Swiss-Cheese action on the totalization of operads under the monoid - PowerPoint PPT Presentation

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Definition A coloured operad over the set of colours S , or S-operad , is given by: a family of spaces { O ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S an operadic composition ◦ i : O ( s 1 , .., s n ; s n +1 ) × O ( s ′ 1 , .., s ′ m ; s i ) → O ( s 1 , .., s i − 1 , s ′ 1 , .., s ′ m , s i +1 , .., s n ; s n +1 ) distinguished elements {∗ s ∈ O ( s ; s ) } . Definition An O -space is a family of spaces { X s } s ∈ S with maps O ( s 1 , . . . , s n ; s n +1 ) × X s 1 × · · · × X s n → X s n +1 If S = { o ; c } then O n ; c ) and O n c = O ( n ; c ) = O ( c , . . . , c o = O ( n + 1; o ) = O ( c , . . . , c , o ; o ) � �� � � �� � n n

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Example: The { o ; c } -operad of unital monoid actions A ct n n − 1 A ct ( n ; c ) = ∗ n ; c for n ≥ 0 ���� ���� A ct ( n ; o ) = ∗ n ; o for n > 0 the empty set otherwise with the obvious operadic composition

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Example: The { o ; c } -operad of unital monoid actions A ct A ct ( n ; c ) = ∗ n ; c for n ≥ 0 A ct ( n ; o ) = ∗ n ; o for n > 0 ∈ A ct / the empty set otherwise with the obvious operadic composition

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Example: The { o ; c } -operad of unital monoid actions A ct A ct ( n ; c ) = ∗ n ; c for n ≥ 0 A ct ( n ; o ) = ∗ n ; o for n > 0 ∈ A ct / the empty set otherwise with the obvious operadic composition X c X o Topological Left module monoid over X c

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Example: The { o ; c } -operad of unital monoid actions A ct A ct ( n ; c ) = ∗ n ; c for n ≥ 0 A ct ( n ; o ) = ∗ n ; o for n > 0 ∈ A ct / the empty set otherwise with the obvious operadic composition Definition An { o ; c } -operad O is pointed if there is a map of operads from A ct to O .

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Example: The { o ; c } -operad of unital monoid actions A ct A ct ( n ; c ) = ∗ n ; c for n ≥ 0 A ct ( n ; o ) = ∗ n ; o for n > 0 ∈ A ct / the empty set otherwise with the obvious operadic composition Definition An { o ; c } -operad O is pointed if there is a map of operads from A ct to O . Example: The { o ; c } -operad of monoid actions A ct > 0 A ct > 0 ( n ; c ) = ∗ n ; c for n > 0 A ct > 0 ( n ; o ) = ∗ n ; o for n > 0 ∈ A ct > 0 / the empty set otherwise with the obvious operadic composition

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Example: The Swiss-Cheese operad SC d If the output is ”close” then all the inputs are also ”close” and SC d ( c , . . . , c ; c ) = C d ( n ). � �� � n If the output is ”open” then we denote by SC d ( n 1 , n 2 ; o ) the space of configurations of nonoverlapping n 1 little disks and n 2 upper semidisks labeled by { 1 , . . . , n 1 + n 2 } in the unit upper semidisk so that the semidisks are all centered about the diameter of the unit semidisk. SC 2 (1 , 2; o )

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Example: The Swiss-Cheese operad SC d If the output is ”close” then all the inputs are also ”close” and SC d ( c , . . . , c ; c ) = C d ( n ). � �� � n If the output is ”open” then we denote by SC d ( n 1 , n 2 ; o ) the space of configurations of nonoverlapping n 1 little disks and n 2 upper semidisks labeled by { 1 , . . . , n 1 + n 2 } in the unit upper semidisk so that the semidisks are all centered about the diameter of the unit semidisk. SC 2 (1 , 2; o ) SC 2 (2; c ) SC 2 (2 , 2; o )

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Example: The Swiss-Cheese operad SC d If the output is ”close” then all the inputs are also ”close” and SC d ( c , . . . , c ; c ) = C d ( n ). � �� � n If the output is ”open” then we denote by SC d ( n 1 , n 2 ; o ) the space of configurations of nonoverlapping n 1 little disks and n 2 upper semidisks labeled by { 1 , . . . , n 1 + n 2 } in the unit upper semidisk so that the semidisks are all centered about the diameter of the unit semidisk. SC 2 (1 , 2; o ) SC 2 (1 , 2; o ) SC 2 (2 , 3; o )

