Lehigh University Geometry and Topology Conference. May 22, 2015

Transcription

1 Lehigh University Geometry and Topology Conference May 22, 2015 University of Virginia Harvard University University of Rochester 1.1

2 Algebraic topologists have been studying spectra for over 50 years 1.2

3 Algebraic topologists have been studying spectra for over 50 years and G-spectra for over 30 years. 1.2

4 Algebraic topologists have been studying spectra for over 50 years and G-spectra for over 30 years. The basic definitions have changed several times, 1.2

5 Algebraic topologists have been studying spectra for over 50 years and G-spectra for over 30 years. The basic definitions have changed several times, yet our intuition about spectra has not. 1.2

6 Algebraic topologists have been studying spectra for over 50 years and G-spectra for over 30 years. The basic definitions have changed several times, yet our intuition about spectra has not. We have made extensive calculations with them from the very beginning. 1.2

7 Algebraic topologists have been studying spectra for over 50 years and G-spectra for over 30 years. The basic definitions have changed several times, yet our intuition about spectra has not. We have made extensive calculations with them from the very beginning. None of these have been affected in the least by the changing foundations of the subject. 1.2

8 Algebraic topologists have been studying spectra for over 50 years and G-spectra for over 30 years. The basic definitions have changed several times, yet our intuition about spectra has not. We have made extensive calculations with them from the very beginning. None of these have been affected in the least by the changing foundations of the subject. This is a peculiar state of affairs! 1.2

9 (continued) Spectra were first defined in a 1959 paper of Lima, 1.3

10 (continued) Spectra were first defined in a 1959 paper of Lima, who is now a very prominent mathematician in Brazil. 1.3

11 (continued) Spectra were first defined in a 1959 paper of Lima, who is now a very prominent mathematician in Brazil. He was a student of Spanier at the University of Chicago. Ed Spanier

12 (continued) George Whitehead

13 (continued) George Whitehead Here is the in a 1962 paper by Whitehead, 1.4

14 (continued) George Whitehead Here is the in a 1962 paper by Whitehead, the earliest online reference I could find. 1.4

15 (continued) George Whitehead Here is the in a 1962 paper by Whitehead, the earliest online reference I could find. 1.4

16 (continued) This definition was adequate for many calculations over the next 20 years. 1.5

17 (continued) This definition was adequate for many calculations over the next 20 years. It was used by Adams in his blue book of Frank Adams

18 (continued) This definition was adequate for many calculations over the next 20 years. It was used by Adams in his blue book of Frank Adams The definition led to a lot of technical problems 1.5

19 (continued) This definition was adequate for many calculations over the next 20 years. It was used by Adams in his blue book of Frank Adams The definition led to a lot of technical problems especially in connection with smash products. 1.5

20 (continued) This definition was adequate for many calculations over the next 20 years. It was used by Adams in his blue book of Frank Adams The definition led to a lot of technical problems especially in connection with smash products. The definition we use today is more categorical. 1.5

21 (continued) Some words you will not hear again in this talk: 1.6

22 (continued) Some words you will not hear again in this talk: up to homotopy 1.6

23 (continued) Some words you will not hear again in this talk: up to homotopy simplicial 1.6

24 (continued) Some words you will not hear again in this talk: up to homotopy simplicial operad 1.6

25 (continued) Some words you will not hear again in this talk: up to homotopy simplicial operad universe 1.6

26 (continued) Some words you will not hear again in this talk: up to homotopy simplicial operad universe -category 1.6

27 (continued) Some words you will not hear again in this talk: up to homotopy simplicial operad universe -category chromatic 1.6

28 (continued) Some words you will not hear again in this talk: up to homotopy simplicial operad universe -category chromatic Mackey functor 1.6

29 (continued) Some words you will not hear again in this talk: up to homotopy simplicial operad universe -category chromatic Mackey functor slice spectral sequence 1.6

30 Some categorical notions: Enrichment, I In a (locally small) category C, 1.7

31 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, 1.7

32 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). 1.7

33 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. 1.7

34 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. 1.7

35 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. 1.7

36 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, 1.7

37 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, 1.7

38 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. 1.7

39 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C 1.7

40 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C induces a map Ab(B, C) Ab(A, B) Ab(A, C). 1.7

41 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C induces a map Ab(B, C) Ab(A, B) Ab(A, C). (ii) Let T be the category of pointed compactly generated weak Hausdorff spaces. 1.7

