 For m ≤ n ≤ p, the composition morphism

p−n n−m p−m jm,n,p : S ∧ S → S

is the standard homeomorphism.

Model categories Spectra as enriched functors and stable homotopy theory Recall that a spectrum X was originally defined to be a sequence of pointed spaces {Xn} with structure maps ΣX → X . We will redefine it to be an enriched T -valued n n+1 Mike Hill functor on a small T -category N. This will make the structure Mike Hopkins J Doug Ravenel maps built in to the functor. Maps between spectra will be enriched natural transformations. Introduction Quillen model categories Definition Cofibrant generation N The indexing category J has natural numbers n ≥ 0 as Bousfield localization objects with Enriched category theory n−m Spectra as enriched S for n ≥ m functors J N(m, n) = ∗ otherwise. Defining the smash product of spectra

Generalizations

1.17 Model categories Spectra as enriched functors and stable homotopy theory Recall that a spectrum X was originally defined to be a sequence of pointed spaces {Xn} with structure maps ΣX → X . We will redefine it to be an enriched T -valued n n+1 Mike Hill functor on a small T -category N. This will make the structure Mike Hopkins J Doug Ravenel maps built in to the functor. Maps between spectra will be enriched natural transformations. Introduction Quillen model categories Definition Cofibrant generation N The indexing category J has natural numbers n ≥ 0 as Bousfield localization objects with Enriched category theory n−m Spectra as enriched S for n ≥ m functors J N(m, n) = ∗ otherwise. Defining the smash product of spectra

Generalizations For m ≤ n ≤ p, the composition morphism

p−n n−m p−m jm,n,p : S ∧ S → S

is the standard homeomorphism.

1.17 We denote its value at n by Xn. Functoriality means that for each m, n ≥ 0 there is a continuous structure map X N m,n : J (m, n) ∧ Xm → Xn. Since Sn−m for n ≥ m J N(m, n) = ∗ otherwise,

n−m for m ≤ n we get the expected map Σ Xm → Xn.

Deﬁnition m For m ≥ 0, the Yoneda spectrum H = S−m is given by

n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

Model categories Spectra as enriched functors (continued) and stable homotopy theory We can deﬁne a spectrum X to be an enriched functor X : J N → T .

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.18 Functoriality means that for each m, n ≥ 0 there is a continuous structure map X N m,n : J (m, n) ∧ Xm → Xn. Since Sn−m for n ≥ m J N(m, n) = ∗ otherwise,

n−m for m ≤ n we get the expected map Σ Xm → Xn.

Deﬁnition m For m ≥ 0, the Yoneda spectrum H = S−m is given by

n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

Model categories Spectra as enriched functors (continued) and stable homotopy theory We can deﬁne a spectrum X to be an enriched functor N X : J → T . We denote its value at n by Xn.

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.18 there is a continuous structure map X N m,n : J (m, n) ∧ Xm → Xn. Since Sn−m for n ≥ m J N(m, n) = ∗ otherwise,

n−m for m ≤ n we get the expected map Σ Xm → Xn.

Deﬁnition m For m ≥ 0, the Yoneda spectrum H = S−m is given by

n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.18 Since Sn−m for n ≥ m J N(m, n) = ∗ otherwise,

n−m for m ≤ n we get the expected map Σ Xm → Xn.

Deﬁnition m For m ≥ 0, the Yoneda spectrum H = S−m is given by

n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.18 n−m for m ≤ n we get the expected map Σ Xm → Xn.

Deﬁnition m For m ≥ 0, the Yoneda spectrum H = S−m is given by

n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.18 Deﬁnition m For m ≥ 0, the Yoneda spectrum H = S−m is given by

n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

Model categories Spectra as enriched functors (continued) and stable homotopy theory We can define a spectrum X to be an enriched functor N X : J → T . We denote its value at n by Xn. Functoriality means that for each m, n ≥ 0 there is a continuous structure Mike Hill map Mike Hopkins X N Doug Ravenel m,n : J (m, n) ∧ Xm → Xn. Introduction Since n−m Quillen model S for n ≥ m categories J N(m, n) = ∗ otherwise, Cofibrant generation Bousfield localization n−m for m ≤ n we get the expected map Σ Xm → Xn. Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.18 m the Yoneda spectrum H = S−m is given by

n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

Deﬁning the smash For m ≥ 0, product of spectra

Generalizations

1.18 n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

Generalizations

1.18 In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

Generalizations n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

1.18 and S−m is its formal mth desuspension.

Generalizations n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum,

1.18 Model categories Spectra as enriched functors (continued) and stable homotopy theory We can define a spectrum X to be an enriched functor N X : J → T . We denote its value at n by Xn. Functoriality means that for each m, n ≥ 0 there is a continuous structure Mike Hill map Mike Hopkins X N Doug Ravenel m,n : J (m, n) ∧ Xm → Xn. Introduction Since n−m Quillen model S for n ≥ m categories J N(m, n) = ∗ otherwise, Cofibrant generation Bousfield localization n−m for m ≤ n we get the expected map Σ Xm → Xn. Enriched category theory Definition Spectra as enriched functors m −m Defining the smash For m ≥ 0, the Yoneda spectrum H = S is given by product of spectra

Generalizations n−m −m N S for n ≥ m (S )n = J (m, n) = ∗ otherwise.

In particular, S−0 is the sphere spectrum, and S−m is its formal mth desuspension.

1.18 Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 T , namely the Yoneda functor H given by

n 7→ J N(0, n) = Sn

T is symmetric monoidal, and there is a twist isomorphism

t : Sm ∧ Sn → Sn ∧ Sm.

0 However this morphism is not in the image of the functor H . There is no twist isomorphism in J N, so its monoidal structure is not symmetric.