Bimodules and Ibimodules Model Structures Main result and applications Coloured operads Example: The Swiss-Cheese operad SC d If the output is ”close” then all the inputs are also ”close” and SC d ( c , . . . , c ; c ) = C d ( n ). � �� � n If the output is ”open” then we denote by SC d ( n 1 , n 2 ; o ) the space of configurations of nonoverlapping n 1 little disks and n 2 upper semidisks labeled by { 1 , . . . , n 1 + n 2 } in the unit upper semidisk so that the semidisks are all centered about the diameter of the unit semidisk. Remark If X is a pointed space and A is a subspace containing the based point then � � � f ( ∂ [0; 1] d ) ⊂ A if t d = 0 f : [0; 1] d → X � Ω d ( X ; A ) = � f ( ∂ [0; 1] d ) = ∗ otherwise � and the pair (Ω d X ; Ω d ( X ; A )) is an algebra over SC d .

Bimodules and Ibimodules Model Structures Main result and applications Bimodules over an S -operad Definition A bimodule M over an S -operad O , or O-bimodule , is given by a family of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S right operations ◦ i : M ( s 1 , .., s n ; s n +1 ) × O ( s ′ 1 , .., s ′ m ; s i ) → M ( s 1 , .., s i − 1 , s ′ 1 , .., s ′ m , s i +1 , .., s n ; s n +1 )

Bimodules and Ibimodules Model Structures Main result and applications Bimodules over an S -operad Definition A bimodule M over an S -operad O , or O-bimodule , is given by a family of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S right operations ◦ i : M ( s 1 , .., s n ; s n +1 ) × O ( s ′ 1 , .., s ′ m ; s i ) → M ( s 1 , .., s i − 1 , s ′ 1 , .., s ′ m , s i +1 , .., s n ; s n +1 ) left operations γ l : O ( s 1 , .., s n ; s n +1 ) × M ( s 1 1 , .., s 1 p 1 ; s 1 ) × .. × M ( s n 1 , .., s n p n ; s n ) → M ( s 1 1 , .., s n p n ; s n +1 )

Bimodules and Ibimodules Model Structures Main result and applications Bimodules over an S -operad Definition A bimodule M over an S -operad O , or O-bimodule , is given by a family of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S right operations ◦ i : M ( s 1 , .., s n ; s n +1 ) × O ( s ′ 1 , .., s ′ m ; s i ) → M ( s 1 , .., s i − 1 , s ′ 1 , .., s ′ m , s i +1 , .., s n ; s n +1 ) left operations γ l : O ( s 1 , .., s n ; s n +1 ) × M ( s 1 1 , .., s 1 p 1 ; s 1 ) × .. × M ( s n 1 , .., s n p n ; s n ) → M ( s 1 1 , .., s n p n ; s n +1 ) satisfying some axioms. Example If η : A ct → O is a map of { o ; c } -operads then η is also a map of A ct > 0 -bimodules.

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Definition An infinitesimal bimodule M over an S -operad O , or O-Ibimodule , is: a family of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S right operations ◦ i : M ( s 1 , .., s n ; s n +1 ) × O ( s ′ 1 , .., s ′ m ; s i ) → M ( s 1 , .., s i − 1 , s ′ 1 , .., s ′ m , s i +1 , .., s n ; s n +1 )

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Definition An infinitesimal bimodule M over an S -operad O , or O-Ibimodule , is: a family of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S right operations ◦ i : M ( s 1 , .., s n ; s n +1 ) × O ( s ′ 1 , .., s ′ m ; s i ) → M ( s 1 , .., s i − 1 , s ′ 1 , .., s ′ m , s i +1 , .., s n ; s n +1 ) left operations ◦ i : O ( s 1 , .., s n ; s n +1 ) × M ( s ′ 1 , .., s ′ m ; s i ) → M ( s 1 , .., s i − 1 , s ′ 1 , .., s ′ m , s i +1 , .., s n ; s n +1 )

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Definition An infinitesimal bimodule M over an S -operad O , or O-Ibimodule , is: a family of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S right operations ◦ i : M ( s 1 , .., s n ; s n +1 ) × O ( s ′ 1 , .., s ′ m ; s i ) → M ( s 1 , .., s i − 1 , s ′ 1 , .., s ′ m , s i +1 , .., s n ; s n +1 ) left operations ◦ i : O ( s 1 , .., s n ; s n +1 ) × M ( s ′ 1 , .., s ′ m ; s i ) → M ( s 1 , .., s i − 1 , s ′ 1 , .., s ′ m , s i +1 , .., s n ; s n +1 ) satisfying some axioims. Example If η : A ct → M is a map between A ct > 0 -bimodules then M is also an A ct > 0 -Ibimodule.