42 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C induces a map Ab(B, C) Ab(A, B) Ab(A, C). (ii) Let T be the category of pointed compactly generated weak Hausdorff spaces. Then for such spaces X and Y, 1.7

43 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C induces a map Ab(B, C) Ab(A, B) Ab(A, C). (ii) Let T be the category of pointed compactly generated weak Hausdorff spaces. Then for such spaces X and Y, the set T (X, Y ) of pointed continuous maps X Y, 1.7

44 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C induces a map Ab(B, C) Ab(A, B) Ab(A, C). (ii) Let T be the category of pointed compactly generated weak Hausdorff spaces. Then for such spaces X and Y, the set T (X, Y ) of pointed continuous maps X Y, is itself a pointed space under the compact open topology, 1.7

45 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C induces a map Ab(B, C) Ab(A, B) Ab(A, C). (ii) Let T be the category of pointed compactly generated weak Hausdorff spaces. Then for such spaces X and Y, the set T (X, Y ) of pointed continuous maps X Y, is itself a pointed space under the compact open topology, the base point being the constant map. 1.7

46 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C induces a map Ab(B, C) Ab(A, B) Ab(A, C). (ii) Let T be the category of pointed compactly generated weak Hausdorff spaces. Then for such spaces X and Y, the set T (X, Y ) of pointed continuous maps X Y, is itself a pointed space under the compact open topology, the base point being the constant map. Here composition leads to a map T (X, Y ) T (W, X) T (W, Y ). 1.7

47 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C induces a map Ab(B, C) Ab(A, B) Ab(A, C). (ii) Let T be the category of pointed compactly generated weak Hausdorff spaces. Then for such spaces X and Y, the set T (X, Y ) of pointed continuous maps X Y, is itself a pointed space under the compact open topology, the base point being the constant map. Here composition leads to a map T (X, Y ) T (W, X) T (W, Y ). (From now on, all topological spaces will be assumed to be compactly generated weak Hausdorff.) 1.7

48 Some categorical notions: Enrichment, I In a (locally small) category C, for each pair of object X and Y, one has a set of morphisms C(X, Y ). It sometimes happens that this set has a richer structure. Here are two examples. (i) Let Ab be the category of abelian groups. Then for abelian groups A and B, the set Ab(A, B) of homomorphisms A B, is itself an abelian group. Composition of morphisms A B C induces a map Ab(B, C) Ab(A, B) Ab(A, C). (ii) Let T be the category of pointed compactly generated weak Hausdorff spaces. Then for such spaces X and Y, the set T (X, Y ) of pointed continuous maps X Y, is itself a pointed space under the compact open topology, the base point being the constant map. Here composition leads to a map T (X, Y ) T (W, X) T (W, Y ). (From now on, all topological spaces will be assumed to be compactly generated weak Hausdorff.) We say that both of these are enriched over themselves. 1.7

49 Some categorical notions: Enrichment, I (continued) Let G be a finite group. 1.8

50 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, 1.8

51 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, 1.8

52 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. 1.8

53 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. 1.8

54 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, 1.8

55 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. 1.8

56 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. (ii) Let T G denote the category of pointed G-spaces and all (not necessarily equivariant) continuous pointed maps. 1.8

57 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. (ii) Let T G denote the category of pointed G-spaces and all (not necessarily equivariant) continuous pointed maps. Then T G (X, Y ) is a pointed G-space. 1.8

58 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. (ii) Let T G denote the category of pointed G-spaces and all (not necessarily equivariant) continuous pointed maps. Then T G (X, Y ) is a pointed G-space. For f : X Y and γ G, 1.8

59 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. (ii) Let T G denote the category of pointed G-spaces and all (not necessarily equivariant) continuous pointed maps. Then T G (X, Y ) is a pointed G-space. For f : X Y and γ G, we define γ(f ) = γf γ 1, 1.8

60 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. (ii) Let T G denote the category of pointed G-spaces and all (not necessarily equivariant) continuous pointed maps. Then T G (X, Y ) is a pointed G-space. For f : X Y and γ G, we define γ(f ) = γf γ 1, the lower composite map in the noncommutative diagram 1.8