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.19 It admits an embedding functor into 0 T , namely the Yoneda functor H given by

n 7→ J N(0, n) = Sn

T is symmetric monoidal, and there is a twist isomorphism

t : Sm ∧ Sn → Sn ∧ Sm.

0 However this morphism is not in the image of the functor H . There is no twist isomorphism in J N, so its monoidal structure is not symmetric.

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.19 0 namely the Yoneda functor H given by

n 7→ J N(0, n) = Sn

T is symmetric monoidal, and there is a twist isomorphism

t : Sm ∧ Sn → Sn ∧ Sm.

0 However this morphism is not in the image of the functor H . There is no twist isomorphism in J N, so its monoidal structure is not symmetric.

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into Mike Hill Mike Hopkins T , Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.19 n 7→ J N(0, n) = Sn

T is symmetric monoidal, and there is a twist isomorphism

t : Sm ∧ Sn → Sn ∧ Sm.

0 However this morphism is not in the image of the functor H . There is no twist isomorphism in J N, so its monoidal structure is not symmetric.

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.19 T is symmetric monoidal, and there is a twist isomorphism

t : Sm ∧ Sn → Sn ∧ Sm.

0 However this morphism is not in the image of the functor H . There is no twist isomorphism in J N, so its monoidal structure is not symmetric.

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

n 7→ J N(0, n) = Sn Introduction Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.19 and there is a twist isomorphism

t : Sm ∧ Sn → Sn ∧ Sm.

0 However this morphism is not in the image of the functor H . There is no twist isomorphism in J N, so its monoidal structure is not symmetric.

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

n 7→ J N(0, n) = Sn Introduction Quillen model categories

T is symmetric monoidal, Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.19 0 However this morphism is not in the image of the functor H . There is no twist isomorphism in J N, so its monoidal structure is not symmetric.

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

n 7→ J N(0, n) = Sn Introduction Quillen model categories

T is symmetric monoidal, and there is a twist isomorphism Coﬁbrant generation

Bousﬁeld localization m n n m t : S ∧ S → S ∧ S . Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.19 There is no twist isomorphism in J N, so its monoidal structure is not symmetric.

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

n 7→ J N(0, n) = Sn Introduction Quillen model categories

T is symmetric monoidal, and there is a twist isomorphism Coﬁbrant generation

Bousﬁeld localization m n n m t : S ∧ S → S ∧ S . Enriched category theory

0 Spectra as enriched However this morphism is not in the image of the functor H . functors Deﬁning the smash product of spectra

Generalizations

1.19 so its monoidal structure is not symmetric.

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

n 7→ J N(0, n) = Sn Introduction Quillen model categories

T is symmetric monoidal, and there is a twist isomorphism Coﬁbrant generation

Bousﬁeld localization m n n m t : S ∧ S → S ∧ S . Enriched category theory

0 Spectra as enriched However this morphism is not in the image of the functor H . functors N Deﬁning the smash There is no twist isomorphism in J , product of spectra

Generalizations

1.19 This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

n 7→ J N(0, n) = Sn Introduction Quillen model categories

T is symmetric monoidal, and there is a twist isomorphism Coﬁbrant generation

Bousﬁeld localization m n n m t : S ∧ S → S ∧ S . Enriched category theory

0 Spectra as enriched However this morphism is not in the image of the functor H . functors N Deﬁning the smash There is no twist isomorphism in J , so its monoidal structure product of spectra is not symmetric. Generalizations

1.19 does not have a convenient smash product. This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

n 7→ J N(0, n) = Sn Introduction Quillen model categories

T is symmetric monoidal, and there is a twist isomorphism Coﬁbrant generation

Bousﬁeld localization m n n m t : S ∧ S → S ∧ S . Enriched category theory

This is the reason that the category of spectra Sp deﬁned in this way

1.19 This was a headache in the subject for decades!

Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

n 7→ J N(0, n) = Sn Introduction Quillen model categories

T is symmetric monoidal, and there is a twist isomorphism Coﬁbrant generation

Bousﬁeld localization m n n m t : S ∧ S → S ∧ S . Enriched category theory

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product.

1.19 Model categories Spectra as enriched functors (continued) and stable homotopy theory

Warning The catgeory J N is monoidal (under addition) but not symmetric monoidal. It admits an embedding functor into 0 Mike Hill Mike Hopkins T , namely the Yoneda functor H given by Doug Ravenel

n 7→ J N(0, n) = Sn Introduction Quillen model categories

T is symmetric monoidal, and there is a twist isomorphism Coﬁbrant generation

Bousﬁeld localization m n n m t : S ∧ S → S ∧ S . Enriched category theory

This is the reason that the category of spectra Sp deﬁned in this way does not have a convenient smash product. This was a headache in the subject for decades!

1.19 To repeat, spectra as originally deﬁned do not have a convenient smash product. A way around this was ﬁrst discovered in 1993.

Tony Igor Mike Peter Elmendorf Kriz Mandell May

An easier way was found a few years later.

Mark Brooke Jeff Hovey Shipley Smith

Model categories Deﬁning the smash product of spectra and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.20 A way around this was ﬁrst discovered in 1993.

Tony Igor Mike Peter Elmendorf Kriz Mandell May

An easier way was found a few years later.

Mark Brooke Jeff Hovey Shipley Smith

Model categories Deﬁning the smash product of spectra and stable homotopy To repeat, spectra as originally deﬁned do not have a theory convenient smash product.

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.20 Tony Igor Mike Peter Elmendorf Kriz Mandell May

An easier way was found a few years later.

Mark Brooke Jeff Hovey Shipley Smith

Model categories Defining the smash product of spectra and stable homotopy To repeat, spectra as originally defined do not have a theory convenient smash product. A way around this was first discovered in 1993.

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.20 An easier way was found a few years later.

