August 2, 2017 Hongyi Chu, joint work with Rune Haugseng and Gijs Heuts 0 enriched ∞ -operads
overview 1. Operads 2. Barwick’s Segal presheaves and dendroidal Segal spaces 4. Applications 1 3. Enriched ∞ -operads
operads In the 70’s Boardman, Vogt and May introduced the notion of operads in order to study algebraic structures in topology such as loop spaces. Operads can be regarded as generalizations of categories where we allow morphisms to have multi-inputs. 2
operads In the 70’s Boardman, Vogt and May introduced the notion of operads in order to study algebraic structures in topology such as loop spaces. Operads can be regarded as generalizations of categories where we allow morphisms to have multi-inputs. 2
operads we want to have 3 Instead of a chain of morphisms x 1 → x n → . . . → x n ,
operads to have 3 Instead of a chain of morphisms x 1 → x n → . . . → x n , we want
2. for finite set x i i I and y , a set Mul x i i I y , 1 j y j y j j J z x i i I z x i i f operads satisfy associativity and unitality conditions. Mul Mul Mul Definition (May) j J 3. compositions 4 A (symmetric coloured) operad O consists of 1. a class of objects Ob ( O ) ,
1 j y j y j j J z x i i I z x i i f operads Definition (May) 3. compositions j J Mul Mul Mul satisfy associativity and unitality conditions. 4 A (symmetric coloured) operad O consists of 1. a class of objects Ob ( O ) , 2. for finite set { x i } i ∈ I and y , a set Mul O ( { x i } i ∈ I ; y ) ,
operads Definition (May) 3. compositions satisfy associativity and unitality conditions. 4 A (symmetric coloured) operad O consists of 1. a class of objects Ob ( O ) , 2. for finite set { x i } i ∈ I and y , a set Mul O ( { x i } i ∈ I ; y ) , ∏ Mul O ( { x i } i ∈ f − 1 ( j ) ; y j ) × Mul O ( { y j } j ∈ J ; z ) → Mul O ( { x i } i ∈ I ; z ) j ∈ J
enrichment Enriched operads 5 Let V be a symmetric monoidal category. We obtain the notion of V - enriched operads by requiring: ∙ Mul O ( { x i } i ; y ) ∈ V , ∙ ⊗ j ∈ J Mul O ( { x i } i ∈ f − 1 ( j ) ; y j ) ⊗ Mul O ( { y j } j ∈ J ; z ) → Mul O ( { x i } i ∈ I ; z ) lies in V .
2. The little n -cube operad E n is a Top-enriched operad with examples one object and Mul E n 1 k the toplogical space of configurations of k -many n -cubes within the unit n -cube in n 3. The operad Lie is a Vect k -eniched operad. 6 1. Vect k is a Vect k -enriched operad with Mul Vect k ( V 1 , ..., V n ; V ) = vector space of n -linear maps V 1 × . . . × V n → V .
examples and Mul E n 1 k the toplogical space of configurations of k -many n -cubes within the unit n -cube in n 3. The operad Lie is a Vect k -eniched operad. 6 1. Vect k is a Vect k -enriched operad with Mul Vect k ( V 1 , ..., V n ; V ) = vector space of n -linear maps V 1 × . . . × V n → V . 2. The little n -cube operad E n is a Top-enriched operad with one object ∗
examples of configurations of k -many n -cubes within the unit n -cube 3. The operad Lie is a Vect k -eniched operad. 6 1. Vect k is a Vect k -enriched operad with Mul Vect k ( V 1 , ..., V n ; V ) = vector space of n -linear maps V 1 × . . . × V n → V . 2. The little n -cube operad E n is a Top-enriched operad with one object ∗ and Mul E n ( ∗ 1 , . . . , ∗ k ; ∗ ) = the toplogical space in R n .
examples of configurations of k -many n -cubes within the unit n -cube 3. The operad Lie is a Vect k -eniched operad. 6 1. Vect k is a Vect k -enriched operad with Mul Vect k ( V 1 , ..., V n ; V ) = vector space of n -linear maps V 1 × . . . × V n → V . 2. The little n -cube operad E n is a Top-enriched operad with one object ∗ and Mul E n ( ∗ 1 , . . . , ∗ k ; ∗ ) = the toplogical space in R n .
Lie algebras = Alg Lie Vect k . Alg E n Top . algebras Lie algebras Loop spaces May’s Recognition Theorem: For X path connected: X is equivalent to an n -fold loop space n Y X 7 For operads O , P , the category of O -algebra in P is given by Fun ( O , P ) .