61 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. (ii) Let T G denote the category of pointed G-spaces and all (not necessarily equivariant) continuous pointed maps. Then T G (X, Y ) is a pointed G-space. For f : X Y and γ G, we define γ(f ) = γf γ 1, the lower composite map in the noncommutative diagram X γ 1 X f Y γ f Y. 1.8

62 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. (ii) Let T G denote the category of pointed G-spaces and all (not necessarily equivariant) continuous pointed maps. Then T G (X, Y ) is a pointed G-space. For f : X Y and γ G, we define γ(f ) = γf γ 1, the lower composite map in the noncommutative diagram X γ 1 f Y γ f Y. T G is enriched T G and hence over itself. X 1.8

63 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. (ii) Let T G denote the category of pointed G-spaces and all (not necessarily equivariant) continuous pointed maps. Then T G (X, Y ) is a pointed G-space. For f : X Y and γ G, we define γ(f ) = γf γ 1, the lower composite map in the noncommutative diagram X γ 1 f Y γ f Y. T G is enriched T G and hence over itself. T G (X, Y ) G X 1.8

64 Some categorical notions: Enrichment, I (continued) Let G be a finite group. There are two whose objects are pointed G-spaces, where the base point is always fixed by G, because there are two types of morphisms to consider. (i) Let T G denote the category of pointed G-spaces and equivariant continuous pointed maps. Then T G (X, Y ) is a pointed topological space, so T G is enriched over T. (ii) Let T G denote the category of pointed G-spaces and all (not necessarily equivariant) continuous pointed maps. Then T G (X, Y ) is a pointed G-space. For f : X Y and γ G, we define γ(f ) = γf γ 1, the lower composite map in the noncommutative diagram X γ 1 f Y γ f Y. T G is enriched T G and hence over itself. T G (X, Y ) G = T G (X, Y ). X 1.8

65 A symmetric monoidal category is a category V equipped with a map : V V V 1.9

66 A symmetric monoidal category is a category V equipped with a map : V V V with natural associativity isomorphisms (X Y ) Z X (Y Z ), 1.9

67 A symmetric monoidal category is a category V equipped with a map : V V V with natural associativity isomorphisms (X Y ) Z X (Y Z ), natural symmetry isomorphisms X Y Y X 1.9

68 A symmetric monoidal category is a category V equipped with a map : V V V with natural associativity isomorphisms (X Y ) Z X (Y Z ), natural symmetry isomorphisms X Y Y X and a unit object 1 with unit isomorphisms ι X : 1 X X. 1.9

69 A symmetric monoidal category is a category V equipped with a map : V V V with natural associativity isomorphisms (X Y ) Z X (Y Z ), natural symmetry isomorphisms X Y Y X and a unit object 1 with unit isomorphisms ι X : 1 X X. We will denote this structure by (V,, 1), 1.9

70 A symmetric monoidal category is a category V equipped with a map : V V V with natural associativity isomorphisms (X Y ) Z X (Y Z ), natural symmetry isomorphisms X Y Y X and a unit object 1 with unit isomorphisms ι X : 1 X X. We will denote this structure by (V,, 1), surpressing the required isomorphisms from the notation. 1.9

71 A symmetric monoidal category is a category V equipped with a map : V V V with natural associativity isomorphisms (X Y ) Z X (Y Z ), natural symmetry isomorphisms X Y Y X and a unit object 1 with unit isomorphisms ι X : 1 X X. We will denote this structure by (V,, 1), surpressing the required isomorphisms from the notation. The monoidal structure is closed if the functor A ( ) has a right adjoint ( ) A, the internal Hom with V(1, X A ) = V(A, X). 1.9

72 (continued) Here are some familiar examples: 1.10

73 (continued) Here are some familiar examples: (i) (Sets,, ), 1.10

74 (continued) Here are some familiar examples: (i) (Sets,, ), the category of sets under Cartesian product, 1.10

75 (continued) Here are some familiar examples: (i) (Sets,, ), the category of sets under Cartesian product, where the unit is a set with one element. 1.10

76 (continued) Here are some familiar examples: (i) (Sets,, ), the category of sets under Cartesian product, where the unit is a set with one element. (ii) (Ab,, Z), 1.10

77 (continued) Here are some familiar examples: (i) (Sets,, ), the category of sets under Cartesian product, where the unit is a set with one element. (ii) (Ab,, Z), the category of abelian groups under tensor product, with the integers Z as unit. 1.10