Mark Brooke Jeff Hovey Shipley Smith

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories Tony Igor Mike Peter Coﬁbrant generation Elmendorf Kriz Mandell May Bousﬁeld localization Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.20 Mark Brooke Jeff Hovey Shipley Smith

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories Tony Igor Mike Peter Coﬁbrant generation Elmendorf Kriz Mandell May Bousﬁeld localization Enriched category theory An easier way was found a few years later. Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.20 Model categories Defining the smash product of spectra and stable homotopy To repeat, spectra as originally defined do not have a theory convenient smash product. A way around this was first discovered in 1993.

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Deﬁning the smash product of spectra

Generalizations

Mark Brooke Jeff Hovey Shipley Smith

1.20 Its objects are the natural numbers as before, but it has bigger morphism objects.

For m ≤ n we have

Σ n−m J (m, n) := Σn+ ∧ S . Σn−m

This is the wedge of copies of Sn−m indexed by inclusions

[m] ,→ [n]

of the set of m elements into that of n elements. The symmetric n−m 1 ∧(n−m) group Σn−m acts on S = (S ) by permuting its smash factors.

This means we have the symmetry morphism

Sm ∧ Sn → Sn ∧ Sm

that we were missing before, so J Σ is symmetric monoidal.

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy The Hovey-Shipley-Smith approach was to enlarge the theory indexing category J N to make it into a symmetric monoidal category J Σ.

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.21 but it has bigger morphism objects.

For m ≤ n we have

Σ n−m J (m, n) := Σn+ ∧ S . Σn−m

This is the wedge of copies of Sn−m indexed by inclusions

[m] ,→ [n]

of the set of m elements into that of n elements. The symmetric n−m 1 ∧(n−m) group Σn−m acts on S = (S ) by permuting its smash factors.

This means we have the symmetry morphism

Sm ∧ Sn → Sn ∧ Sm

that we were missing before, so J Σ is symmetric monoidal.

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.21 For m ≤ n we have

Σ n−m J (m, n) := Σn+ ∧ S . Σn−m

This is the wedge of copies of Sn−m indexed by inclusions

[m] ,→ [n]

of the set of m elements into that of n elements. The symmetric n−m 1 ∧(n−m) group Σn−m acts on S = (S ) by permuting its smash factors.

This means we have the symmetry morphism

Sm ∧ Sn → Sn ∧ Sm

that we were missing before, so J Σ is symmetric monoidal.

but it has bigger morphism objects. Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.21 This is the wedge of copies of Sn−m indexed by inclusions

[m] ,→ [n]

of the set of m elements into that of n elements. The symmetric n−m 1 ∧(n−m) group Σn−m acts on S = (S ) by permuting its smash factors.

This means we have the symmetry morphism

Sm ∧ Sn → Sn ∧ Sm

that we were missing before, so J Σ is symmetric monoidal.

but it has bigger morphism objects. Mike Hill Mike Hopkins Doug Ravenel For m ≤ n we have Introduction

Σ n−m Quillen model J (m, n) := Σn+ ∧ S . categories Σn−m Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.21 The symmetric n−m 1 ∧(n−m) group Σn−m acts on S = (S ) by permuting its smash factors.

This means we have the symmetry morphism

Sm ∧ Sn → Sn ∧ Sm

that we were missing before, so J Σ is symmetric monoidal.

but it has bigger morphism objects. Mike Hill Mike Hopkins Doug Ravenel For m ≤ n we have Introduction

Σ n−m Quillen model J (m, n) := Σn+ ∧ S . categories Σn−m Coﬁbrant generation This is the wedge of copies of Sn−m indexed by inclusions Bousﬁeld localization Enriched category theory

[m] ,→ [n] Spectra as enriched functors of the set of m elements into that of n elements. Deﬁning the smash product of spectra

Generalizations

1.21 This means we have the symmetry morphism

Sm ∧ Sn → Sn ∧ Sm

that we were missing before, so J Σ is symmetric monoidal.

but it has bigger morphism objects. Mike Hill Mike Hopkins Doug Ravenel For m ≤ n we have Introduction

Σ n−m Quillen model J (m, n) := Σn+ ∧ S . categories Σn−m Coﬁbrant generation This is the wedge of copies of Sn−m indexed by inclusions Bousﬁeld localization Enriched category theory

[m] ,→ [n] Spectra as enriched functors of the set of m elements into that of n elements. The symmetric Deﬁning the smash product of spectra group Σ acts on Sn−m = (S1)∧(n−m) by permuting its n−m Generalizations smash factors.

1.21 Sm ∧ Sn → Sn ∧ Sm

that we were missing before, so J Σ is symmetric monoidal.

but it has bigger morphism objects. Mike Hill Mike Hopkins Doug Ravenel For m ≤ n we have Introduction

Σ n−m Quillen model J (m, n) := Σn+ ∧ S . categories Σn−m Coﬁbrant generation This is the wedge of copies of Sn−m indexed by inclusions Bousﬁeld localization Enriched category theory

This means we have the symmetry morphism

1.21 so J Σ is symmetric monoidal.

but it has bigger morphism objects. Mike Hill Mike Hopkins Doug Ravenel For m ≤ n we have Introduction

Σ n−m Quillen model J (m, n) := Σn+ ∧ S . categories Σn−m Coﬁbrant generation This is the wedge of copies of Sn−m indexed by inclusions Bousﬁeld localization Enriched category theory

This means we have the symmetry morphism

Sm ∧ Sn → Sn ∧ Sm

that we were missing before, 1.21 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy The Hovey-Shipley-Smith approach was to enlarge the theory indexing category J N to make it into a symmetric monoidal category J Σ. Its objects are the natural numbers as before,

but it has bigger morphism objects. Mike Hill Mike Hopkins Doug Ravenel For m ≤ n we have Introduction

Σ n−m Quillen model J (m, n) := Σn+ ∧ S . categories Σn−m Coﬁbrant generation This is the wedge of copies of Sn−m indexed by inclusions Bousﬁeld localization Enriched category theory

This means we have the symmetry morphism

Sm ∧ Sn → Sn ∧ Sm

Σ that we were missing before, so J is symmetric monoidal. 1.21 • the category of pointed topological spaces T is closed symmetric monoidal, and • the indexing category J Σ is symmetric monoidal and enriched over T .