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o .

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o .

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . ∈ A ct > 0 (2; c )

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o .

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o .

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . ← n operations { d 1 , . . . , d n }

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . ← n operations { d 1 , . . . , d n } ← 2 operations { d 0 , d n +1 }

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o .

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o .

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o .

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . ∈ A ct > 0 (2; o )

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . ← n + 1 operations { d 1 , . . . , d n +1 }

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . ← n + 1 operations { d 1 , . . . , d n +1 } ← 1 operation d 0

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o .

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . ∈ M n o = M ( n + 1; o )

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . Corollary 1 If η : A ct → M is a map of A ct > 0 -bimodules then: there exists a product: M n c × M m c → M n + m c and an action: M n c × M m o → M n + m o

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . Corollary 1 If η : A ct → M is a map of A ct > 0 -bimodules then: there exists a product: M n c × M m c → M n + m c and an action: M n c × M m o → M n + m o ∈ A ct > 0 (2; c )

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . Corollary 1 If η : A ct → M is a map of A ct > 0 -bimodules then: there exists a product: M n c × M m c → M n + m c and an action: M n c × M m o → M n + m o ∈ A ct > 0 (2; o )

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . Corollary 2 If η : A ct → M is a map of { o ; c } -operads then: M c is a multiplicative operad there exists an action: M n c × M m o → M n + m o and a product: M n o × M m o → M n + m o

Bimodules and Ibimodules Model Structures Main result and applications Infinitesimal bimodules over an S -operad Theorem (D) If M is an A ct > 0 -Ibimodule then M • c := { M n c } and M • o := { M n o } are semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : M c → M o . Corollary 2 If η : A ct → M is a map of { o ; c } -operads then: M c is a multiplicative operad there exists an action: M n c × M m o → M n + m o and a product: M n o × M m o → M n + m o

Bimodules and Ibimodules Model Structures Main result and applications Model category L : C 1 ⇄ C 2 : R A model category cofibrantly generated ( I , J )

Bimodules and Ibimodules Model Structures Main result and applications Model category L : C 1 ⇄ C 2 : R A model category A model category cofibrantly generated cofibrantly generated ( I , J ) ( LI , LJ )

Bimodules and Ibimodules Model Structures Main result and applications Free S -operad F : Coll ( S ) ⇄ Operad S : U • objects are families of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N Coll ( S ) : s i ∈ S • maps are families of continuous maps

� � � Bimodules and Ibimodules Model Structures Main result and applications Free S -operad F : Coll ( S ) ⇄ Operad S : U • objects are families of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N Coll ( S ) : s i ∈ S • maps are families of continuous maps T op has a cofibrantly generated model category : f : X → Y is a weak equivalence iff f ∗ n : π n ( X ) → π n ( Y ) are isomorphisms f : X → Y is a ”Serre” fibration iff for every CW -complex A there is a lift in every commutative diagram: A × { 0 } X � Y A × [0 ; 1]

� � � Bimodules and Ibimodules Model Structures Main result and applications Free S -operad F : Coll ( S ) ⇄ Operad S : U • objects are families of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N Coll ( S ) : s i ∈ S • maps are families of continuous maps T op has a cofibrantly generated model category : f : X → Y is a weak equivalence iff f ∗ n : π n ( X ) → π n ( Y ) are isomorphisms f : X → Y is a ”Serre” fibration iff for every CW -complex A there is a lift in every commutative diagram: A × { 0 } X � Y A × [0 ; 1] k → ∆ n | n ∈ N , k ≤ n } I = { ∂ ∆ n → ∆ n | n ∈ N } J = {∧ n all the objects are fibrant

� � � Bimodules and Ibimodules Model Structures Main result and applications Free S -operad F : Coll ( S ) ⇄ Operad S : U • objects are families of spaces { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N Coll ( S ) : s i ∈ S • maps are families of continuous maps T op has a cofibrantly generated model category : f : X → Y is a weak equivalence iff f ∗ n : π n ( X ) → π n ( Y ) are isomorphisms f : X → Y is a ”Serre” fibration iff for every CW -complex A there is a lift in every commutative diagram: A × { 0 } X � Y A × [0 ; 1] k → ∆ n | n ∈ N , k ≤ n } I = { ∂ ∆ n → ∆ n | n ∈ N } J = {∧ n all the objects are fibrant ⇒ Coll ( S ) inherits a cofibrantly generated model category.

