Swiss-Cheese action on the totalization of operads under the monoid - - PowerPoint PPT Presentation
Bimodules and Ibimodules Model Structures Main result and applications Swiss-Cheese action on the totalization of operads under the monoid actions operad Julien Ducoulombier Universit e paris 13 23/04/2015 Bimodules and Ibimodules Model
Bimodules and Ibimodules Model Structures Main result and applications
Julien Ducoulombier
Universit´ e paris 13
23/04/2015
Bimodules and Ibimodules Model Structures Main result and applications
Definition A semi-cosimplicial space X • is a family of spaces {X n}n≥0 equipped with maps di : X n → X n+1, for i ∈ {0, . . . , n + 1}, satisfying the relations djdi = didj−1 if i < j.
Bimodules and Ibimodules Model Structures Main result and applications
Definition A semi-cosimplicial space X • is a family of spaces {X n}n≥0 equipped with maps di : X n → X n+1, for i ∈ {0, . . . , n + 1}, satisfying the relations djdi = didj−1 if i < j. Example The simplices ∆n := {(t1 ≤ · · · ≤ tn) | ti ∈ [0 ; 1]} with the cofaces: di : (t1, . . . , tn) → (0, t1, . . . , tn) if i = 0 (t1, . . . , ti, ti, . . . , tn) if i ∈ {1 . . . , n} (t1, . . . , tn, 1) if i = n + 1
Bimodules and Ibimodules Model Structures Main result and applications
Definition A semi-cosimplicial space X • is a family of spaces {X n}n≥0 equipped with maps di : X n → X n+1, for i ∈ {0, . . . , n + 1}, satisfying the relations djdi = didj−1 if i < j. Example The simplices ∆n := {(t1 ≤ · · · ≤ tn) | ti ∈ [0 ; 1]} with the cofaces: di : (t1, . . . , tn) → (0, t1, . . . , tn) if i = 0 (t1, . . . , ti, ti, . . . , tn) if i ∈ {1 . . . , n} (t1, . . . , tn, 1) if i = n + 1 Definition The semi-totalization, sTot(X), is the space of natural transformations from ∆• to X •.
Bimodules and Ibimodules Model Structures Main result and applications
Q1 : Structure on X • such that sTot(X •) is an En-space?
Bimodules and Ibimodules Model Structures Main result and applications
Q1 : Structure on X • such that sTot(X •) is an En-space? ◮ McClure-Smith (2004)
⇐ X • is equipped with maps: η : X n × X m → X n+m satisfying η(d|x|+1x ; y) = η(x ; d0y).
⇐ X • comes from a multiplicative operad.
Bimodules and Ibimodules Model Structures Main result and applications
Q1 : Structure on X • such that sTot(X •) is an En-space? ◮ McClure-Smith (2004) Q2: Can we identify the loop spaces?
Bimodules and Ibimodules Model Structures Main result and applications
Q1 : Structure on X • such that sTot(X •) is an En-space? ◮ McClure-Smith (2004) Q2: Can we identify the loop spaces? ◮ Turchin and independently Dwyer-Hess (2012)
E1 ⇒ sTot(X •) ≃ ΩBimodh
As>0(As>0 ; X).
E2 ⇒ sTot(X •) ≃ Ω2Operadh(As>0 ; X).
Bimodules and Ibimodules Model Structures Main result and applications
Q1 : Structure on X • such that sTot(X •) is an En-space? ◮ McClure-Smith (2004) Q2: Can we identify the loop spaces? ◮ Turchin and independently Dwyer-Hess (2012)
E2 ⇒ sTot(X •) ≃ Ω2Operadh(As>0 ; X).
Q3: The Gerstenhaber structures on H∗(sTot(X)) and HH∗(X). ◮ Bousfield , Voronov , Gerstenhaber , Salvatore , Sakai
Bimodules and Ibimodules Model Structures Main result and applications
Q1 : Structure on X • such that sTot(X •) is an En-space? ◮ McClure-Smith (2004) Q2: Can we identify the loop spaces? ◮ Turchin and independently Dwyer-Hess (2012)
E2 ⇒ sTot(X •) ≃ Ω2Operadh(As>0 ; X).
Q3: The Gerstenhaber structures on H∗(sTot(X)) and HH∗(X). ◮ Bousfield , Voronov , Gerstenhaber , Salvatore , Sakai Theorem (D) If a pair of semi-cosimplicial spaces (Xc ; Xo) arises from a coloured
(sTot(Xc) ; sTot(Xo)) is weakly equivalent to the SC2-space:
{o ; c}(Act>0 ; X)
Bimodules and Ibimodules Model Structures Main result and applications
1
Bimodules and Ibimodules
2
Model Structures
3
Main result and applications
Bimodules and Ibimodules Model Structures Main result and applications
Definition An operad O is a collection of spaces {O(n)}n≥0 endowed with operations:
and a distinguished element ∗ ∈ O(1) satisfying some axioms.