78 (continued) Here are some familiar examples: (i) (Sets,, ), the category of sets under Cartesian product, where the unit is a set with one element. (ii) (Ab,, Z), the category of abelian groups under tensor product, with the integers Z as unit. (ii) (Ab,, 0), 1.10

79 (continued) Here are some familiar examples: (i) (Sets,, ), the category of sets under Cartesian product, where the unit is a set with one element. (ii) (Ab,, Z), the category of abelian groups under tensor product, with the integers Z as unit. (ii) (Ab,, 0), the category of abelian groups under direct sum, with the trivial group as unit. 1.10

80 (continued) Here are some familiar examples: (i) (Sets,, ), the category of sets under Cartesian product, where the unit is a set with one element. (ii) (Ab,, Z), the category of abelian groups under tensor product, with the integers Z as unit. (ii) (Ab,, 0), the category of abelian groups under direct sum, with the trivial group as unit. (iv) (T op,, ), the category of topological spaces (without base point) under Cartesian product with the one point space as unit. 1.10

81 (continued) Here are some familiar examples: (i) (Sets,, ), the category of sets under Cartesian product, where the unit is a set with one element. (ii) (Ab,, Z), the category of abelian groups under tensor product, with the integers Z as unit. (ii) (Ab,, 0), the category of abelian groups under direct sum, with the trivial group as unit. (iv) (T op,, ), the category of topological spaces (without base point) under Cartesian product with the one point space as unit. (v) (T G,, S 0 ), the category of pointed G-spaces and nonequivariant maps under smash product with the 0-sphere S 0 as unit. 1.10

82 (continued) Here are some familiar examples: (i) (Sets,, ), the category of sets under Cartesian product, where the unit is a set with one element. (ii) (Ab,, Z), the category of abelian groups under tensor product, with the integers Z as unit. (ii) (Ab,, 0), the category of abelian groups under direct sum, with the trivial group as unit. (iv) (T op,, ), the category of topological spaces (without base point) under Cartesian product with the one point space as unit. (v) (T G,, S 0 ), the category of pointed G-spaces and nonequivariant maps under smash product with the 0-sphere S 0 as unit. (vi) (T G,, S 0 ), the category of pointed G-spaces and equivariant maps under smash product with S 0 as unit. 1.10

83 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly

84 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. 1.11

85 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. A V-category C 1.11

86 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. A V-category C (or a category enriched over V) 1.11

87 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. A V-category C (or a category enriched over V) has a collection of objects ob(c) 1.11

88 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. A V-category C (or a category enriched over V) has a collection of objects ob(c) and for each pair of objects X, Y an object C(X, Y ) in V 0, 1.11

89 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. A V-category C (or a category enriched over V) has a collection of objects ob(c) and for each pair of objects X, Y an object C(X, Y ) in V 0, instead of a morphism set. 1.11

90 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. A V-category C (or a category enriched over V) has a collection of objects ob(c) and for each pair of objects X, Y an object C(X, Y ) in V 0, instead of a morphism set. For each object X in C we have a morphism 1 C(X, X) in V

91 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. A V-category C (or a category enriched over V) has a collection of objects ob(c) and for each pair of objects X, Y an object C(X, Y ) in V 0, instead of a morphism set. For each object X in C we have a morphism 1 C(X, X) in V 0 instead of an identity morphism. 1.11

92 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. A V-category C (or a category enriched over V) has a collection of objects ob(c) and for each pair of objects X, Y an object C(X, Y ) in V 0, instead of a morphism set. For each object X in C we have a morphism 1 C(X, X) in V 0 instead of an identity morphism. For each triple of objects X, Y, Z in C, 1.11

93 The following definitions were first published by Eilenberg-Kelly in Sammy Eilenberg Max Kelly Let V = (V 0,, 1) be a symmetric monoidal category. A V-category C (or a category enriched over V) has a collection of objects ob(c) and for each pair of objects X, Y an object C(X, Y ) in V 0, instead of a morphism set. For each object X in C we have a morphism 1 C(X, X) in V 0 instead of an identity morphism. For each triple of objects X, Y, Z in C, we have composition morphism C(Y, Z ) C(X, Y ) C(X, Z ) in V

94 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C 1.12

95 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). 1.12

96 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- 1.12

97 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- consists of a function F : ob(c) ob(d), 1.12

98 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- consists of a function F : ob(c) ob(d), and for each pair of objects X and Y in C, a morphism C(X, Y ) D(FX, FY ) in V