Bimodules and Ibimodules Model Structures Main result and applications Free S -operad F : Coll ( S ) ⇄ Operad S : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. For instance a point in F ( M )( s 1 , s 2 , s 3 , s 4 ; s 5 ) is given by: s 1 s 2 s 3 s 4 s 7 s 6 s 1 s 8 s 5

Bimodules and Ibimodules Model Structures Main result and applications Free S -operad F : Coll ( S ) ⇄ Operad S : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. For instance a point in F ( M )( s 1 , s 2 , s 3 , s 4 ; s 5 ) is given by: s 1 s 2 s 3 s 4 M ( ; s 7 ) M ( s 2 , s 3 ; s 6 ) s 7 s 6 ∗ s 1 M ( s 6 , s 4 , s 7 ; s 8 ) s 1 s 8 M ( s 1 , s 8 ; s 5 ) s 5

Bimodules and Ibimodules Model Structures Main result and applications Free S -operad F : Coll ( S ) ⇄ Operad S : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. The operadic composition: F ( M )( s 1 , s 2 , s 3 : s 4 ) F ( M )( s 6 , s 7 : s 1 ) F ( M )( s 6 , s 7 , s 2 , s 3 : s 4 )

Bimodules and Ibimodules Model Structures Main result and applications Free bimodule over an S -operad O B O : Coll ( S ) ⇄ Bimod O : U

Bimodules and Ibimodules Model Structures Main result and applications Free bimodule over an S -operad O B O : Coll ( S ) ⇄ Bimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. For instance a point in B O ( M )( s 1 , s 2 , s 3 ; s 4 ) is given by:

Bimodules and Ibimodules Model Structures Main result and applications Free bimodule over an S -operad O B O : Coll ( S ) ⇄ Bimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. For instance a point in B O ( M )( s 1 , s 2 , s 3 ; s 4 ) is given by:

Bimodules and Ibimodules Model Structures Main result and applications Free bimodule over an S -operad O B O : Coll ( S ) ⇄ Bimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. For instance a point in B O ( M )( s 1 , s 2 , s 3 ; s 4 ) is given by:

Bimodules and Ibimodules Model Structures Main result and applications Free bimodule over an S -operad O B O : Coll ( S ) ⇄ Bimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. For instance a point in B O ( M )( s 1 , s 2 , s 3 ; s 4 ) is given by:

Bimodules and Ibimodules Model Structures Main result and applications Free bimodule over an S -operad O B O : Coll ( S ) ⇄ Bimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. The right operations ◦ i : B O ( M )( s 1 , s 2 , s 3 ; s 4 ) O ( s 6 , s 7 ; s 1 ) B O ( M )( s 6 , s 7 , s 2 , s 3 ; s 4 )

Bimodules and Ibimodules Model Structures Main result and applications Free bimodule over an S -operad O B O : Coll ( S ) ⇄ Bimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. The right operations ◦ i : B O ( M )( s 1 , s 2 , s 3 ; s 4 ) O ( s 6 , s 7 ; s 1 ) B O ( M )( s 1 , s 6 , s 7 , s 3 ; s 4 )

Bimodules and Ibimodules Model Structures Main result and applications Free bimodule over an S -operad O B O : Coll ( S ) ⇄ Bimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. The left operations γ l :

Bimodules and Ibimodules Model Structures Main result and applications Free bimodule over an S -operad O B O : Coll ( S ) ⇄ Bimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. The left operations γ l :

Bimodules and Ibimodules Model Structures Main result and applications Free infinitesimal bimodule over an S -operad O I b O : Coll ( S ) ⇄ Ibimod O : U

Bimodules and Ibimodules Model Structures Main result and applications Free infinitesimal bimodule over an S -operad O I b O : Coll ( S ) ⇄ Ibimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. For instance a point in I b O ( M )( s 1 , s 2 , s 3 , s 4 ; s 5 ) is given by:

Bimodules and Ibimodules Model Structures Main result and applications Free infinitesimal bimodule over an S -operad O I b O : Coll ( S ) ⇄ Ibimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. For instance a point in I b O ( M )( s 1 , s 2 , s 3 , s 4 ; s 5 ) is given by:

Bimodules and Ibimodules Model Structures Main result and applications Free infinitesimal bimodule over an S -operad O I b O : Coll ( S ) ⇄ Ibimod O : U Let { M ( s 1 , . . . , s n ; s n +1 ) } n ∈ N s i ∈ S be a family of spaces. The left operations ◦ i : O ( s 1 , s 2 ; s 3 ) I b O ( M )( s 4 , s 5 , s 6 ; s 2 ) I b O ( M )( s 1 , s 4 , s 5 , s 6 ; s 3 )

Bimodules and Ibimodules Model Structures Main result and applications Cofibrant replacements � A s > 0 − Ibimodules � A s > 0 − Ibimodules maps ⇔ � semi-cosimplicial spaces � semi-cosimplicial maps

Bimodules and Ibimodules Model Structures Main result and applications Cofibrant replacements � A s > 0 − Ibimodules � A s > 0 − Ibimodules maps ⇔ � semi-cosimplicial spaces � semi-cosimplicial maps Theorem (V.Turchin) A cofibrant replacement of A s as an A s > 0 -Ibimodule is given by the semi-cosimplicial space ∆.

Bimodules and Ibimodules Model Structures Main result and applications Cofibrant replacements � A s > 0 − Ibimodules � A s > 0 − Ibimodules maps ⇔ � semi-cosimplicial spaces � semi-cosimplicial maps Theorem (V.Turchin) A cofibrant replacement of A s as an A s > 0 -Ibimodule is given by the semi-cosimplicial space ∆. � A ct > 0 − Ibimodule M ⇒ � a semi-cosimplicial map h : M • c → M • o

Bimodules and Ibimodules Model Structures Main result and applications Cofibrant replacements � A s > 0 − Ibimodules � A s > 0 − Ibimodules maps ⇔ � semi-cosimplicial spaces � semi-cosimplicial maps Theorem (V.Turchin) A cofibrant replacement of A s as an A s > 0 -Ibimodule is given by the semi-cosimplicial space ∆. � A ct > 0 − Ibimodule M ⇒ � a semi-cosimplicial map h : M • c → M • o Consequence If M is an A ct > 0 -Ibimodule then: Ibimod h A s > 0 ( A s ; M c ) ≃ Ibimod A s > 0 (∆; M c ) = Nat (∆ ; M c ) = sTot ( M c ) Ibimod h A s > 0 ( A s ; M o ) ≃ Ibimod A s > 0 (∆; M o ) = Nat (∆ ; M o ) = sTot ( M o )

Bimodules and Ibimodules Model Structures Main result and applications Cofibrant replacements � A ct > 0 − Ibimodule M ⇒ � a semi-cosimplicial map h : M • c → M • o Theorem (D) △ of A ct as an A ct > 0 -Ibimodule is given by the A cofibrant replacement � pair of semi-cosimplicial spaces: △ ( n + 1; o ) = ∆ n × [0 ; 1] △ ( n ; c ) = ∆ n � and � where the semi-cosimplicial map h : ∆ n → ∆ n × [0 ; 1] is defined by: ( t 1 ≤ · · · ≤ t n ) �→ ( t 1 ≤ · · · ≤ t n ) × { 1 }

Bimodules and Ibimodules Model Structures Main result and applications Cofibrant replacements � A ct > 0 − Ibimodule M ⇒ � a semi-cosimplicial map h : M • c → M • o Theorem (D) △ of A ct as an A ct > 0 -Ibimodule is given by the A cofibrant replacement � pair of semi-cosimplicial spaces: △ ( n + 1; o ) = ∆ n × [0 ; 1] △ ( n ; c ) = ∆ n � and � where the semi-cosimplicial map h : ∆ n → ∆ n × [0 ; 1] is defined by: ( t 1 ≤ · · · ≤ t n ) �→ ( t 1 ≤ · · · ≤ t n ) × { 1 } Theorem (D) Let M be an A ct > 0 -Ibimodule. One has: Ibimod h △ ; M ) A ct > 0 ( A ct ; M ) ≃ Ibimod A ct > 0 ( � ≃ Ibimod A s > 0 (∆; M c ) ≃ sTot ( M c )

Bimodules and Ibimodules Model Structures Main result and applications Cofibrant replacements � M is an A ct > 0 − bimodule ⇒ � M c is an A s > 0 − bimodule Theorem (V.Turchin) If η : A ct → M is a map of A ct > 0 -bimodules and M (0; c ) ≃ ∗ then: sTot ( M c ) ≃ Ω Bimod h A s > 0 ( A s > 0 ; M c ) sTot ( M o ) ≃ Ω???