Bimodules and Ibimodules Model Structures Main result and applications
Definition An operad O is a collection of spaces {O(n)}n≥0 endowed with operations:
and a distinguished element ∗ ∈ O(1) satisfying some axioms.
Bimodules and Ibimodules Model Structures Main result and applications
Definition An operad O is a collection of spaces {O(n)}n≥0 endowed with operations:
and a distinguished element ∗ ∈ O(1) satisfying some axioms. Definition An O-space is a topological space X equipped with maps: O(n) × X ×n → X.
Bimodules and Ibimodules Model Structures Main result and applications
Definition An operad O is a collection of spaces {O(n)}n≥0 endowed with operations:
and a distinguished element ∗ ∈ O(1) satisfying some axioms. Definition An O-space is a topological space X equipped with maps: O(n) × X ×n → X. Example: The associative operad As
As(n) = ∗n for n ≥ 0 with the obvious operadic composition
Bimodules and Ibimodules Model Structures Main result and applications
Definition An operad O is a collection of spaces {O(n)}n≥0 endowed with operations:
and a distinguished element ∗ ∈ O(1) satisfying some axioms. Definition An O-space is a topological space X equipped with maps: O(n) × X ×n → X. Example: The strict associative operad As>0
As>0(n) = ∗n for n > 0 As>0(0) = ∅ with the obvious operadic composition
Bimodules and Ibimodules Model Structures Main result and applications
Example: The little disk operad Cd
Cd(n) is the configurations of n nonoverlapping little disks of dimension d labelled by {1, . . . , n} in the unit disk. For instance a point in C2(3) is the following:
Bimodules and Ibimodules Model Structures Main result and applications
Example: The little disk operad Cd
Cd(n) is the configurations of n nonoverlapping little disks of dimension d labelled by {1, . . . , n} in the unit disk. The operadic composition:
Bimodules and Ibimodules Model Structures Main result and applications
Example: The little disk operad Cd
Cd(n) is the configurations of n nonoverlapping little disks of dimension d labelled by {1, . . . , n} in the unit disk. The operadic composition:
Remark For any pointed topological space X, ΩdX is a Cd-space.
Bimodules and Ibimodules Model Structures Main result and applications
Definition A coloured operad over the set of colours S, or S-operad, is given by:
a family of spaces {O(s1, . . . , sn; sn+1)}n∈N
si∈S
an operadic composition ◦i: O(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → O(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
distinguished elements {∗s ∈ O(s; s)}.
Bimodules and Ibimodules Model Structures Main result and applications
Definition A coloured operad over the set of colours S, or S-operad, is given by:
a family of spaces {O(s1, . . . , sn; sn+1)}n∈N
si∈S
an operadic composition ◦i: O(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → O(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
distinguished elements {∗s ∈ O(s; s)}.
Bimodules and Ibimodules Model Structures Main result and applications
Definition A coloured operad over the set of colours S, or S-operad, is given by:
a family of spaces {O(s1, . . . , sn; sn+1)}n∈N
si∈S
an operadic composition ◦i: O(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → O(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
distinguished elements {∗s ∈ O(s; s)}.
Definition An O-space is a family of spaces {Xs}s∈S with maps O(s1, . . . , sn; sn+1) × Xs1 × · · · × Xsn → Xsn+1
Bimodules and Ibimodules Model Structures Main result and applications
Definition A coloured operad over the set of colours S, or S-operad, is given by:
a family of spaces {O(s1, . . . , sn; sn+1)}n∈N
si∈S
an operadic composition ◦i: O(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → O(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
distinguished elements {∗s ∈ O(s; s)}.
Definition An O-space is a family of spaces {Xs}s∈S with maps O(s1, . . . , sn; sn+1) × Xs1 × · · · × Xsn → Xsn+1 If S = {o ; c} then On
c = O(n; c) = O(c, . . . , c n
; c) and On
n
, o; o)
Bimodules and Ibimodules Model Structures Main result and applications
Example: The {o; c}-operad of unital monoid actions Act
Act(n; c) = ∗n;c for n ≥ 0 Act(n; o) = ∗n;o for n > 0 the empty set otherwise with the obvious operadic composition
n−1
Bimodules and Ibimodules Model Structures Main result and applications
Example: The {o; c}-operad of unital monoid actions Act
Act(n; c) = ∗n;c for n ≥ 0 Act(n; o) = ∗n;o for n > 0 the empty set otherwise with the obvious operadic composition
/ ∈ Act
Bimodules and Ibimodules Model Structures Main result and applications
Example: The {o; c}-operad of unital monoid actions Act
Act(n; c) = ∗n;c for n ≥ 0 Act(n; o) = ∗n;o for n > 0 the empty set otherwise with the obvious operadic composition
/ ∈ Act Xc Xo Topological monoid Left module
Bimodules and Ibimodules Model Structures Main result and applications
Example: The {o; c}-operad of unital monoid actions Act
Act(n; c) = ∗n;c for n ≥ 0 Act(n; o) = ∗n;o for n > 0 the empty set otherwise with the obvious operadic composition
/ ∈ Act Definition An {o; c}-operad O is pointed if there is a map of operads from Act to O.