Now for some categorical magic!

Day Convolution Theorem (1970) Let V be a closed symmetric monoidal category, and let J be a symmetric monoidal V-category. Then the category of functors from J to V is also closed symmetric monoidal.

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

A symmetric spectrum is an T -enriched functor J Σ → T . Note that

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.22 • the indexing category J Σ is symmetric monoidal and enriched over T .

Now for some categorical magic!

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

A symmetric spectrum is an T -enriched functor J Σ → T . Note that

Mike Hill • the category of pointed topological spaces T is closed Mike Hopkins symmetric monoidal, and Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.22 Now for some categorical magic!

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

A symmetric spectrum is an T -enriched functor J Σ → T . Note that

Mike Hill • the category of pointed topological spaces T is closed Mike Hopkins symmetric monoidal, and Doug Ravenel • the indexing category J Σ is symmetric monoidal and Introduction Quillen model enriched over T . categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.22 Day Convolution Theorem (1970) Let V be a closed symmetric monoidal category, and let J be a symmetric monoidal V-category. Then the category of functors from J to V is also closed symmetric monoidal.

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

A symmetric spectrum is an T -enriched functor J Σ → T . Note that

Coﬁbrant generation

Bousﬁeld localization

Enriched category Now for some categorical magic! theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.22 and let J be a symmetric monoidal V-category. Then the category of functors from J to V is also closed symmetric monoidal.

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

A symmetric spectrum is an T -enriched functor J Σ → T . Note that

Coﬁbrant generation

Bousﬁeld localization

Enriched category Now for some categorical magic! theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations Day Convolution Theorem (1970) Let V be a closed symmetric monoidal category,

1.22 Then the category of functors from J to V is also closed symmetric monoidal.

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

A symmetric spectrum is an T -enriched functor J Σ → T . Note that

Coﬁbrant generation

Bousﬁeld localization

Enriched category Now for some categorical magic! theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations Day Convolution Theorem (1970) Let V be a closed symmetric monoidal category, and let J be a symmetric monoidal V-category.

1.22 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

A symmetric spectrum is an T -enriched functor J Σ → T . Note that

Coﬁbrant generation

Bousﬁeld localization

Enriched category Now for some categorical magic! theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations Day Convolution Theorem (1970) Let V be a closed symmetric monoidal category, and let J be a symmetric monoidal V-category. Then the category of functors from J to V is also closed symmetric monoidal.

1.22 How does this work? Let X and Y be symmetric spectra. Then we have

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Day Convolution Theorem (1970) Introduction Quillen model Let V be a closed symmetric monoidal category, and let J be categories a symmetric monoidal V-category. Then the category of Coﬁbrant generation Bousﬁeld localization functors from to V is also also symmetric monoidal. J Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.23 Let X and Y be symmetric spectra. Then we have

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Deﬁning the smash product of spectra

Generalizations

1.23 Then we have

Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Deﬁning the smash product of spectra

Generalizations

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

we have Deﬁning the smash product of spectra

Generalizations

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Mike Hill Mike Hopkins Day Convolution Theorem (1970) Doug Ravenel Let V be a closed symmetric monoidal category, and let J be Introduction Quillen model a symmetric monoidal V-category. Then the category of categories functors from J to V is also also symmetric monoidal. Coﬁbrant generation Bousﬁeld localization How does this work? Let X and Y be symmetric spectra. Then Enriched category theory

we have Spectra as enriched functors X×Y Deﬁning the smash J Σ × J Σ / T × T product of spectra Generalizations

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

we have Spectra as enriched functors X×Y ∧ Deﬁning the smash J Σ × J Σ / T × T / T product of spectra Generalizations

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Mike Hill Mike Hopkins Day Convolution Theorem (1970) Doug Ravenel

Let V be a closed symmetric monoidal category, and let J be Introduction a symmetric monoidal V-category. Then the category of Quillen model categories functors from J to V is also also symmetric monoidal. Coﬁbrant generation

Bousﬁeld localization How does this work? Let X and Y be symmetric spectra. Then Enriched category we have theory Spectra as enriched functors (A, B) / A ∧ B Deﬁning the smash product of spectra Σ Σ X×Y ∧ J × J / T × T / T Generalizations

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Day Convolution Theorem (1970) Mike Hill Mike Hopkins Let V be a closed symmetric monoidal category, and let J be Doug Ravenel

a symmetric monoidal V-category. Then the category of Introduction functors from J to V is also also symmetric monoidal. Quillen model categories How does this work? Let X and Y be symmetric spectra. Then Cofibrant generation Bousfield localization we have Enriched category theory Spectra as enriched (A, B) / A ∧ B functors Σ Σ X×Y ∧ Defining the smash J × J / T × T / T product of spectra Generalizations + * J Σ

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Day Convolution Theorem (1970)

Mike Hill Let V be a closed symmetric monoidal category, and let J be Mike Hopkins a symmetric monoidal V-category. Then the category of Doug Ravenel functors from J to V is also also symmetric monoidal. Introduction Quillen model How does this work? Let X and Y be symmetric spectra. Then categories Coﬁbrant generation

we have Bousﬁeld localization

Enriched category (A, B) / A ∧ B theory Spectra as enriched X×Y functors Σ × Σ T × T ∧ T J J / / Deﬁning the smash product of spectra

(m, n) Generalizations + ) Σ J ) m + n

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Day Convolution Theorem (1970)

we have Bousﬁeld localization

Enriched category (A, B) / A ∧ B theory Spectra as enriched X×Y ∧ functors J Σ × J Σ / T × T / T 6 Deﬁning the smash product of spectra