99 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- consists of a function F : ob(c) ob(d), and for each pair of objects X and Y in C, a morphism C(X, Y ) D(FX, FY ) in V 0 satisfying suitable naturality conditions. 1.12

100 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- consists of a function F : ob(c) ob(d), and for each pair of objects X and Y in C, a morphism C(X, Y ) D(FX, FY ) in V 0 satisfying suitable naturality conditions. A symmetric monoidal category is closed iff it is enriched over itself. 1.12

101 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- consists of a function F : ob(c) ob(d), and for each pair of objects X and Y in C, a morphism C(X, Y ) D(FX, FY ) in V 0 satisfying suitable naturality conditions. A symmetric monoidal category is closed iff it is enriched over itself. When V = (T,, S 0 ), we say, C is a topological category. 1.12

102 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- consists of a function F : ob(c) ob(d), and for each pair of objects X and Y in C, a morphism C(X, Y ) D(FX, FY ) in V 0 satisfying suitable naturality conditions. A symmetric monoidal category is closed iff it is enriched over itself. When V = (T,, S 0 ), we say, C is a topological category. When V = (T G,, S 0 ), we say, C is a topological G-category. 1.12

103 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- consists of a function F : ob(c) ob(d), and for each pair of objects X and Y in C, a morphism C(X, Y ) D(FX, FY ) in V 0 satisfying suitable naturality conditions. A symmetric monoidal category is closed iff it is enriched over itself. When V = (T,, S 0 ), we say, C is a topological category. When V = (T G,, S 0 ), we say, C is a topological G-category. It is also enriched over T G, 1.12

104 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- consists of a function F : ob(c) ob(d), and for each pair of objects X and Y in C, a morphism C(X, Y ) D(FX, FY ) in V 0 satisfying suitable naturality conditions. A symmetric monoidal category is closed iff it is enriched over itself. When V = (T,, S 0 ), we say, C is a topological category. When V = (T G,, S 0 ), we say, C is a topological G-category. It is also enriched over T G, since T G has the same objects as T G, 1.12

105 (continued) A V-category C is underlain by an ordinary category C 0 having the same objects as C and morphism sets C 0 (X, Y ) = V 0 (1, C(X, Y )). A functor F : C D between V- consists of a function F : ob(c) ob(d), and for each pair of objects X and Y in C, a morphism C(X, Y ) D(FX, FY ) in V 0 satisfying suitable naturality conditions. A symmetric monoidal category is closed iff it is enriched over itself. When V = (T,, S 0 ), we say, C is a topological category. When V = (T G,, S 0 ), we say, C is a topological G-category. It is also enriched over T G, since T G has the same objects as T G, and more morphisms. 1.12

106 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. 1.13

107 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G

108 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. 1.13

109 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. 1.13

110 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. For each such embedding we have an orthogonal complement W V, 1.13

111 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. For each such embedding we have an orthogonal complement W V, giving us a vector bundle over O(V, W ). 1.13

112 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. For each such embedding we have an orthogonal complement W V, giving us a vector bundle over O(V, W ). The morphism object J G (V, W ) is its Thom space, 1.13

113 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. For each such embedding we have an orthogonal complement W V, giving us a vector bundle over O(V, W ). The morphism object J G (V, W ) is its Thom space, which is a pointed G-space. 1.13

114 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. For each such embedding we have an orthogonal complement W V, giving us a vector bundle over O(V, W ). The morphism object J G (V, W ) is its Thom space, which is a pointed G-space. Informally, J G (V, W ) is a wedge of spheres S W V 1.13

115 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. For each such embedding we have an orthogonal complement W V, giving us a vector bundle over O(V, W ). The morphism object J G (V, W ) is its Thom space, which is a pointed G-space. Informally, J G (V, W ) is a wedge of spheres S W V (where W V denotes the orthogonal complement of V embedded in W ) 1.13

116 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. For each such embedding we have an orthogonal complement W V, giving us a vector bundle over O(V, W ). The morphism object J G (V, W ) is its Thom space, which is a pointed G-space. Informally, J G (V, W ) is a wedge of spheres S W V (where W V denotes the orthogonal complement of V embedded in W ) parametrized by the orthogonal embeddings V W. 1.13