Bimodules and Ibimodules Model Structures Main result and applications
Example: The {o; c}-operad of unital monoid actions Act
Act(n; c) = ∗n;c for n ≥ 0 Act(n; o) = ∗n;o for n > 0 the empty set otherwise with the obvious operadic composition
/ ∈ Act Definition An {o; c}-operad O is pointed if there is a map of operads from Act to O. Example: The {o; c}-operad of monoid actions Act>0
Act>0(n; c) = ∗n;c for n > 0 Act>0(n; o) = ∗n;o for n > 0 the empty set otherwise with the obvious operadic composition
/ ∈ Act>0
Bimodules and Ibimodules Model Structures Main result and applications
Example: The Swiss-Cheese operad SCd
If the output is ”close” then all the inputs are also ”close” and SCd(c, . . . , c
n
; c) = Cd(n). If the output is ”open” then we denote by SCd(n1, n2; o) the space of configurations of nonoverlapping n1 little disks and n2 upper semidisks labeled by {1, . . . , n1 + n2} in the unit upper semidisk so that the semidisks are all centered about the diameter of the unit semidisk.
SC2(1, 2; o)
Bimodules and Ibimodules Model Structures Main result and applications
Example: The Swiss-Cheese operad SCd
If the output is ”close” then all the inputs are also ”close” and SCd(c, . . . , c
n
; c) = Cd(n). If the output is ”open” then we denote by SCd(n1, n2; o) the space of configurations of nonoverlapping n1 little disks and n2 upper semidisks labeled by {1, . . . , n1 + n2} in the unit upper semidisk so that the semidisks are all centered about the diameter of the unit semidisk.
SC2(1, 2; o) SC2(2; c) SC2(2, 2; o)
Bimodules and Ibimodules Model Structures Main result and applications
Example: The Swiss-Cheese operad SCd
If the output is ”close” then all the inputs are also ”close” and SCd(c, . . . , c
n
; c) = Cd(n). If the output is ”open” then we denote by SCd(n1, n2; o) the space of configurations of nonoverlapping n1 little disks and n2 upper semidisks labeled by {1, . . . , n1 + n2} in the unit upper semidisk so that the semidisks are all centered about the diameter of the unit semidisk.
SC2(1, 2; o) SC2(1, 2; o) SC2(2, 3; o)
Bimodules and Ibimodules Model Structures Main result and applications
Example: The Swiss-Cheese operad SCd
If the output is ”close” then all the inputs are also ”close” and SCd(c, . . . , c
n
; c) = Cd(n). If the output is ”open” then we denote by SCd(n1, n2; o) the space of configurations of nonoverlapping n1 little disks and n2 upper semidisks labeled by {1, . . . , n1 + n2} 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 Ωd(X; A) =
if td = 0 f (∂[0; 1]d) = ∗
Bimodules and Ibimodules Model Structures Main result and applications
Definition A bimodule M over an S-operad O, or O-bimodule, is given by
a family of spaces {M(s1, . . . , sn; sn+1)}n∈N
si∈S
right operations ◦i: M(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → M(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
Bimodules and Ibimodules Model Structures Main result and applications
Definition A bimodule M over an S-operad O, or O-bimodule, is given by
a family of spaces {M(s1, . . . , sn; sn+1)}n∈N
si∈S
right operations ◦i: M(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → M(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
left operations γl: O(s1, .., sn; sn+1)×M(s1
1, .., s1 p1; s1)×..×M(sn 1 , .., sn pn; sn) → M(s1 1, .., sn pn; sn+1)
Bimodules and Ibimodules Model Structures Main result and applications
Definition A bimodule M over an S-operad O, or O-bimodule, is given by
a family of spaces {M(s1, . . . , sn; sn+1)}n∈N
si∈S
right operations ◦i: M(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → M(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
left operations γl: O(s1, .., sn; sn+1)×M(s1
1, .., s1 p1; s1)×..×M(sn 1 , .., sn pn; sn) → M(s1 1, .., sn pn; sn+1)
satisfying some axioms. Example If η : Act → O is a map of {o; c}-operads then η is also a map of Act>0-bimodules.