Bimodules and Ibimodules Model Structures Main result and applications Cofibrant replacements � M is an A ct > 0 − bimodule ⇒ � M c is an A s > 0 − bimodule Theorem (V.Turchin) If η : A ct → M is a map of A ct > 0 -bimodules and M (0; c ) ≃ ∗ then: sTot ( M c ) ≃ Ω Bimod h A s > 0 ( A s > 0 ; M c ) sTot ( M o ) ≃ Ω??? � M is an { o ; c } − operad pointed ⇒ � M c is a multiplicative operad � M o is an A s > 0 − bimodule

Bimodules and Ibimodules Model Structures Main result and applications Cofibrant replacements � M is an A ct > 0 − bimodule ⇒ � M c is an A s > 0 − bimodule Theorem (V.Turchin) If η : A ct → M is a map of A ct > 0 -bimodules and M (0; c ) ≃ ∗ then: sTot ( M c ) ≃ Ω Bimod h A s > 0 ( A s > 0 ; M c ) sTot ( M o ) ≃ Ω??? � M is an { o ; c } − operad pointed ⇒ � M c is a multiplicative operad � M o is an A s > 0 − bimodule Theorem (V.Turchin) If η : A ct → M is a map of { o ; c } -operads and M 0 c ≃ M 1 c ≃ M 0 o ≃ ∗ then: sTot ( M c ) ≃ Ω 2 Operad h ( A s > 0 ; M c ) A s > 0 ( A s > 0 ; M o ) ≃ Ω 2 ??? sTot ( M o ) ≃ Ω Bimod h

Bimodules and Ibimodules Model Structures Main result and applications Main result Theorem (D) If η : A ct → M is an A ct > 0 -bimodules map then sTot ( M o ) is weakly equivalent to: � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h Ω A ct > 0 ( A ct > 0 ; M )

Bimodules and Ibimodules Model Structures Main result and applications Main result Theorem (D) If η : A ct → M is an A ct > 0 -bimodules map then sTot ( M o ) is weakly equivalent to: � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h Ω A ct > 0 ( A ct > 0 ; M ) Sketch of proof: Determine a cofibrant replacement � of A ct > 0 as an A ct > 0 -bimodule

Bimodules and Ibimodules Model Structures Main result and applications Main result Theorem (D) If η : A ct → M is an A ct > 0 -bimodules map then sTot ( M o ) is weakly equivalent to: � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h Ω A ct > 0 ( A ct > 0 ; M ) Sketch of proof: Determine a cofibrant replacement � of A ct > 0 as an A ct > 0 -bimodule Determine a sub-operad M ∗ such that: sTot ( M o ) ≃ Bimod h A ct > 0 ( A ct > 0 ; M ∗ )

Bimodules and Ibimodules Model Structures Main result and applications Main result Theorem (D) If η : A ct → M is an A ct > 0 -bimodules map then sTot ( M o ) is weakly equivalent to: � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h Ω A ct > 0 ( A ct > 0 ; M ) Sketch of proof: Determine a cofibrant replacement � of A ct > 0 as an A ct > 0 -bimodule Determine a sub-operad M ∗ such that: sTot ( M o ) ≃ Bimod h A ct > 0 ( A ct > 0 ; M ∗ ) Prove that Bimod h A ct > 0 ( A ct > 0 ; M ∗ ) is equipped with an inclusion into � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h Ω A ct > 0 ( A ct > 0 ; M ) which is a weak equivalence.

Bimodules and Ibimodules Model Structures Main result and applications Main result Theorem (D) If η : A ct → M is an A ct > 0 -bimodules map then sTot ( M o ) is weakly equivalent to: � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h Ω A ct > 0 ( A ct > 0 ; M ) sTot ( M c ) ≃ Ω Bimod h A s > 0 ( A s > 0 ; M c ) � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h sTot ( M o ) ≃ Ω A ct > 0 ( A ct > 0 ; M ) ⇒ which is an algebra over SC 1 .

