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. forget 1 uAs V As V Closed symmetric monoidal category 4
The space of units Being unital is a property rather than a structure. forget 1 uAs V As V Unital associative algebras in V 4
The space of units Being unital is a property rather than a structure. forget 1 uAs V As V Associative algebras in V 4
The space of units Being unital is a property rather than a structure. forget 1 uAs V As V Faithful 4
The space of units Being unital is a property rather than a structure. forget 1 uAs iso As iso V V 4
The space of units Being unital is a property rather than a structure. forget 1 uAs iso As iso V V Fully faithful! 4
The space of units Being unital is a property rather than a structure. forget 1 uAs iso As iso V V Closed symmetric monoidal model category 4
The space of units Being unital is a property rather than a structure. forget 1 uAs we As we V V 4
The space of units Being unital is a property rather than a structure. forget 1 | uAs we | As we V | V | 4
The space of units Being unital is a property rather than a structure. forget 1 | uAs we | As we V | V | A vertex is an associative algebra A 4
The space of units Being unital is a property rather than a structure. forget 1 Space of fiber at A | uAs we | As we V | V | units of A 4
The space of units Being unital is a property rather than a structure. forget 1 Space of fiber at A | uAs we | As we V | V | units of A Theorem The space of units is either empty or contractible if V is sim- plicial, complicial, or spectral. 4
How to compute it
The space of units uAs V V forgetful functors As V V 6
The space of units |V we | | uAs we V | forgetful functors |V we | | As we V | 6
The space of units fiber at X |V we | | uAs we V | ? htpy. pullback forgetful functors fiber at X |V we | | As we V | ? Here A is an associative algebra with underlying object X . 6
The space of units Space of units of A fiber at A fiber at X |V we | | uAs we V | ? htpy. pullback forgetful functors fiber at X |V we | | As we V | ? Here A is an associative algebra with underlying object X . 6
The space of algebra structures forget As iso V iso V Faithful 7
The space of algebra structures forget |V we | | As we V | 7
The space of algebra structures Space of associative forget fiber at X |V we | | As we V | algebra structures on X 7
The space of algebra structures Space of associative forget fiber at X |V we | | As we V | algebra structures on X Theorem (Rezk’96, M’15) The space of associative algebra structures on a fibrant and cofibrant object X is Map Op ( As , End ( X )) . Op � the category of nonsymmetric operads in V . As � the associative operad in V . End ( X ) � the endomorphism operad of X . 7
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. ◦ 2 � arity 3 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. ◦ 2 � arity 3 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. ◦ 2 � arity 3 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. associativity identity 8
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 ) � Hom V ( X ⊗ n , X ) . 11
Operads Example The endomorphism operad of an object X in V , End ( X )( n ) � Hom V ( X ⊗ n , X ) . Definition A P -algebra is a map of operads P → End ( X ) . 11
Operads Example The endomorphism operad of an object X in V , End ( X )( n ) � Hom V ( 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. 11
The space of algebra structures forget Space of P -algebra fiber at X | P we V | |V we | structures on X Theorem (Rezk’96, M’15) The space of P -algebra structures on a fibrant and cofibrant object X is Map Op ( P , End ( X )) . P � a nonsymmetric operad in V with cofibrant components, e.g. P � As or uAs , the unital associative operad. 12
The spaces of units and algebra structures Space of units of A fiber at A fiber at X | uAs we Map Op ( uAs , End ( X )) |V we | V | htpy. pullback forgetful functors fiber at X | As we Map Op ( As , End ( X )) |V we | V | Here A is an associative algebra with underlying object X . 13
The spaces of units and algebra structures Space of units of A fiber at A fiber at X | uAs we Map Op ( uAs , End ( X )) |V we | V | φ ∗ htpy. pullback forgetful functors fiber at X | As we Map Op ( As , End ( X )) |V we | V | Here A is an associative algebra with underlying object X . The red map is induced by the canonical map φ : As −→ uAs . 13
A homotopy epimorphism Theorem For any operad P in V , the fibers of the following map are either empty or contractible, φ ∗ : Map Op ( uAs , P ) −→ Map Op ( As , P ) . 14
A homotopy epimorphism Theorem For any operad P in V , the fibers of the following map are either empty or contractible, φ ∗ : Map Op ( uAs , P ) −→ Map Op ( As , P ) . A map f : X → Y in a category C is an epimorphism if any of these equivalent statements holds: � f ∗ : Hom C ( Y , Z ) → Hom C ( X , Z ) is injective for any Z in C . � In the following pushout the red arrows are isomorphisms f Y X pushout f Y � X Y Y 14
A homotopy epimorphism Theorem For any operad P in V , the fibers of the following map are either empty or contractible, φ ∗ : Map Op ( uAs , P ) −→ Map Op ( As , P ) . A map f : X → Y in a model category C is a homotopy epimorphism if any of these equivalent statements holds: � f ∗ : Map C ( Y , Z ) → Map C ( X , Z ) has empty or contactible fibers for any Z in C . � In the following homotopy pushout the red arrows are w.e. f X Y htpy. push. f Y � L Y X Y 14
A homotopy epimorphism Theorem For any operad P in V , the fibers of the following map are either empty or contractible, φ ∗ : Map Op ( uAs , P ) −→ Map Op ( As , P ) . Equivalently, one (and hence both) of the two red maps in the following homotopy pushout in Op is a weak equivalence, φ As uAs φ htpy. pushout uAs � L uAs As uAs 14
The homotopy pushout If Op were left proper , the previous homotopy pushout would be the following pushout φ λ ∼ u ∞ As As uAs φ pushout uAs � uAs As u ∞ As ψ 15
The homotopy pushout If Op were left proper , the previous homotopy pushout would be the following pushout φ λ ∼ u ∞ As As uAs φ pushout uAs � uAs As u ∞ As ψ The u ∞ associative operad . 15
Relative left properness Theorem Given a pushout diagram in Op P R g f pushout Q � Q P R such that the components of P and Q are cofibrant in V , if f is a weak equivalence then so is g . 16
Specific cases
In the category of groupoids What are algebras over these operads? φ λ ∼ As u ∞ As uAs φ pushout uAs � As u ∞ As uAs ψ 18
In the category of groupoids What are algebras over these operads? φ λ ∼ As u ∞ As uAs φ pushout uAs � As u ∞ As uAs ψ Strictly associative and strictly unital monoidal groupoids 18
In the category of groupoids What are algebras over these operads? φ λ ∼ As u ∞ As uAs φ pushout uAs � As u ∞ As uAs ψ Strictly associative monoidal groupoids 18
In the category of groupoids What are algebras over these operads? φ λ ∼ As u ∞ As uAs φ pushout uAs � As u ∞ As uAs ψ The tensor unit can be canonically strictified 18
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