Bimodules and Ibimodules Model Structures Main result and applications
Definition An infinitesimal bimodule M over an S-operad O, or O-Ibimodule, is:
a family of spaces {M(s1, . . . , sn; sn+1)}n∈N
si∈S
right operations ◦i: M(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → M(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
Bimodules and Ibimodules Model Structures Main result and applications
Definition An infinitesimal bimodule M over an S-operad O, or O-Ibimodule, is:
a family of spaces {M(s1, . . . , sn; sn+1)}n∈N
si∈S
right operations ◦i: M(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → M(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
left operations ◦i: O(s1, .., sn; sn+1) × M(s′
1, .., s′ m; si) → M(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
Bimodules and Ibimodules Model Structures Main result and applications
Definition An infinitesimal bimodule M over an S-operad O, or O-Ibimodule, is:
a family of spaces {M(s1, . . . , sn; sn+1)}n∈N
si∈S
right operations ◦i: M(s1, .., sn; sn+1) × O(s′
1, .., s′ m; si) → M(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
left operations ◦i: O(s1, .., sn; sn+1) × M(s′
1, .., s′ m; si) → M(s1, .., si−1, s′ 1, .., s′ m, si+1, .., sn; sn+1)
satisfying some axioims. Example If η : Act → M is a map between Act>0-bimodules then M is also an Act>0-Ibimodule.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. ∈ Act>0(2; c)
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. ← n operations {d1, . . . , dn}
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. ← n operations {d1, . . . , dn} ← 2 operations {d0, dn+1}
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. ∈ Act>0(2; o)
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. ← n + 1 operations {d1, . . . , dn+1}
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. ← n + 1 operations {d1, . . . , dn+1} ← 1 operation d0
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. ∈ Mn
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. Corollary 1 If η : Act → M is a map of Act>0-bimodules then:
there exists a product: Mn
c × Mm c → Mn+m c
and an action: Mn
c × Mm
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. Corollary 1 If η : Act → M is a map of Act>0-bimodules then:
there exists a product: Mn
c × Mm c → Mn+m c
and an action: Mn
c × Mm
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. Corollary 1 If η : Act → M is a map of Act>0-bimodules then:
there exists a product: Mn
c × Mm c → Mn+m c
and an action: Mn
c × Mm
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. Corollary 2 If η : Act → M is a map of {o; c}-operads then:
Mc is a multiplicative operad there exists an action: Mn
c × Mm
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If M is an Act>0-Ibimodule then M•
c := {Mn c } and M•
semi-cosimplicial spaces. Furthermore there exists a semi-cosimplicial map h : Mc → Mo. Corollary 2 If η : Act → M is a map of {o; c}-operads then:
Mc is a multiplicative operad there exists an action: Mn
c × Mm
Bimodules and Ibimodules Model Structures Main result and applications
L : C1 ⇄ C2 : R A model category cofibrantly generated (I, J)
Bimodules and Ibimodules Model Structures Main result and applications
L : C1 ⇄ C2 : R A model category cofibrantly generated (I, J) A model category cofibrantly generated (LI, LJ)
Bimodules and Ibimodules Model Structures Main result and applications
F : Coll(S) ⇄ OperadS : U Coll(S) :
si∈S
Bimodules and Ibimodules Model Structures Main result and applications
F : Coll(S) ⇄ OperadS : U Coll(S) :
si∈S
T op has a cofibrantly generated model category :
f : X → Y is a weak equivalence iff f ∗
n : πn(X) → πn(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}
Y
Bimodules and Ibimodules Model Structures Main result and applications
F : Coll(S) ⇄ OperadS : U Coll(S) :
si∈S
T op has a cofibrantly generated model category :
f : X → Y is a weak equivalence iff f ∗
n : πn(X) → πn(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}
Y I = {∂∆n → ∆n | n ∈ N} J = {∧n
k → ∆n | n ∈ N, k ≤ n}
all the objects are fibrant
Bimodules and Ibimodules Model Structures Main result and applications
F : Coll(S) ⇄ OperadS : U Coll(S) :
si∈S
T op has a cofibrantly generated model category :
f : X → Y is a weak equivalence iff f ∗
n : πn(X) → πn(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}
Y I = {∂∆n → ∆n | n ∈ N} J = {∧n
k → ∆n | n ∈ N, k ≤ n}
all the objects are fibrant
⇒ Coll(S) inherits a cofibrantly generated model category.
