Overview 1. The space of units 2. How to compute it 3. Specific - - PowerPoint PPT Presentation

HOW MANY UNITS MAY AN A∞-ALGEBRA HAVE?

Workshop in Category Theory and Algebraic Topology, Louvain-la-Neuve, 10–12 September 2015. Fernando Muro Universidad de Sevilla

Overview

• 1. The space of units
• 2. How to compute it
• 3. Specific cases

3.1 Groupoids 3.2 Chain complexes

• 4. Transfer to other categories

2

The space of units

The space of units Being unital is a property rather than a structure.

4

The space of units Being unital is a property rather than a structure. uAsV AsV

forget 1

Closed symmetric monoidal category

4

The space of units Being unital is a property rather than a structure. uAsV AsV

forget 1

Unital associative algebras in V

4

The space of units Being unital is a property rather than a structure. uAsV AsV

forget 1

Associative algebras in V

4

The space of units Being unital is a property rather than a structure. uAsV AsV

forget 1

Faithful

4

The space of units Being unital is a property rather than a structure. uAs iso

V

As iso

V forget 1

4

The space of units Being unital is a property rather than a structure. uAs iso

V

As iso

V forget 1

Fully faithful!

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The space of units Being unital is a property rather than a structure. uAs iso

V

As iso

V forget 1

Closed symmetric monoidal model category

4

The space of units Being unital is a property rather than a structure. uAswe

V

Aswe

V forget 1

4

The space of units Being unital is a property rather than a structure. |uAswe

V |

|Aswe

V | forget 1

4

The space of units Being unital is a property rather than a structure. |uAswe

V |

|Aswe

V | forget 1

A vertex is an associative algebra A

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The space of units Being unital is a property rather than a structure. Space of units of A |uAswe

V |

|Aswe

V | forget 1 fiber at A

4

The space of units Being unital is a property rather than a structure. Space of units of A |uAswe

V |

|Aswe

V | forget 1 fiber at A

Theorem The space of units is either empty or contractible if V is sim- plicial, complicial, or spectral.

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How to compute it

The space of units uAsV V AsV V

forgetful functors

6

The space of units |uAs we

V |

|V we | |As we

V |

|V we |

forgetful functors

6

The space of units |uAs we

V |

|V we | |As we

V |

|V we |

forgetful functors

? ?

fiber at X fiber at X

• htpy. pullback

Here A is an associative algebra with underlying object X.

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The space of units |uAs we

V |

|V we | |As we

V |

|V we |

forgetful functors

? ?

fiber at X fiber at X

• htpy. pullback

Space of units of A

fiber at A

Here A is an associative algebra with underlying object X.

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The space of algebra structures As iso

V

V iso

forget

Faithful

7

The space of algebra structures |Aswe

V |

|V we |

forget

7

The space of algebra structures |Aswe

V |

|V we |

forget

Space of associative algebra structures on X

fiber at X

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The space of algebra structures |Aswe

V |

|V we |

forget

Space of associative algebra structures on X

fiber at X

Theorem (Rezk’96, M’15) The space of associative algebra structures on a fibrant and cofibrant object X is MapOp(As, End(X)). Op the category of nonsymmetric operads in V. As the associative operad in V. End(X) the endomorphism operad of X.

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Operads Definition An operad P {P(n)}n≥0 is a sequence of objects in V equipped with composition operations

• i : P(s) ⊗ P(t) −→ P(s + t − 1),

1 ≤ i ≤ s, and an identity in arity 1 satisfying the laws of tree grafting. arity 3

• 2

arity 2

• arity 4

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Operads Definition An operad P {P(n)}n≥0 is a sequence of objects in V equipped with composition operations

• i : P(s) ⊗ P(t) −→ P(s + t − 1),

1 ≤ i ≤ s, and an identity in arity 1 satisfying the laws of tree grafting. arity 3

• 2

arity 2

• arity 4

8

Operads Definition An operad P {P(n)}n≥0 is a sequence of objects in V equipped with composition operations

• i : P(s) ⊗ P(t) −→ P(s + t − 1),

1 ≤ i ≤ s, and an identity in arity 1 satisfying the laws of tree grafting. arity 3

• 2

arity 2

• arity 4

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Operads Definition An operad P {P(n)}n≥0 is a sequence of objects in V equipped with composition operations

• i : P(s) ⊗ P(t) −→ P(s + t − 1),

1 ≤ i ≤ s, and an identity in arity 1 satisfying the laws of tree grafting. associativity identity

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Operads Example The associative operad As consists of · · · Composition away from the identity is given by grafting and contracting

• 9

Operads Example The unital associative operad uAs consists of · · · Composition away from the identity is given by grafting and contracting, except for

• 10

Operads Example The endomorphism operad of an object X in V, End(X)(n) HomV(X⊗n, X).

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Operads Example The endomorphism operad of an object X in V, End(X)(n) HomV(X⊗n, X). Definition A P-algebra is a map of operads P → End(X).

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Operads Example The endomorphism operad of an object X in V, End(X)(n) HomV(X⊗n, X). Definition A P-algebra is a map of operads P → End(X). Theorem (Rezk, Hinich, Berger–Moerdijk...Lyubashenko, M.) The category Op of operads in V inherits a model structure.