(m, n) Generalizations + X∧Y ) Σ J ) m + n

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Day Convolution Theorem (1970)

Let V be a closed symmetric monoidal category, and let J be Mike Hill Mike Hopkins a symmetric monoidal V-category. Then the category of Doug Ravenel

functors from J to V is also also symmetric monoidal. Introduction

Quillen model How does this work? Let X and Y be symmetric spectra. Then categories we have Coﬁbrant generation Bousﬁeld localization

Enriched category (A, B) / A ∧ B theory X×Y Spectra as enriched Σ Σ ∧ functors J × J / T × T 6/ T Deﬁning the smash (m, n) product of spectra + X∧Y Generalizations ) Σ J ) m + n The red arrow is a left Kan extension,

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Day Convolution Theorem (1970) Let V be a closed symmetric monoidal category, and let J be Mike Hill a symmetric monoidal V-category. Then the category of Mike Hopkins Doug Ravenel functors from J to V is also also symmetric monoidal. Introduction

Quillen model How does this work? Let X and Y be symmetric spectra. Then categories

we have Coﬁbrant generation

Bousﬁeld localization (A, B) A ∧ B Enriched category / theory Σ Σ X×Y ∧ Spectra as enriched J × J / T × T 6/ T functors Deﬁning the smash (m, n) product of spectra + X∧Y Generalizations ) Σ J ) m + n The red arrow is a left Kan extension, a categorical construction known to exist

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Day Convolution Theorem (1970) Let V be a closed symmetric monoidal category, and let J be Mike Hill a symmetric monoidal V-category. Then the category of Mike Hopkins functors from J to V is also also symmetric monoidal. Doug Ravenel

Introduction

How does this work? Let X and Y be symmetric spectra. Then Quillen model we have categories Cofibrant generation Bousfield localization (A, B) / A ∧ B Enriched category theory Σ Σ X×Y ∧ J × J / T × T / T Spectra as enriched 6 functors (m, n) Defining the smash + X∧Y product of spectra ) Σ Generalizations J ) m + n The red arrow is a left Kan extension, a categorical construction known to exist when the source category J Σ is small

1.23 Model categories Deﬁning the smash product of spectra (continued) and stable homotopy theory

Day Convolution Theorem (1970) Let V be a closed symmetric monoidal category, and let J be a symmetric monoidal V-category. Then the category of Mike Hill Mike Hopkins functors from J to V is also also symmetric monoidal. Doug Ravenel

Introduction

How does this work? Let X and Y be symmetric spectra. Then Quillen model we have categories Coﬁbrant generation

Bousﬁeld localization (A, B) A ∧ B / Enriched category theory Σ Σ X×Y ∧ J × J / T × T / T Spectra as enriched 6 functors (m, n) Deﬁning the smash + X∧Y product of spectra ) Σ Generalizations J ) m + n The red arrow is a left Kan extension, a categorical construction known to exist when the source category J Σ is small and the target category T is cocomplete.

1.23 from the symmetric monoidal T -category J Σ to the closed symmetric monoidal category T can be generalized in three different ways.

Generalizing the target category

For each ﬁnite group G, we can replace the category T of pointed topological spaces with T G, the category of pointed G-spaces and equivariant maps.

It is enriched over T . It has a model structure in which fibrations and weak equivalences are equivariant maps X → Y inducing ordinary fibrations and weak equivalences of fixed point sets X H → Y H for each subgroup H ⊆ G. Its cofibrant generating sets are

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Model categories Generalizations and stable homotopy theory The construction of symmetric spectra as functors

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.24 to the closed symmetric monoidal category T can be generalized in three different ways.

Generalizing the target category

For each ﬁnite group G, we can replace the category T of pointed topological spaces with T G, the category of pointed G-spaces and equivariant maps.

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Model categories Generalizations and stable homotopy theory The construction of symmetric spectra as functors from the symmetric monoidal T -category J Σ

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.24 can be generalized in three different ways.

Generalizing the target category

For each ﬁnite group G, we can replace the category T of pointed topological spaces with T G, the category of pointed G-spaces and equivariant maps.

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Model categories Generalizations and stable homotopy theory The construction of symmetric spectra as functors from the symmetric monoidal T -category J Σ to the closed symmetric monoidal category T Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.24 Generalizing the target category

For each ﬁnite group G, we can replace the category T of pointed topological spaces with T G, the category of pointed G-spaces and equivariant maps.

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.24 For each ﬁnite group G, we can replace the category T of pointed topological spaces with T G, the category of pointed G-spaces and equivariant maps.

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Generalizing the target category Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.24 we can replace the category T of pointed topological spaces with T G, the category of pointed G-spaces and equivariant maps.

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Generalizing the target category Introduction

Quillen model categories For each ﬁnite group G, Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.24 with T G, the category of pointed G-spaces and equivariant maps.

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Generalizing the target category Introduction

Quillen model categories For each ﬁnite group G, we can replace the category T of Coﬁbrant generation

pointed topological spaces Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.24 It is enriched over T . It has a model structure in which fibrations and weak equivalences are equivariant maps X → Y inducing ordinary fibrations and weak equivalences of fixed point sets X H → Y H for each subgroup H ⊆ G. Its cofibrant generating sets are

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Generalizing the target category Introduction

Quillen model categories For each finite group G, we can replace the category T of Cofibrant generation G pointed topological spaces with T , the category of pointed Bousfield localization

G-spaces and equivariant maps. Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.24 inducing ordinary fibrations and weak equivalences of fixed point sets X H → Y H for each subgroup H ⊆ G. Its cofibrant generating sets are

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Generalizing the target category Introduction

Quillen model categories For each finite group G, we can replace the category T of Cofibrant generation G pointed topological spaces with T , the category of pointed Bousfield localization

G-spaces and equivariant maps. Enriched category theory

Spectra as enriched It is enriched over T . It has a model structure in which functors ﬁbrations and weak equivalences are equivariant maps X → Y Deﬁning the smash product of spectra

Generalizations

1.24 Its coﬁbrant generating sets are

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Generalizing the target category Introduction

Quillen model categories For each finite group G, we can replace the category T of Cofibrant generation G pointed topological spaces with T , the category of pointed Bousfield localization

G-spaces and equivariant maps. Enriched category theory

Spectra as enriched It is enriched over T . It has a model structure in which functors fibrations and weak equivalences are equivariant maps X → Y Defining the smash product of spectra inducing ordinary fibrations and weak equivalences of fixed Generalizations point sets X H → Y H for each subgroup H ⊆ G.