Bimodules and Ibimodules Model Structures Main result and applications
F : Coll(S) ⇄ OperadS : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
For instance a point in F(M)(s1, s2, s3, s4; s5) is given by: s1 s2 s3 s4 s6 s7 s1 s8 s5
Bimodules and Ibimodules Model Structures Main result and applications
F : Coll(S) ⇄ OperadS : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
For instance a point in F(M)(s1, s2, s3, s4; s5) is given by: s1 s2 s3 s4 s6 s7 s1 s8 s5 M( ; s7) ∗s1 M(s2, s3; s6) M(s6, s4, s7; s8) M(s1, s8; s5)
Bimodules and Ibimodules Model Structures Main result and applications
F : Coll(S) ⇄ OperadS : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
The operadic composition: F(M)(s1, s2, s3 : s4) F(M)(s6, s7 : s1) F(M)(s6, s7, s2, s3 : s4)
Bimodules and Ibimodules Model Structures Main result and applications
BO : Coll(S) ⇄ BimodO : U
Bimodules and Ibimodules Model Structures Main result and applications
BO : Coll(S) ⇄ BimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
For instance a point in BO(M)(s1, s2, s3; s4) is given by:
Bimodules and Ibimodules Model Structures Main result and applications
BO : Coll(S) ⇄ BimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
For instance a point in BO(M)(s1, s2, s3; s4) is given by:
Bimodules and Ibimodules Model Structures Main result and applications
BO : Coll(S) ⇄ BimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
For instance a point in BO(M)(s1, s2, s3; s4) is given by:
Bimodules and Ibimodules Model Structures Main result and applications
BO : Coll(S) ⇄ BimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
For instance a point in BO(M)(s1, s2, s3; s4) is given by:
Bimodules and Ibimodules Model Structures Main result and applications
BO : Coll(S) ⇄ BimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
The right operations ◦i: BO(M)(s1, s2, s3; s4) O(s6, s7; s1) BO(M)(s6, s7, s2, s3; s4)
Bimodules and Ibimodules Model Structures Main result and applications
BO : Coll(S) ⇄ BimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
The right operations ◦i: BO(M)(s1, s2, s3; s4) O(s6, s7; s1) BO(M)(s1, s6, s7, s3; s4)
Bimodules and Ibimodules Model Structures Main result and applications
BO : Coll(S) ⇄ BimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
The left operations γl:
Bimodules and Ibimodules Model Structures Main result and applications
BO : Coll(S) ⇄ BimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
The left operations γl:
Bimodules and Ibimodules Model Structures Main result and applications
IbO : Coll(S) ⇄ IbimodO : U
Bimodules and Ibimodules Model Structures Main result and applications
IbO : Coll(S) ⇄ IbimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
For instance a point in IbO(M)(s1, s2, s3, s4; s5) is given by:
Bimodules and Ibimodules Model Structures Main result and applications
IbO : Coll(S) ⇄ IbimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
For instance a point in IbO(M)(s1, s2, s3, s4; s5) is given by:
Bimodules and Ibimodules Model Structures Main result and applications
IbO : Coll(S) ⇄ IbimodO : U Let {M(s1, . . . , sn; sn+1)}n∈N
si∈S be a family of spaces.
The left operations ◦i: O(s1, s2; s3) IbO(M)(s4, s5, s6; s2) IbO(M)(s1, s4, s5, s6; s3)
Bimodules and Ibimodules Model Structures Main result and applications
As>0 − Ibimodules As>0 − Ibimodules maps ⇔ semi-cosimplicial spaces semi-cosimplicial maps
Bimodules and Ibimodules Model Structures Main result and applications
As>0 − Ibimodules As>0 − Ibimodules maps ⇔ semi-cosimplicial spaces semi-cosimplicial maps Theorem (V.Turchin) A cofibrant replacement of As as an As>0-Ibimodule is given by the semi-cosimplicial space ∆.
Bimodules and Ibimodules Model Structures Main result and applications
As>0 − Ibimodules As>0 − Ibimodules maps ⇔ semi-cosimplicial spaces semi-cosimplicial maps Theorem (V.Turchin) A cofibrant replacement of As as an As>0-Ibimodule is given by the semi-cosimplicial space ∆. Act>0 − Ibimodule M ⇒ a semi-cosimplicial map h : M•
c → M•
Bimodules and Ibimodules Model Structures Main result and applications
As>0 − Ibimodules As>0 − Ibimodules maps ⇔ semi-cosimplicial spaces semi-cosimplicial maps Theorem (V.Turchin) A cofibrant replacement of As as an As>0-Ibimodule is given by the semi-cosimplicial space ∆. Act>0 − Ibimodule M ⇒ a semi-cosimplicial map h : M•
c → M•
If M is an Act>0-Ibimodule then:
Ibimodh
As>0(As; Mc) ≃ IbimodAs>0(∆; Mc) = Nat(∆ ; Mc) = sTot(Mc)
Ibimodh
As>0(As; Mo) ≃ IbimodAs>0(∆; Mo) = Nat(∆ ; Mo) = sTot(Mo)
Bimodules and Ibimodules Model Structures Main result and applications
Act>0 − Ibimodule M ⇒ a semi-cosimplicial map h : M•
c → M•
A cofibrant replacement
△ of Act as an Act>0-Ibimodule is given by the
pair of semi-cosimplicial spaces:
and
where the semi-cosimplicial map h : ∆n → ∆n × [0 ; 1] is defined by: (t1 ≤ · · · ≤ tn) → (t1 ≤ · · · ≤ tn) × {1}
Bimodules and Ibimodules Model Structures Main result and applications
Act>0 − Ibimodule M ⇒ a semi-cosimplicial map h : M•
c → M•
A cofibrant replacement
△ of Act as an Act>0-Ibimodule is given by the
pair of semi-cosimplicial spaces:
and
where the semi-cosimplicial map h : ∆n → ∆n × [0 ; 1] is defined by: (t1 ≤ · · · ≤ tn) → (t1 ≤ · · · ≤ tn) × {1} Theorem (D) Let M be an Act>0-Ibimodule. One has: Ibimodh
Act>0(Act; M)
≃ IbimodAct>0(
△; M)
≃ IbimodAs>0(∆; Mc) ≃ sTot(Mc)
Bimodules and Ibimodules Model Structures Main result and applications
M is an Act>0 − bimodule ⇒ Mc is an As>0 − bimodule Theorem (V.Turchin) If η : Act → M is a map of Act>0-bimodules and M(0; c) ≃ ∗ then:
sTot(Mc) ≃ ΩBimodh
As>0(As>0; Mc)
sTot(Mo) ≃ Ω???