Bimodules and Ibimodules Model Structures Main result and applications Main result Theorem (D) If η : A ct → M is an A ct > 0 -bimodules map then sTot ( M o ) is weakly equivalent to: � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h Ω A ct > 0 ( A ct > 0 ; M ) Theorem (D) If η : A ct → M is a map of { o ; c } -operads and M (1; c ) ≃ M (1; o ) ≃ ∗ then sTot ( M o ) is weakly equivalent to: Ω 2 � � Operad h ( A s > 0 ; M c ) ; Operad h { o ; c } ( A ct > 0 ; M )

Bimodules and Ibimodules Model Structures Main result and applications Main result Theorem (D) If η : A ct → M is an A ct > 0 -bimodules map then sTot ( M o ) is weakly equivalent to: � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h Ω A ct > 0 ( A ct > 0 ; M ) Theorem (D) If η : A ct → M is a map of { o ; c } -operads and M (1; c ) ≃ M (1; o ) ≃ ∗ then sTot ( M o ) is weakly equivalent to: Ω 2 � � Operad h ( A s > 0 ; M c ) ; Operad h { o ; c } ( A ct > 0 ; M ) Sketch of proof: Bimod h A s > 0 ( A s > 0 ; M c ) ≃ Ω Operad h ( A s > 0 ; M c ) Bimod h A ct > 0 ( A ct > 0 ; M ) ≃ Ω Operad h { o ; c } ( A ct > 0 ; M )

Bimodules and Ibimodules Model Structures Main result and applications Main result Theorem (D) If η : A ct → M is an A ct > 0 -bimodules map then sTot ( M o ) is weakly equivalent to: � � Bimod h A s > 0 ( A s > 0 ; M c ) ; Bimod h Ω A ct > 0 ( A ct > 0 ; M ) Theorem (D) If η : A ct → M is a map of { o ; c } -operads and M (1; c ) ≃ M (1; o ) ≃ ∗ then sTot ( M o ) is weakly equivalent to: Ω 2 � � Operad h ( A s > 0 ; M c ) ; Operad h { o ; c } ( A ct > 0 ; M ) sTot ( M c ) ≃ Ω 2 Operad h ( A s > 0 ; M c ) sTot ( M o ) ≃ Ω 2 � � Operad h ( A s > 0 ; M c ) ; Operad h { o ; c } ( A ct > 0 ; M ) ⇒ which is an algebra over SC 2 .

Bimodules and Ibimodules Model Structures Main result and applications Structure on the singular homology Consequence If M is a pointed { o ; c } -operad ( ∃ η : A ct → M ) then the pair � � H ∗ ( sTot ( M c )) ; H ∗ ( sTot ( M o )) is an sc 2 -algebra.

Bimodules and Ibimodules Model Structures Main result and applications Structure on the singular homology Consequence If M is a pointed { o ; c } -operad ( ∃ η : A ct → M ) then the pair � � H ∗ ( sTot ( M c )) ; H ∗ ( sTot ( M o )) is an sc 2 -algebra. ◮ a bracket of degre 1: [ − ; − ] : H p ( sTot ( M c )) ⊗ H q ( sTot ( M c )) → H p + q +1 ( sTot ( M c )) ◮ a commutative product of degre 0: − ∗ − : H p ( sTot ( M c )) ⊗ H q ( sTot ( M c )) → H p + q ( sTot ( M c ))

Bimodules and Ibimodules Model Structures Main result and applications Structure on the singular homology Consequence If M is a pointed { o ; c } -operad ( ∃ η : A ct → M ) then the pair � � H ∗ ( sTot ( M c )) ; H ∗ ( sTot ( M o )) is an sc 2 -algebra. ◮ a bracket of degre 1: [ − ; − ] : H p ( sTot ( M c )) ⊗ H q ( sTot ( M c )) → H p + q +1 ( sTot ( M c )) ◮ a commutative product of degre 0: − ∗ − : H p ( sTot ( M c )) ⊗ H q ( sTot ( M c )) → H p + q ( sTot ( M c )) ◮ an associative product of degre 0: − ∗ i − : H p ( sTot ( M o )) ⊗ H q ( sTot ( M o )) → H p + q ( sTot ( M o )) ◮ a module structure of degre 0 over the commutative product: − ∗ e − : H p ( sTot ( M c )) ⊗ H q ( sTot ( M o )) → H p + q ( sTot ( M o ))