1.24 G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

Generalizing the target category Introduction

Quillen model categories For each finite group G, we can replace the category T of Cofibrant generation G pointed topological spaces with T , the category of pointed Bousfield localization

G-spaces and equivariant maps. Enriched category theory

1.24 Model categories Generalizations and stable homotopy theory The construction of symmetric spectra as functors from the symmetric monoidal T -category J Σ to the closed symmetric monoidal category T can be generalized in three different ways. Mike Hill Mike Hopkins Doug Ravenel

Generalizing the target category Introduction

Quillen model categories For each finite group G, we can replace the category T of Cofibrant generation G pointed topological spaces with T , the category of pointed Bousfield localization

G-spaces and equivariant maps. Enriched category theory

G [ G [ I = (G/H)+ ∧ I+ and J = (G/H)+ ∧ J+. H⊆G H⊆G

1.24 Here the objects are still natural numbers, and J O(m, n) is a point for m > n. For m ≤ n, J O(m, n) is a wedge of copies of Sn−m parametrized by orthogonal embeddings of Rm into Rn. T -valued functors on it are called orthogonal spectra.

For a finite group G we can define a similar category J G in which the objects are finite dimensional orthogonal real representations V of G. It is enriched over T G. T G-valued functors on it are called orthogonal G-spectra.

In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

Model categories Generalizing the indexing category and stable homotopy theory

We can replace J Σ by an orthogonal analog J O, the Mike Hill Mandell-May category. Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.25 For m ≤ n, J O(m, n) is a wedge of copies of Sn−m parametrized by orthogonal embeddings of Rm into Rn. T -valued functors on it are called orthogonal spectra.

In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

Model categories Generalizing the indexing category and stable homotopy theory

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.25 parametrized by orthogonal embeddings of Rm into Rn. T -valued functors on it are called orthogonal spectra.

In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

Model categories Generalizing the indexing category and stable homotopy theory

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.25 T -valued functors on it are called orthogonal spectra.

In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

Model categories Generalizing the indexing category and stable homotopy theory

We can replace J Σ by an orthogonal analog J O, the Mike Hill Mandell-May category. Here the objects are still natural Mike Hopkins Doug Ravenel numbers, and J O(m, n) is a point for m > n. For m ≤ n, J O(m, n) is a wedge of copies of Sn−m parametrized by Introduction m n Quillen model orthogonal embeddings of R into R . categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.25 For a finite group G we can define a similar category J G in which the objects are finite dimensional orthogonal real representations V of G. It is enriched over T G. T G-valued functors on it are called orthogonal G-spectra.

In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

Model categories Generalizing the indexing category and stable homotopy theory

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.25 in which the objects are ﬁnite dimensional orthogonal real representations V of G. It is enriched over T G. T G-valued functors on it are called orthogonal G-spectra.

In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

Model categories Generalizing the indexing category and stable homotopy theory

G Enriched category For a ﬁnite group G we can deﬁne a similar category J theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.25 It is enriched over T G. T G-valued functors on it are called orthogonal G-spectra.

In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

Model categories Generalizing the indexing category and stable homotopy theory

G Enriched category For a ﬁnite group G we can deﬁne a similar category J in theory

which the objects are ﬁnite dimensional orthogonal real Spectra as enriched representations V of G. functors Deﬁning the smash product of spectra

Generalizations

1.25 T G-valued functors on it are called orthogonal G-spectra.

In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

Model categories Generalizing the indexing category and stable homotopy theory

G Enriched category For a ﬁnite group G we can deﬁne a similar category J in theory

which the objects are ﬁnite dimensional orthogonal real Spectra as enriched representations V of G. It is enriched over T G. functors Deﬁning the smash product of spectra

Generalizations

1.25 In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

Model categories Generalizing the indexing category and stable homotopy theory

G Enriched category For a ﬁnite group G we can deﬁne a similar category J in theory

which the objects are ﬁnite dimensional orthogonal real Spectra as enriched representations V of G. It is enriched over T G. T G-valued functors Deﬁning the smash functors on it are called orthogonal G-spectra. product of spectra

Generalizations

1.25 Model categories Generalizing the indexing category and stable homotopy theory

G Enriched category For a ﬁnite group G we can deﬁne a similar category J in theory

Generalizations In each case one has a smash product of spectra deﬁned using the Day Convolution as before.

1.25 We have now seen four categories of spectra, each deﬁned as the category of enriched functors from an enriched indexing category J to a pointed topological model category M. We denote such a functor category by [J , M]. Given a spectrum X and an object V in J , we denote the value of X on V by XV .

Our categories are • Sp = [J N, T ], the original category of spectra, • SpΣ = [J Σ, T ], the category of symmetric spectra of Hovey-Shipley-Smith, • SpO = [J O, T ], the category of orthogonal spectra of Mandell-May, and • SpG = [J G, T G], the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Model categories Generalizing the model structure and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 each deﬁned as the category of enriched functors from an enriched indexing category J to a pointed topological model category M. We denote such a functor category by [J , M]. Given a spectrum X and an object V in J , we denote the value of X on V by XV .