Bimodules and Ibimodules Model Structures Main result and applications
M is an Act>0 − bimodule ⇒ Mc is an As>0 − bimodule Theorem (V.Turchin) If η : Act → M is a map of Act>0-bimodules and M(0; c) ≃ ∗ then:
sTot(Mc) ≃ ΩBimodh
As>0(As>0; Mc)
sTot(Mo) ≃ Ω???
M is an {o; c} − operad pointed ⇒ Mc is a multiplicative operad Mo is an As>0 − bimodule
Bimodules and Ibimodules Model Structures Main result and applications
M is an Act>0 − bimodule ⇒ Mc is an As>0 − bimodule Theorem (V.Turchin) If η : Act → M is a map of Act>0-bimodules and M(0; c) ≃ ∗ then:
sTot(Mc) ≃ ΩBimodh
As>0(As>0; Mc)
sTot(Mo) ≃ Ω???
M is an {o; c} − operad pointed ⇒ Mc is a multiplicative operad Mo is an As>0 − bimodule Theorem (V.Turchin) If η : Act → M is a map of {o; c}-operads and M0
c ≃ M1 c ≃ M0
sTot(Mc) ≃ Ω2Operadh(As>0; Mc) sTot(Mo) ≃ ΩBimodh
As>0(As>0; Mo) ≃ Ω2 ???
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If η : Act → M is an Act>0-bimodules map then sTot(Mo) is weakly equivalent to: Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If η : Act → M is an Act>0-bimodules map then sTot(Mo) is weakly equivalent to: Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
Determine a cofibrant replacement of Act>0 as an Act>0-bimodule
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If η : Act → M is an Act>0-bimodules map then sTot(Mo) is weakly equivalent to: Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
Determine a cofibrant replacement of Act>0 as an Act>0-bimodule Determine a sub-operad M∗ such that: sTot(Mo) ≃ Bimodh
Act>0(Act>0; M∗)
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If η : Act → M is an Act>0-bimodules map then sTot(Mo) is weakly equivalent to: Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
Determine a cofibrant replacement of Act>0 as an Act>0-bimodule Determine a sub-operad M∗ such that: sTot(Mo) ≃ Bimodh
Act>0(Act>0; M∗)
Prove that Bimodh
Act>0(Act>0; M∗) is equipped with an inclusion into
Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If η : Act → M is an Act>0-bimodules map then sTot(Mo) is weakly equivalent to: Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
As>0(As>0; Mc)
sTot(Mo) ≃ Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If η : Act → M is an Act>0-bimodules map then sTot(Mo) is weakly equivalent to: Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
If η : Act → M is a map of {o; c}-operads and M(1; c) ≃ M(1; o) ≃ ∗ then sTot(Mo) is weakly equivalent to: Ω2 Operadh(As>0; Mc) ; Operadh
{o;c}(Act>0; M)
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If η : Act → M is an Act>0-bimodules map then sTot(Mo) is weakly equivalent to: Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
If η : Act → M is a map of {o; c}-operads and M(1; c) ≃ M(1; o) ≃ ∗ then sTot(Mo) is weakly equivalent to: Ω2 Operadh(As>0; Mc) ; Operadh
{o;c}(Act>0; M)
Bimodh
As>0(As>0; Mc) ≃ ΩOperadh(As>0; Mc)
Bimodh
Act>0(Act>0; M) ≃ ΩOperadh {o;c}(Act>0; M)
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) If η : Act → M is an Act>0-bimodules map then sTot(Mo) is weakly equivalent to: Ω
As>0(As>0; Mc) ; Bimodh Act>0(Act>0; M)
If η : Act → M is a map of {o; c}-operads and M(1; c) ≃ M(1; o) ≃ ∗ then sTot(Mo) is weakly equivalent to: Ω2 Operadh(As>0; Mc) ; Operadh
{o;c}(Act>0; M)
sTot(Mo) ≃ Ω2 Operadh(As>0; Mc) ; Operadh
{o;c}(Act>0; M)
Bimodules and Ibimodules Model Structures Main result and applications
Consequence If M is a pointed {o; c}-operad (∃ η : Act → M) then the pair
Bimodules and Ibimodules Model Structures Main result and applications
Consequence If M is a pointed {o; c}-operad (∃ η : Act → M) then the pair