117 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. For each such embedding we have an orthogonal complement W V, giving us a vector bundle over O(V, W ). The morphism object J G (V, W ) is its Thom space, which is a pointed G-space. Informally, J G (V, W ) is a wedge of spheres S W V (where W V denotes the orthogonal complement of V embedded in W ) parametrized by the orthogonal embeddings V W. Main Definition An orthogonal G-spectrum E is a functor J G T G. 1.13

118 The definition of a G-spectrum We will define spectra as functors to T G from a certain indexing category J G. Both are topological G-. Definition The indexing category J G is the topological G-category whose objects are finite dimensional real orthogonal representations V of G. Let O(V, W ) denote the Stiefel manifold of (possibly nonequivariant) orthogonal embeddings V W. For each such embedding we have an orthogonal complement W V, giving us a vector bundle over O(V, W ). The morphism object J G (V, W ) is its Thom space, which is a pointed G-space. Informally, J G (V, W ) is a wedge of spheres S W V (where W V denotes the orthogonal complement of V embedded in W ) parametrized by the orthogonal embeddings V W. Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. 1.13

119 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. 1.14

120 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. Mike Mandell Peter May 1.14

121 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. Mike Mandell Peter May This definition is due to Mandell May 1.14

122 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. Mike Mandell Peter May This definition is due to Mandell May and can be found in their book, Equivariant orthogonal spectra and S-modules,

123 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. Mike Mandell Peter May This definition is due to Mandell May and can be found in their book, Equivariant orthogonal spectra and S-modules, There are similar definitions by other authors, 1.14

124 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. Mike Mandell Peter May This definition is due to Mandell May and can be found in their book, Equivariant orthogonal spectra and S-modules, There are similar definitions by other authors, such as that of symmetric spectra by Jeff Smith et al in 2000, 1.14

125 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. Mike Mandell Peter May This definition is due to Mandell May and can be found in their book, Equivariant orthogonal spectra and S-modules, There are similar definitions by other authors, such as that of symmetric spectra by Jeff Smith et al in 2000, in which J G is replaced by other symmetric monoidal. 1.14

126 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. 1.15

127 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! 1.15

128 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). 1.15

129 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), 1.15

130 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, 1.15

131 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. 1.15

132 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), 1.15

133 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, 1.15

134 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, 1.15

135 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, the orthogonal group (equipped with a G-action) 1.15

136 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, the orthogonal group (equipped with a G-action) with a disjoint base point. 1.15

137 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, the orthogonal group (equipped with a G-action) with a disjoint base point. When dim(v ) = 0, 1.15

138 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, the orthogonal group (equipped with a G-action) with a disjoint base point. When dim(v ) = 0, the embedding space is a point, 1.15

139 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, the orthogonal group (equipped with a G-action) with a disjoint base point. When dim(v ) = 0, the embedding space is a point, so J G (0, W ) = S W, 1.15

140 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, the orthogonal group (equipped with a G-action) with a disjoint base point. When dim(v ) = 0, the embedding space is a point, so J G (0, W ) = S W, the one point compactification of W. 1.15

141 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, the orthogonal group (equipped with a G-action) with a disjoint base point. When dim(v ) = 0, the embedding space is a point, so J G (0, W ) = S W, the one point compactification of W. When dim(v ) = 1, 1.15

142 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, the orthogonal group (equipped with a G-action) with a disjoint base point. When dim(v ) = 0, the embedding space is a point, so J G (0, W ) = S W, the one point compactification of W. When dim(v ) = 1, the embedding space is the unit sphere S(W ), 1.15

143 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. This definition requires some unpacking! First we examine the indexing spaces J G (V, W ). When dim(v ) > dim(w ), the embedding space O(V, W ) is empty, so J G (V, W ) =. When dim(v ) = dim(w ), the vector bundle is 0-dimensional, so J G (V, W ) = O(V, W ) +, the orthogonal group (equipped with a G-action) with a disjoint base point. When dim(v ) = 0, the embedding space is a point, so J G (0, W ) = S W, the one point compactification of W. When dim(v ) = 1, the embedding space is the unit sphere S(W ), and J G (V, W ) is its tangent Thom space. 1.15

144 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. 1.16

145 The definition of a G-spectrum (continued) Main Definition An orthogonal G-spectrum E is a functor J G T G. We will denote its value on V by E V. There are equivariant structure maps J G (V, W ) J G (U, V ) J G (U, W ) (composition in J G ) 1.16