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The space of algebra structures Space of P-algebra structures on X |Pwe

V |

|Vwe|

forget fiber at X

Theorem (Rezk’96, M’15) The space of P-algebra structures on a fibrant and cofibrant

• bject X is

MapOp(P, End(X)). P a nonsymmetric operad in V with cofibrant components, e.g. P As or uAs, the unital associative operad.

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The spaces of units and algebra structures |uAswe

V |

|Vwe| |Aswe

V |

|Vwe|

forgetful functors

MapOp(uAs, End(X)) MapOp(As, End(X))

fiber at X fiber at X

• htpy. pullback

Space of units of A

fiber at A

Here A is an associative algebra with underlying object X.

13

The spaces of units and algebra structures |uAswe

V |

|Vwe| |Aswe

V |

|Vwe|

forgetful functors

MapOp(uAs, End(X)) MapOp(As, End(X))

fiber at X fiber at X φ∗

• htpy. pullback

Space of units of A

fiber at A

Here A is an associative algebra with underlying object X. The red map is induced by the canonical map φ: As −→ uAs.

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A homotopy epimorphism Theorem For any operad P in V, the fibers of the following map are either empty or contractible, φ∗ : MapOp(uAs, P) −→ MapOp(As, P).

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A homotopy epimorphism Theorem For any operad P in V, the fibers of the following map are either empty or contractible, φ∗ : MapOp(uAs, P) −→ MapOp(As, P). A map f : X → Y in a category C is an epimorphism if any of these equivalent statements holds: f ∗ : HomC(Y, Z) → HomC(X, Z) is injective for any Z in C. In the following pushout the red arrows are isomorphisms X Y Y Y

X Y pushout f f

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A homotopy epimorphism Theorem For any operad P in V, the fibers of the following map are either empty or contractible, φ∗ : MapOp(uAs, P) −→ MapOp(As, P). A map f : X → Y in a model category C is a homotopy epimorphism if any of these equivalent statements holds: f ∗ : MapC(Y, Z) → MapC(X, Z) has empty or contactible fibers for any Z in C. In the following homotopy pushout the red arrows are w.e. X Y Y Y L

X Y

• htpy. push.

f f

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A homotopy epimorphism Theorem For any operad P in V, the fibers of the following map are either empty or contractible, φ∗ : MapOp(uAs, P) −→ MapOp(As, P). Equivalently, one (and hence both) of the two red maps in the following homotopy pushout in Op is a weak equivalence, As uAs uAs uAs L

As uAs

• htpy. pushout

φ φ

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The homotopy pushout If Op were left proper, the previous homotopy pushout would be the following pushout As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

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The homotopy pushout If Op were left proper, the previous homotopy pushout would be the following pushout As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

The u∞ associative operad.

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Relative left properness Theorem Given a pushout diagram in Op P R Q Q

P R pushout f g

such that the components of P and Q are cofibrant in V, if f is a weak equivalence then so is g.

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Specific cases

In the category of groupoids What are algebras over these operads? As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

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In the category of groupoids What are algebras over these operads? As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

Strictly associative and strictly unital monoidal groupoids

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In the category of groupoids What are algebras over these operads? As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

Strictly associative monoidal groupoids

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In the category of groupoids What are algebras over these operads? As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

The tensor unit can be canonically strictified

18

In the category of groupoids What are algebras over these operads? As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

Strictly associative non-unital monoidal groupoids

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In the category of groupoids What are algebras over these operads? As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

Forgetting the unit

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In the category of groupoids What are algebras over these operads? As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

Strictly associative monoidal groupoids equipped with an extra strict unit

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In the category of groupoids What are algebras over these operads? As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

Strictly associative monoidal groupoids equipped with an extra strict unit This is the same as a strictly associative and strictly unital monoidal groupoid equipped with an isomorphism 1 I.

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In the category of groupoids What are algebras over these operads? As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

is a weak equivalence!

18

In the category of chain complexes As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

Generated by in degree 0 with trivial differential and relation

• 19

In the category of chain complexes As u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

Generated by in degree 0 with trivial differential and relations

• 19

In the category of chain complexes As

?

u∞As uAs uAs uAs

As u∞As pushout λ φ ψ ∼ φ

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In the category of chain complexes By relative left properness, λ can be obtained as A∞ uA∞ As u∞As uAs

pushout ∼ ∼ λ ∼ φ ∼

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In the category of chain complexes By relative left properness, λ can be obtained as A∞ uA∞ As u∞As uAs

pushout ∼ ∼ λ ∼ φ ∼

Stasheff’s operad, cellular chains on associahedra · · ·

20

In the category of chain complexes By relative left properness, λ can be obtained as A∞ uA∞ As u∞As uAs

pushout ∼ ∼ λ ∼ φ ∼

Fukaya–Oh–Ohta–Ono’s operad, cellular chains on unital associahedra [M.–Tonks’14] · · ·

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In the category of chain complexes The operad u∞As is generated by and all corked corollas like The first two generators have degree 0 and trivial differential. The degree of a corked corolla is 2 · #{corks} + #{leaves} − 2. The only relation is the one in As, hence the inclusion λ: As ֌ u∞As is a cofibration. The ideal generated by corked corollas is contractible and its quotient is uAs, so u∞As

∼

−→ uAs.