◮ a bracket of degre 1: [− ; −] : Hp(sTot(Mc)) ⊗ Hq(sTot(Mc)) → Hp+q+1(sTot(Mc)) ◮ a commutative product of degre 0: − ∗ − : Hp(sTot(Mc)) ⊗ Hq(sTot(Mc)) → Hp+q(sTot(Mc))
Bimodules and Ibimodules Model Structures Main result and applications
Consequence If M is a pointed {o; c}-operad (∃ η : Act → M) then the pair
◮ a bracket of degre 1: [− ; −] : Hp(sTot(Mc)) ⊗ Hq(sTot(Mc)) → Hp+q+1(sTot(Mc)) ◮ a commutative product of degre 0: − ∗ − : Hp(sTot(Mc)) ⊗ Hq(sTot(Mc)) → Hp+q(sTot(Mc)) ◮ an associative product of degre 0: − ∗i − : Hp(sTot(Mo)) ⊗ Hq(sTot(Mo)) → Hp+q(sTot(Mo)) ◮ a module structure of degre 0 over the commutative product: − ∗e − : Hp(sTot(Mc)) ⊗ Hq(sTot(Mo)) → Hp+q(sTot(Mo))
Bimodules and Ibimodules Model Structures Main result and applications
The commutative product − ∗ −:
Bimodules and Ibimodules Model Structures Main result and applications
The commutative product − ∗ −: λ1 : sTot(Mc) × sTot(Mc) → sTot(Mc) (f1 ; f2) → h1
Bimodules and Ibimodules Model Structures Main result and applications
The commutative product − ∗ −: λ1 : sTot(Mc) × sTot(Mc) → sTot(Mc) (f1 ; f2) → h1 hn
1 : (t1 ≤ · · · ≤ tn) −
→ ∗2 ; c
1 (2t1, . . . , 2tℓ) ; f n−ℓ 2
(2tℓ+1 − 1, . . . , 2tn − 1)
Bimodules and Ibimodules Model Structures Main result and applications
The commutative product − ∗ −: λ1 : sTot(Mc) × sTot(Mc) → sTot(Mc) (f1 ; f2) → h1 hn
1 : (t1 ≤ · · · ≤ tn) −
→ ∗2 ; c
1 (2t1, . . . , 2tℓ) ; f n−ℓ 2
(2tℓ+1 − 1, . . . , 2tn − 1)
Bimodules and Ibimodules Model Structures Main result and applications
The module structure − ∗e −: λ2 : sTot(Mc) × sTot(Mo) → sTot(Mo) (f1 ; f2) → h2 hn
2 : (t1 ≤ · · · ≤ tn) −
→ ∗2 ; o
1 (2t1, . . . , 2tℓ) ; f n−ℓ 2
(2tℓ+1−1, . . . , 2tn−1)
Bimodules and Ibimodules Model Structures Main result and applications
The assaciative product − ∗i −: λ3 : sTot(Mo) × sTot(Mo) → sTot(Mo) (f1 ; f2) → h3 hn
3 : (t1 ≤ · · · ≤ tn) −
→ f ℓ
1 (2t1, . . . , 2tℓ) ◦ℓ+1 f n−ℓ 2
(2tℓ+1 − 1, . . . , 2tn − 1) if tℓ ≤ 1/2 < tℓ+1
Bimodules and Ibimodules Model Structures Main result and applications
Let X • be a semi-cosimplicial space.
∂ : C s
p(X q) → C s p−1(X q)
δ : C s
p(X q) → C s p(X q+1)
Bimodules and Ibimodules Model Structures Main result and applications
Let X • be a semi-cosimplicial space.
∂ : C s
p(X q) → C s p−1(X q)
δ : C s
p(X q) → C s p(X q+1)
Definition The Hochschild homology of X •, HH∗(X), is the homology of the chain complex (C∗(X), d) where: Cp(X) =
ℓ≥0
C s
p+ℓ(X ℓ)
and d = ∂ + (−1)p+1δ
Bimodules and Ibimodules Model Structures Main result and applications
Let X • be a semi-cosimplicial space.
∂ : C s
p(X q) → C s p−1(X q)
δ : C s
p(X q) → C s p(X q+1)
Definition The Hochschild homology of X •, HH∗(X), is the homology of the chain complex (C∗(X), d) where: Cp(X) =
ℓ≥0
C s
p+ℓ(X ℓ)
and d = ∂ + (−1)p+1δ From Bousfield, there exists an isomorphism: ϕ∗ : H∗(sTot(X)) → HH∗(X)
Bimodules and Ibimodules Model Structures Main result and applications
Let X • be a semi-cosimplicial space.
∂ : C s
p(X q) → C s p−1(X q)
δ : C s
p(X q) → C s p(X q+1)
Definition The Hochschild homology of X •, HH∗(X), is the homology of the chain complex (C∗(X), d) where: Cp(X) =
ℓ≥0
C s
p+ℓ(X ℓ)
and d = ∂ + (−1)p+1δ From Bousfield, there exists an isomorphism: ϕ∗ : H∗(sTot(X)) → HH∗(X) Consequence If M is a pointed {o; c}-operad then the pair (HH∗(Mc) ; HH∗(Mo)) is also an sc2-algebra.
Bimodules and Ibimodules Model Structures Main result and applications
Let M be pointed {o; c}-operad in the category of graded vector spaces. [−; −] : ± + ± − ∗ − :
Bimodules and Ibimodules Model Structures Main result and applications
− ∗e − : − ∗i − :
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) The isomorphism introduced by Bousfield sends the structure introduced
Hochschild homologies.