Model categories Generalizing the model structure and stable homotopy theory We have now seen four categories of spectra,

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 from an enriched indexing category J to a pointed topological model category M. We denote such a functor category by [J , M]. Given a spectrum X and an object V in J , we denote the value of X on V by XV .

Model categories Generalizing the model structure and stable homotopy theory We have now seen four categories of spectra, each deﬁned as the category of enriched functors

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 to a pointed topological model category M. We denote such a functor category by [J , M]. Given a spectrum X and an object V in J , we denote the value of X on V by XV .

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 We denote such a functor category by [J , M]. Given a spectrum X and an object V in J , we denote the value of X on V by XV .

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 Given a spectrum X and an object V in J , we denote the value of X on V by XV .

Model categories Generalizing the model structure and stable homotopy theory We have now seen four categories of spectra, each deﬁned as the category of enriched functors from an enriched indexing category J to a pointed topological model category M. We denote such a functor category by [J , M]. Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 we denote the value of X on V by XV .

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 Our categories are • Sp = [J N, T ], the original category of spectra, • SpΣ = [J Σ, T ], the category of symmetric spectra of Hovey-Shipley-Smith, • SpO = [J O, T ], the category of orthogonal spectra of Mandell-May, and • SpG = [J G, T G], the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 • Sp = [J N, T ], the original category of spectra, • SpΣ = [J Σ, T ], the category of symmetric spectra of Hovey-Shipley-Smith, • SpO = [J O, T ], the category of orthogonal spectra of Mandell-May, and • SpG = [J G, T G], the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Introduction

Our categories are Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 the original category of spectra, • SpΣ = [J Σ, T ], the category of symmetric spectra of Hovey-Shipley-Smith, • SpO = [J O, T ], the category of orthogonal spectra of Mandell-May, and • SpG = [J G, T G], the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Introduction

Our categories are Quillen model N categories • Sp = [J , T ], Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 • SpΣ = [J Σ, T ], the category of symmetric spectra of Hovey-Shipley-Smith, • SpO = [J O, T ], the category of orthogonal spectra of Mandell-May, and • SpG = [J G, T G], the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Introduction

Our categories are Quillen model N categories • Sp = [J , T ], the original category of spectra, Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 the category of symmetric spectra of Hovey-Shipley-Smith, • SpO = [J O, T ], the category of orthogonal spectra of Mandell-May, and • SpG = [J G, T G], the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Introduction

Our categories are Quillen model N categories • Sp = [J , T ], the original category of spectra, Coﬁbrant generation Σ Σ • Sp = [J , T ], Bousﬁeld localization Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 • SpO = [J O, T ], the category of orthogonal spectra of Mandell-May, and • SpG = [J G, T G], the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Introduction

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.26 the category of orthogonal spectra of Mandell-May, and • SpG = [J G, T G], the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Introduction

Deﬁning the smash product of spectra

Generalizations

1.26 • SpG = [J G, T G], the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Introduction

Our categories are Quillen model N categories • Sp = [J , T ], the original category of spectra, Cofibrant generation Σ Σ • Sp = [J , T ], the category of symmetric spectra of Bousfield localization Enriched category Hovey-Shipley-Smith, theory O O • Sp = [J , T ], the category of orthogonal spectra of Spectra as enriched functors Mandell-May, and Defining the smash product of spectra

Generalizations

1.26 the category of orthogonal G-spectra for a ﬁnite group G, also introduced by Mandell and May.

Introduction

1.26 We will refer to an object in any but the ﬁrst of these as a structured spectrum. Each category of structured spectra has a closed symmetric monoidal smash product deﬁned using the Day Convolution.

Introduction

1.26 Each category of structured spectra has a closed symmetric monoidal smash product deﬁned using the Day Convolution.

Introduction

We will refer to an object in any but the ﬁrst of these as a structured spectrum.

1.26 deﬁned using the Day Convolution.

Introduction

We will refer to an object in any but the ﬁrst of these as a structured spectrum. Each category of structured spectra has a closed symmetric monoidal smash product

1.26 Model categories Generalizing the model structure and stable homotopy theory We have now seen four categories of spectra, each deﬁned as the category of enriched functors from an enriched indexing category J to a pointed topological model category M. We denote such a functor category by [J , M]. Given a spectrum Mike Hill Mike Hopkins X and an object V in J , we denote the value of X on V by XV . Doug Ravenel

Introduction

We will refer to an object in any but the first of these as a structured spectrum. Each category of structured spectra has a closed symmetric monoidal smash product defined using the Day Convolution. 1.26 For each object V in the indexing category J , we define the V Yoneda spectrum S−V = H by

−V (S )W := J (V , W ).

In the structured cases one can show that for objects V 0 and V 00 in J , 0 00 0 00 S−(V +V ) =∼ S−V ∧ S−V .

Definition Let I and J be cofibrant generating sets for M. In the projective model structure on the category of spectra [J , M], the cofibrant generating sets are [ [ Ie = S−V ∧ I and Je = S−V ∧ J . V ∈J V ∈J

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.27 we deﬁne the V Yoneda spectrum S−V = H by

−V (S )W := J (V , W ).

In the structured cases one can show that for objects V 0 and V 00 in J , 0 00 0 00 S−(V +V ) =∼ S−V ∧ S−V .

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J ,

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.27 −V (S )W := J (V , W ).

In the structured cases one can show that for objects V 0 and V 00 in J , 0 00 0 00 S−(V +V ) =∼ S−V ∧ S−V .

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.27 In the structured cases one can show that for objects V 0 and V 00 in J , 0 00 0 00 S−(V +V ) =∼ S−V ∧ S−V .

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

−V (S ) := (V , W ). Mike Hill W J Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.27 0 00 0 00 S−(V +V ) =∼ S−V ∧ S−V .

