Operads and Cochain Models in Algebraic Topology

Operads and Cochain Models in Algebraic Topology

Benoit Fresse

Laboratoire Paul Painlev´e - Universit´e de Lille

23 April 2009

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX, and the dual complex

N^∗(X) = HomdgkMod(N∗(X),k). For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative, but not symmetric though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1}

we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX, and the dual complex

N^∗(X) = HomdgkMod(N∗(X),k). For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative, but not symmetric though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices,

the normalized chain complex ofX, and the dual complex N^∗(X) = HomdgkMod(N∗(X),k).

For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative, but not symmetric though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX,

and the dual complex N^∗(X) = HomdgkMod(N∗(X),k).

For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative, but not symmetric though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX, and the dual complex

N^∗(X) = HomdgkMod(N∗(X),k).

For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative, but not symmetric though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX, and the dual complex

N^∗(X) = HomdgkMod(N∗(X),k).

For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y)

defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative, but not symmetric though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX, and the dual complex

N^∗(X) = HomdgkMod(N∗(X),k).

For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative, but not symmetric though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX, and the dual complex

N^∗(X) = HomdgkMod(N∗(X),k).

For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital,

associative, but not symmetric though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX, and the dual complex

N^∗(X) = HomdgkMod(N∗(X),k).

For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative,

but not symmetric though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX, and the dual complex

N^∗(X) = HomdgkMod(N∗(X),k).

For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative, but not symmetric

though it becomes apparently symmetric in homology.

Reminder

For a simplicial set

X ={X_n,d_i :X_n→Xn−1,s_j :X_n→X_n+1} we form

N_d(X) =k{X_d}/degenerate simplices, the normalized chain complex ofX, and the dual complex

N^∗(X) = HomdgkMod(N∗(X),k).

For simplicial setsX andY, we have a map

AW :N∗(X ×Y)→N∗(X)⊗N∗(Y) defined by

AW(x×y) =

n

X

m=0

x(0. . .m)⊗y(m. . .n) for anyx×y ∈X_d×Y_d.

Issue: this mapAW is unital, associative, but not symmetric though it becomes apparently symmetric in homology.

Options

1. Explain that theE_m-operadEm, is in a sense Koszul self-dual, so that

H_∗^Em(A) =H∗(Dm(A), ∂),

where Dm = Λ^−mE^∨m, and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

2. Revisit the definition of the bar complex of associative algebras, explain that any algebra over an E_n-operad has a well-defined iterated bar complexB^m(A), for every m≤n, such that

H_∗^Em(A) =H∗(B^m(A)), and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

Options

1. Explain that theE_m-operadEm, is in a sense Koszul self-dual, so that

H_∗^Em(A) =H∗(Dm(A), ∂), where Dm = Λ^−mE^∨m,

and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

2. Revisit the definition of the bar complex of associative algebras, explain that any algebra over an E_n-operad has a well-defined iterated bar complexB^m(A), for every m≤n, such that

H_∗^Em(A) =H∗(B^m(A)), and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

Options

1. Explain that theE_m-operadEm, is in a sense Koszul self-dual, so that

H_∗^Em(A) =H∗(Dm(A), ∂), where Dm = Λ^−mE^∨m, and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) =

H¯_p−prof^∗ (Σ^mΩ^mX).

2. Revisit the definition of the bar complex of associative algebras, explain that any algebra over an E_n-operad has a well-defined iterated bar complexB^m(A), for every m≤n, such that

H_∗^Em(A) =H∗(B^m(A)), and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

Options

1. Explain that theE_m-operadEm, is in a sense Koszul self-dual, so that

H_∗^Em(A) =H∗(Dm(A), ∂), where Dm = Λ^−mE^∨m, and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

2. Revisit the definition of the bar complex of associative algebras, explain that any algebra over an E_n-operad has a well-defined iterated bar complexB^m(A), for every m≤n, such that

H_∗^Em(A) =H∗(B^m(A)), and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

Options

1. Explain that theE_m-operadEm, is in a sense Koszul self-dual, so that

H_∗^Em(A) =H∗(Dm(A), ∂), where Dm = Λ^−mE^∨m, and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

2. Revisit the definition of the bar complex of associative algebras,

explain that any algebra over an E_n-operad has a well-defined iterated bar complexB^m(A), for every m≤n, such that

H_∗^Em(A) =H∗(B^m(A)), and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

Options

1. Explain that theE_m-operadEm, is in a sense Koszul self-dual, so that

H_∗^Em(A) =H∗(Dm(A), ∂), where Dm = Λ^−mE^∨m, and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

2. Revisit the definition of the bar complex of associative algebras, explain that any algebra over an E_n-operad has a well-defined iterated bar complexB^m(A), for every m≤n,

such that

H_∗^Em(A) =H∗(B^m(A)), and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

Options

1. Explain that theE_m-operadEm, is in a sense Koszul self-dual, so that

H_∗^Em(A) =H∗(Dm(A), ∂), where Dm = Λ^−mE^∨m, and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

2. Revisit the definition of the bar complex of associative algebras, explain that any algebra over an E_n-operad has a well-defined iterated bar complexB^m(A), for every m≤n, such that

H_∗^Em(A) =H∗(B^m(A)),

and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

Options

1. Explain that theE_m-operadEm, is in a sense Koszul self-dual, so that

H_∗^Em(A) =H∗(Dm(A), ∂), where Dm = Λ^−mE^∨m, and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

2. Revisit the definition of the bar complex of associative algebras, explain that any algebra over an E_n-operad has a well-defined iterated bar complexB^m(A), for every m≤n, such that

H_∗^Em(A) =H∗(B^m(A)), and for a cochain algebra, we have:

H_∗^Em( ¯N^∗(X)) = ¯H_p−prof^∗ (Σ^mΩ^mX).

References

1.

I “Homology of partition posets and Koszul duality of operads”, in Contemporary Mathematics346, American Mathematical Society.

I “Koszul duality ofEn-operads”.

2.

I “Modules over operads and functors”, Lecture Notes in Mathematics, Springer-Verlag.

I “The bar complex series”.

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