21

In the category of chain complexes We define the exhaustive filtration by cofibrations As u0As ⊂ · · · ⊂ unAs ⊂ · · · ⊂ u∞As where, for n ≥ 1, unAs is the suboperad of unAs generated by and all corked corollas with ≤ n corks.

22

In the category of chain complexes We define the exhaustive filtration by cofibrations As u0As ⊂ · · · ⊂ unAs ⊂ · · · ⊂ u∞As where, for n ≥ 1, unAs is the suboperad of unAs generated by and all corked corollas with ≤ n corks. Lemma The inclusion uAs

As un−1As ⊂ uAs As unAs is always a weak

equivalence. In particular ψ: uAs uAs

As u0As ⊂ uAs As u∞As too.

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In the category of chain complexes By induction on n and relative left properness, uAs

As un−1As

uAs

As unAs

uAs Q

pushout ∼ ∼

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In the category of chain complexes By induction on n and relative left properness, uAs

As un−1As

uAs

As unAs

uAs Q

pushout ∼ ∼

The red retraction is defined by → , → 0.

23

In the category of chain complexes By induction on n and relative left properness, uAs

As un−1As

uAs

As unAs

uAs Q

pushout ∼ ∼

For n 1, the operad Q is generated by and all corked corollas with 1 cork. The only relations are the two ones in uAs.

23

In the category of chain complexes By induction on n and relative left properness, uAs

As un−1As

uAs

As unAs

uAs Q

pushout ∼ ∼

For n 1, the operad Q is generated by and all corked corollas with 1 cork. The only relations are the two ones in uAs. The differential is d

• −

, d

• −

.

23

In the category of chain complexes By induction on n and relative left properness, uAs

As un−1As

uAs

As unAs

uAs Q

pushout ∼ ∼

For n 1, the operad Q is generated by and all corked corollas with 1 cork. The only relations are the two ones in uAs. The differential is d

• ± terms of the form

, , .

23

In the category of chain complexes By induction on n and relative left properness, uAs

As un−1As

uAs

As unAs

uAs Q

pushout ∼ ∼

For n > 1, the operad Q is generated by and all corked corollas with n corks. The only relations are the two ones in uAs. The differential is d

• ± terms of the form

, .

23

In the category of chain complexes Theorem Any DG-operad of the form P (F(S), d) has a cylinder IP (F(i0S ∐ ΣS ∐ i1S), d) such that i0, i1 : P → IP are DG-maps and d(Σx) i0x − i1x + extra terms.

24

In the category of chain complexes Theorem Any DG-operad of the form P (F(S), d) has a cylinder IP (F(i0S ∐ ΣS ∐ i1S), d) such that i0, i1 : P → IP are DG-maps and d(Σx) i0x − i1x + extra terms. Q is free and linear relative to uAs and there is a strong deformation retraction uAs ⇄ Q h h

• Σ
• ±

, h

• Σ
• ±

.

24

Transfer to other categories

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs.

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

Ch(k) ⇆ Ch(k)≥0 ⇄ Mod(k)∆op ⇆ Set∆op ⇄ Grd .

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

Ch(k) ⇆ Ch(k)≥0 ⇄ Mod(k)∆op ⇆ Set∆op ⇄ Grd . Inclusion ⊣ truncation.

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

Ch(k) ⇆ Ch(k)≥0 ⇄ Mod(k)∆op ⇆ Set∆op ⇄ Grd . Dold–Kan equivalence [Schwede–Shipley’03].

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

Ch(k) ⇆ Ch(k)≥0 ⇄ Mod(k)∆op ⇆ Set∆op ⇄ Grd . Free module ⊣ forgetful, fundamental groupoid ⊣ nerve.

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

Ch(k) ⇆ Ch(k)≥0 ⇄ Mod(k)∆op ⇆ Set∆op ⇄ Grd . Free module ⊣ forgetful, fundamental groupoid ⊣ nerve. In simplicial sets, the map ψ: uAs → uAs

As u∞As induces an

equivalence on fundamental groupoids and a quasi-isomorphism in homology, so it is a weak equivalence.

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

Set∆op ⇄ Spectra Infinite suspension ⊣ 0th term.

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

V ⇆ Set∆op . Simplicial V.

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

V ⇆ Ch(k). Complicial V.

26

Transfer to other categories Theorem Any weak symmetric monoidal Quillen pair V ⇄ W induces a Quillen pair OpV ⇄ OpW and the derived left Quillen func- tor in Ho OpV ⇄ Ho OpW preserves φ: As → uAs. Derived left Quillen functors preserve homotopy

• epimorphisms. They also reflect them if they are fully faithful.

V ⇆ Spectra . Spectral V.

26

HOW MANY UNITS MAY AN A∞-ALGEBRA HAVE?

Workshop in Category Theory and Algebraic Topology, Louvain-la-Neuve, 10–12 September 2015. Fernando Muro Universidad de Sevilla