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

−V (S ) := (V , W ). Mike Hill W J Mike Hopkins Doug Ravenel 0 In the structured cases one can show that for objects V and Introduction 00 V in J , Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.27 Definition Let I and J be cofibrant generating sets for M. In the projective model structure on the category of spectra [J , M], the cofibrant generating sets are [ [ Ie = S−V ∧ I and Je = S−V ∧ J . V ∈J V ∈J

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

−V (S ) := (V , W ). Mike Hill W J Mike Hopkins Doug Ravenel 0 In the structured cases one can show that for objects V and Introduction 00 V in J , Quillen model 0 00 0 00 S−(V +V ) =∼ S−V ∧ S−V . categories Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.27 In the projective model structure on the category of spectra [J , M], the coﬁbrant generating sets are [ [ Ie = S−V ∧ I and Je = S−V ∧ J . V ∈J V ∈J

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

Bousﬁeld localization

Deﬁnition Enriched category Let I and J be coﬁbrant generating sets for M. theory Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.27 the coﬁbrant generating sets are [ [ Ie = S−V ∧ I and Je = S−V ∧ J . V ∈J V ∈J

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

Bousﬁeld localization

Generalizations

1.27 [ [ Ie = S−V ∧ I and Je = S−V ∧ J . V ∈J V ∈J

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

Bousﬁeld localization

Definition Enriched category Let I and J be cofibrant generating sets for M. In the theory Spectra as enriched projective model structure on the category of spectra [J , M], functors Defining the smash the cofibrant generating sets are product of spectra

Generalizations

1.27 A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

Bousﬁeld localization

1.27 Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

Bousﬁeld localization

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

1.27 Model categories Generalizing the indexing category (continued) and stable homotopy theory For each object V in the indexing category J , we deﬁne the V Yoneda spectrum S−V = H by

Bousﬁeld localization

A map f : X → Y is a projective (or strict) weak equivalence if fV is a weak equivalence in M for each V .

1.27 We can obtain the stable model structure on the category of spectra [J , M] from the projective one by Bousﬁeld localization. We expand the class of weak equivalences by including certain maps

−V V −0 sV : S ∧ S → S for each V ∈ J .

Here SV denotes the sphere object J (0, V ) in M, and S−0 is the sphere spectrum deﬁned by

−0 W (S )W := S = J (0, W ).

The W th component of the map sV is

j0,V ,W : J (V , W ) ∧ J (0, V ) → J (0, W ),

a composition morphism in J .

Model categories Generalizing the indexing category (continued) and stable homotopy theory [ [ Ie = S−V ∧ I and Je = S−V ∧ J . V ∈J V ∈J Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.28 We expand the class of weak equivalences by including certain maps

−V V −0 sV : S ∧ S → S for each V ∈ J .

Here SV denotes the sphere object J (0, V ) in M, and S−0 is the sphere spectrum deﬁned by

−0 W (S )W := S = J (0, W ).

The W th component of the map sV is

j0,V ,W : J (V , W ) ∧ J (0, V ) → J (0, W ),

a composition morphism in J .

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.28 −V V −0 sV : S ∧ S → S for each V ∈ J .

Here SV denotes the sphere object J (0, V ) in M, and S−0 is the sphere spectrum deﬁned by

−0 W (S )W := S = J (0, W ).

The W th component of the map sV is

j0,V ,W : J (V , W ) ∧ J (0, V ) → J (0, W ),

a composition morphism in J .

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.28 Here SV denotes the sphere object J (0, V ) in M, and S−0 is the sphere spectrum deﬁned by

−0 W (S )W := S = J (0, W ).

The W th component of the map sV is

j0,V ,W : J (V , W ) ∧ J (0, V ) → J (0, W ),

a composition morphism in J .

Model categories Generalizing the indexing category (continued) and stable homotopy theory [ [ Ie = S−V ∧ I and Je = S−V ∧ J . V ∈J V ∈J Mike Hill Mike Hopkins We can obtain the stable model structure on the category of Doug Ravenel spectra [J , M] from the projective one by Bousfield localization. We expand the class of weak equivalences by Introduction Quillen model including certain maps categories Cofibrant generation −V V −0 sV : S ∧ S → S for each V ∈ J . Bousfield localization Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.28 and S−0 is the sphere spectrum deﬁned by

−0 W (S )W := S = J (0, W ).

The W th component of the map sV is

j0,V ,W : J (V , W ) ∧ J (0, V ) → J (0, W ),

a composition morphism in J .

Deﬁning the smash product of spectra

Generalizations

1.28 The W th component of the map sV is

j0,V ,W : J (V , W ) ∧ J (0, V ) → J (0, W ),

a composition morphism in J .

1.28 Model categories Generalizing the indexing category (continued) and stable homotopy theory [ [ Ie = S−V ∧ I and Je = S−V ∧ J . V ∈J V ∈J Mike Hill Mike Hopkins We can obtain the stable model structure on the category of Doug Ravenel spectra [J , M] from the projective one by Bousfield localization. We expand the class of weak equivalences by Introduction Quillen model including certain maps categories Cofibrant generation −V V −0 sV : S ∧ S → S for each V ∈ J . Bousfield localization Enriched category V −0 theory Here S denotes the sphere object J (0, V ) in M, and S is Spectra as enriched the sphere spectrum defined by functors Defining the smash product of spectra −0 W (S )W := S = J (0, W ). Generalizations

The W th component of the map sV is

j0,V ,W : J (V , W ) ∧ J (0, V ) → J (0, W ),

a composition morphism in J .

1.28 Model categories and stable homotopy theory

Mike Hill Mike Hopkins Doug Ravenel

Introduction

Quillen model categories

Coﬁbrant generation

Bousﬁeld localization

Enriched category theory

Spectra as enriched functors

Deﬁning the smash product of spectra

Generalizations

1.29