Alg E n Top . algebras Lie algebras Loop spaces May’s Recognition Theorem: For X path connected: X is equivalent to an n -fold loop space n Y X 7 For operads O , P , the category of O -algebra in P is given by Fun ( O , P ) . Lie algebras = Alg Lie ( Vect k ) .
algebras Lie algebras Loop spaces May’s Recognition Theorem: For X path connected: 7 For operads O , P , the category of O -algebra in P is given by Fun ( O , P ) . Lie algebras = Alg Lie ( Vect k ) . X is equivalent to an n -fold loop space Ω n Y ⇔ X ∈ Alg E n ( Top ) .
topological operads describe compositions of paths in a topological space which are only unique up to equivalence and associative up to a coherent homotopy. Such an operad is called -operad . -operads are used to describe algebraic structures in categories with homotopies such as model categories, relative categories, topological categories... 8 In the case of E n the enrichment in spaces is important to
topological operads describe compositions of paths in a topological space which are only unique up to equivalence and associative up to a coherent homotopy. -operads are used to describe algebraic structures in categories with homotopies such as model categories, relative categories, topological categories... 8 In the case of E n the enrichment in spaces is important to Such an operad is called ∞ -operad .
topological operads describe compositions of paths in a topological space which are only unique up to equivalence and associative up to a coherent homotopy. categories with homotopies such as model categories, relative categories, topological categories... 8 In the case of E n the enrichment in spaces is important to Such an operad is called ∞ -operad . ∞ -operads are used to describe algebraic structures in
topological operads But topological operads are often too rigid: ∙ Weakly equivalent topological operads and do not need to induce equivalent homotopy theories of -algebras and -algebras. ∙ Homotopy-invariant constructions such as homotopy (co)limits are difficult to set up. Therefore many different approaches to homotopy-coherent operads are invented. 9
topological operads But topological operads are often too rigid: ∙ Homotopy-invariant constructions such as homotopy (co)limits are difficult to set up. Therefore many different approaches to homotopy-coherent operads are invented. 9 ∙ Weakly equivalent topological operads O and P do not need to induce equivalent homotopy theories of O -algebras and P -algebras.
topological operads But topological operads are often too rigid: ∙ Homotopy-invariant constructions such as homotopy (co)limits are difficult to set up. Therefore many different approaches to homotopy-coherent operads are invented. 9 ∙ Weakly equivalent topological operads O and P do not need to induce equivalent homotopy theories of O -algebras and P -algebras.
topological operads But topological operads are often too rigid: ∙ Homotopy-invariant constructions such as homotopy (co)limits are difficult to set up. Therefore many different approaches to homotopy-coherent operads are invented. 9 ∙ Weakly equivalent topological operads O and P do not need to induce equivalent homotopy theories of O -algebras and P -algebras.
Barwick’s Segal presheaves ∙ Ob( ): sequence of finite sets I 0 I 1 I m . Objects in are “levelwise” trees. I 0 1 2 3 I 1 1 2 I 2 10 The category ∆ F Let F be the category of finite sets, define ∆ F as follows:
Barwick’s Segal presheaves Objects in are “levelwise” trees. I 0 1 2 3 I 1 1 2 I 2 10 The category ∆ F Let F be the category of finite sets, define ∆ F as follows: ∙ Ob( ∆ F ): sequence of finite sets I 0 → I 1 → ... → I m = ∗ .
Barwick’s Segal presheaves 10 The category ∆ F Let F be the category of finite sets, define ∆ F as follows: ∙ Ob( ∆ F ): sequence of finite sets I 0 → I 1 → ... → I m = ∗ . Objects in ∆ F are “levelwise” trees. ❅ ❅ ⑥ ❅ ⑥ ❅ ⑥ ❅ ⑥ I 0 = { 1 , 2 , 3 } ❅ ⑥ ❅ ⑥ ❅ ⑥ ⑥ . . ❆ ❆ ⑥ ❆ ⑥ ❆ ⑥ ❆ ⑥ I 1 = { 1 , 2 } ❆ ⑥ ❆ ⑥ ❆ ⑥ ⑥ . I 2 = ∗
Barwick’s Segal presheaves inclusions of edges which respects the tree structure. 11 ∙ A morphism in ∆ F ( I 0 → ... → I m = ∗ ) → ( J 0 → ... → J n = ∗ ) is given by a map f : [ m ] → [ n ] in ∆ together with levelwise
k m is a limit of the canonical diagram of k m . Barwick’s Segal presheaves which satisfies the Segal condition, -category of Segal presheaves. ∙ We write Seg for the corollas and edges in k 0 i.e. F k 0 op Definition is a presheaf -category of spaces. A Segal presheaf F denote the ∙ Let corolla . 12 ∙ We write c n for the tree ( { 1 , . . . , n } → ∗ ) and we call it a
Barwick’s Segal presheaves Definition corolla . ∙ We write Seg for the -category of Segal presheaves. 12 ∙ We write c n for the tree ( { 1 , . . . , n } → ∗ ) and we call it a ∙ Let S denote the ∞ -category of spaces. A Segal presheaf F is a presheaf ∆ op F → S which satisfies the Segal condition, i.e. F ( k 0 → ... → k m ) is a limit of the canonical diagram of corollas and edges in ( k 0 → ... → k m ) .
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