The nerve theorem and Grothendieck's hypothesis on homotopy types

Clemens Berger

Description

The nerve theorem and Grothendieck's hypothesis on homotopy types The nerve theorem and Grothendieck's hypothesis on homotopy types Clemens Berger University of Nice CT2010 Genova, June 20-26, 2010

• monadic squares


• ct2010 genova


• cell ?


• grothendieck's hypothesis


• commutative diagram


• higher categories

Sujets

• Berger
• Commutative diagram
• Higher category theory

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

The nerve theorem and Grothendieck’s hypothesis on homotopy types

The

nerve

theorem and Grothendieck’s on homotopy types

Clemens

Berger

University of Nice

CT2010 Genova, June 20-26, 2010

hypothesis

The nerve theorem and Grothendieck’s hypothesis on homotopy types

1

2

3

4

Monadic squares

Nerves and theories

Higher categories and wreath products

Grothendieck’s hypothesis and Θ n

-spaces

The nerve theorem and Grothendieck’s hypothesis on homotopy types Monadic squares

Amonadic squareis a commutative diagram of functors E 0 G 0 E 2 0 1 ✲

U 1 ❄ E 1

=

U 2 ❄ ✲ E 2 G

such that (i)U 1 U 2 are monadic functors with left adjointsF 1 F 2 ; (ii)the induced 2-cellφ= 2 G 0 F 1 ◦F 2 Gη 1 (the “mate”) 0 E 1 G 0 ✲ E 0 2 ✻ ✻ F 1 φ ⇐F 2 E 1 G ✲ E 2

is invertible.

The nerve theorem and Grothendieck’s hypothesis on homotopy types Monadic squares

Amonadic squareis a commutative diagram of functors 0 E 0 1 G E 0 2 ✲

U 1 ❄ E 1

=

U 2 ❄ G ✲ E 2

such that (i)U 1 U 2 are monadic functors with left adjointsF 1 F 2 ; (ii)the induced 2-cellφ= 2 G 0 F 1 ◦F 2 Gη 1 (the “mate”) 0 G 0 E 1 ✲ E 0 2 ✻ ✻ F 1 φ ⇐F 2 E 1 G ✲ E 2

is invertible.

The nerve theorem and Grothendieck’s hypothesis on homotopy types Monadic squares

Amonadic squareis a commutative diagram of functors E 0 1 G 0 ✲ E 0 2

U 1 ❄ E 1

=

U 2 ❄ G ✲ E 2

such that (i)U 1 U 2 are monadic functors with left adjointsF 1 F 2 ; (ii)the induced 2-cellφ= 2 G 0 F 1 ◦F 2 Gη 1 (the “mate”) E 1 G 0 ✲ E 2 0 0 ✻ ✻ F 1 φ ⇐F 2 E 1 G ✲ E 2

is invertible.

The nerve theorem and Grothendieck’s hypothesis on homotopy types Monadic squares

Let (T 1  µ 1  η 1 )(T 2  µ 2  η 2 ) be monads onE 1 E 2 respectively. A(strong) monad morphism(G ψ) : (E 1 T 1 )→(E 2 T 2 ) is a functorG:E 1 → E 2 together with an (invertible) 2-cell ψ:T 2 G⇒GT 1 such thatGη 1 =ψ◦η 2 Gand ψ◦µ 2 G=Gµ 1 ◦GψT 1 ◦T 2 ψG. A strong monad morphism (G ψ) induces a monadic square

Alg T 1

U 1 ❄ E 1

G 0 ✲ Alg T 2

=U 2 ❄ ✲ G E 2

withG 0 (X ξ:T 1 X→X) = (GXGξ◦ψ:T 2 GX→GX) Conversely, a monadic square induces a strong monad morphism from which it derives up to canonical equivalence.