//img.uscri.be/pth/99294efcc4c8d3e04a5ae6d1c2829b02a722c03d
Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

The nerve theorem and Grothendieck's hypothesis on homotopy types

De
61 pages
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


Voir plus Voir moins
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 Θn
-spaces
The nerve theorem and Grothendieck’s hypothesis on homotopy types Monadic squares
Amonadic squareis a commutative diagram of functors E0G0E20 1
U1 E1
=
U2 E2 G
such that (i)U1U2are monadic functors with left adjointsF1F2; (ii)the induced 2-cellφ=2G0F1F2Gη1(the “mate”) 0 E1G0E02 ✻ ✻ F1φF2 E1GE2
is invertible.
The nerve theorem and Grothendieck’s hypothesis on homotopy types Monadic squares
Amonadic squareis a commutative diagram of functors 0 E01GE02
U1 E1
=
U2 GE2
such that (i)U1U2are monadic functors with left adjointsF1F2; (ii)the induced 2-cellφ=2G0F1F2Gη1(the “mate”) 0G0 E1E02 ✻ ✻ F1φF2 E1GE2
is invertible.
The nerve theorem and Grothendieck’s hypothesis on homotopy types Monadic squares
Amonadic squareis a commutative diagram of functors E01G0E02
U1 E1
=
U2 GE2
such that (i)U1U2are monadic functors with left adjointsF1F2; (ii)the induced 2-cellφ=2G0F1F2Gη1(the “mate”) E1G0E2 0 0 ✻ ✻ F1φF2 E1GE2
is invertible.
The nerve theorem and Grothendieck’s hypothesis on homotopy types Monadic squares
Let (T1 µ1 η1)(T2 µ2 η2) be monads onE1E2respectively. A(strong) monad morphism(G ψ) : (E1T1)(E2T2) is a functorG:E1→ E2together with an (invertible) 2-cell ψ:T2GGT1such thatGη1=ψη2Gand ψµ2G=Gµ1GψT1T2ψG. A strong monad morphism (G ψ) induces a monadic square
AlgT1
U1 E1
G0 AlgT2
=U2 GE2
withG0(X ξ:T1XX) = (GXGξψ:T2GXGX) Conversely, a monadic square induces a strong monad morphism from which it derives up to canonical equivalence.