Bimodules and Ibimodules Model Structures Main result and applications
Theorem (D) The isomorphism introduced by Bousfield sends the structure introduced
Hochschild homologies. Theorem (Salvatore) Let O be a multiplicative operad. Then the Bousfield spectral sequence for H∗(sTot(O)) given by: E 1
−p;q = Hq(Op)
and
p+1
(−1)idi : Hq(Op) → Hq(Op+1) is a spectrale sequence of Gerstenhaber algebras. Theorem (D) The Bousfield spectral sequences for (H∗(sTot(Mc)) ; H∗(sTot(Mo))) are spectral sequences of sc2-algebras.
Bimodules and Ibimodules Model Structures Main result and applications
Emb(R ; Rm) := hofib(Emb(R ; Rm) → Imm(R ; Rm))
Bimodules and Ibimodules Model Structures Main result and applications
Emb(R ; Rm) := hofib(Emb(R ; Rm) → Imm(R ; Rm)) Theorem (Dev. Sinha) If m > 3 then Emb(R ; Rm) ≃ sTot(Km) ≃ Ω2Operadh(As>0 ; Km).
Bimodules and Ibimodules Model Structures Main result and applications
Emb(R ; Rm) := hofib(Emb(R ; Rm) → Imm(R ; Rm)) Theorem (Dev. Sinha) If m > 3 then Emb(R ; Rm) ≃ sTot(Km) ≃ Ω2Operadh(As>0 ; Km).
Bimodules and Ibimodules Model Structures Main result and applications
Emb(R ; Rm) := hofib(Emb(R ; Rm) → Imm(R ; Rm)) Theorem (Dev. Sinha) If m > 3 then Emb(R ; Rm) ≃ sTot(Km) ≃ Ω2Operadh(As>0 ; Km).
Bimodules and Ibimodules Model Structures Main result and applications
Ln;m := hofib(Emb(
n
R ; Rm) → Imm(
n
R ; Rm))
Bimodules and Ibimodules Model Structures Main result and applications
Ln;m := hofib(Emb(
n
R ; Rm) → Imm(
n
R ; Rm)) Theorem (Munson-Volic and Dwyer-Hess) If m > 3 then Ln;m ≃ sTot(Φn(Km)) ≃ ΩBimodh
As>0(As>0 ; Φn(Km)).
Bimodules and Ibimodules Model Structures Main result and applications
Ln;m := hofib(Emb(
n
R ; Rm) → Imm(
n
R ; Rm)) Theorem (Munson-Volic and Dwyer-Hess) If m > 3 then Ln;m ≃ sTot(Φn(Km)) ≃ ΩBimodh
As>0(As>0 ; Φn(Km)).
Theorem (D) If m > 3 then the pair (Emb(R ; Rm) ; Ln;m) is an SC2-space.
Bimodules and Ibimodules Model Structures Main result and applications
Ln;m := hofib(Emb(
n
R ; Rm) → Imm(
n
R ; Rm)) Theorem (Munson-Volic and Dwyer-Hess) If m > 3 then Ln;m ≃ sTot(Φn(Km)) ≃ ΩBimodh
As>0(As>0 ; Φn(Km)).
Theorem (D) If m > 3 then the pair (Emb(R ; Rm) ; Ln;m) is an SC2-space.
Bimodules and Ibimodules Model Structures Main result and applications
Immk(R ; Rm) := hofib(Immk(
n
R ; Rm) → Imm(
n
R ; Rm))
Bimodules and Ibimodules Model Structures Main result and applications
Immk(R ; Rm) := hofib(Immk(
n
R ; Rm) → Imm(
n
R ; Rm)) Conjecture (Dobrinskaya , Turchin) 1) If m >?? then Immk(R ; Rm) has the homotopy type of a loop space.
Bimodules and Ibimodules Model Structures Main result and applications
Immk(R ; Rm) := hofib(Immk(
n
R ; Rm) → Imm(
n
R ; Rm)) Conjecture (Dobrinskaya , Turchin and D) 1) If m >?? then Immk(R ; Rm) has the homotopy type of a loop space. 2) If m >?? then the pair (Emb(R ; Rm) ; Immk(R ; Rm)) is an SC2-space.
Bimodules and Ibimodules Model Structures Main result and applications
Immk(R ; Rm) := hofib(Immk(
n
R ; Rm) → Imm(
n
R ; Rm)) Conjecture (Dobrinskaya , Turchin and D) 1) If m >?? then Immk(R ; Rm) has the homotopy type of a loop space. 2) If m >?? then the pair (Emb(R ; Rm) ; Immk(R ; Rm)) is an SC2-space. Problems 1) To find a cosimplicial replacement with a Km-bimodule structure. 2) The converge of the Taylor tower associated to the functor Immk(−; Rm).
Bimodules and Ibimodules Model Structures Main result and applications