enriched -operads overview 1. Operads 2. Barwicks Segal - - PowerPoint PPT Presentation

enriched ∞-operads

Hongyi Chu, joint work with Rune Haugseng and Gijs Heuts August 2, 2017

• verview
• 1. Operads
• 2. Barwick’s Segal presheaves and dendroidal Segal spaces
• 3. Enriched ∞-operads
• 4. Applications

1

• perads

In the 70’s Boardman, Vogt and May introduced the notion of

• perads 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

• perads

In the 70’s Boardman, Vogt and May introduced the notion of

• perads 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

• perads

Instead of a chain of morphisms x1 → xn → . . . → xn, we want to have

3

• perads

Instead of a chain of morphisms x1 → xn → . . . → xn, we want to have

3

• perads

Definition (May) A (symmetric coloured) operad O consists of

• 1. a class of objects Ob(O),
• 2. for finite set xi i I and y, a set Mul

xi i I y ,

• 3. compositions

j J

Mul xi i f

1 j yj

Mul yj j J z Mul xi i I z satisfy associativity and unitality conditions.

4

• perads

Definition (May) A (symmetric coloured) operad O consists of

• 1. a class of objects Ob(O),
• 2. for finite set {xi}i∈I and y, a set MulO({xi}i∈I; y),
• 3. compositions

j J

Mul xi i f

1 j yj

Mul yj j J z Mul xi i I z satisfy associativity and unitality conditions.

4

• perads

Definition (May) A (symmetric coloured) operad O consists of

• 1. a class of objects Ob(O),
• 2. for finite set {xi}i∈I and y, a set MulO({xi}i∈I; y),
• 3. compositions

∏

j∈J

MulO({xi}i∈f−1(j); yj) × MulO({yj}j∈J; z) → MulO({xi}i∈I; z) satisfy associativity and unitality conditions.

4

enrichment

Enriched operads Let V be a symmetric monoidal category. We obtain the notion

• f V-enriched operads by requiring:

∙ MulO({xi}i; y) ∈ V, ∙ ⊗

j∈J MulO({xi}i∈f−1(j); yj) ⊗ MulO({yj}j∈J; z) → MulO({xi}i∈I; z)

lies in V.

5

examples

• 1. Vectk is a Vectk-enriched operad with MulVectk(V1, ..., Vn; V) =

vector space of n-linear maps V1 × . . . × Vn → V.

• 2. The little n-cube operad En is a Top-enriched operad with
• ne object

and MulEn

1 k

the toplogical space

• f configurations of k-many n-cubes within the unit n-cube

in

n

• 3. The operad Lie is a Vectk-eniched operad.

6

examples

• 1. Vectk is a Vectk-enriched operad with MulVectk(V1, ..., Vn; V) =

vector space of n-linear maps V1 × . . . × Vn → V.

• 2. The little n-cube operad En is a Top-enriched operad with
• ne object ∗

and MulEn

1 k

the toplogical space

• f configurations of k-many n-cubes within the unit n-cube

in

n

• 3. The operad Lie is a Vectk-eniched operad.

6

examples

• 1. Vectk is a Vectk-enriched operad with MulVectk(V1, ..., Vn; V) =

vector space of n-linear maps V1 × . . . × Vn → V.

• 2. The little n-cube operad En is a Top-enriched operad with
• ne object ∗ and MulEn(∗1, . . . , ∗k; ∗) = the toplogical space
• f configurations of k-many n-cubes within the unit n-cube

in Rn.

• 3. The operad Lie is a Vectk-eniched operad.

6

examples

• 1. Vectk is a Vectk-enriched operad with MulVectk(V1, ..., Vn; V) =

vector space of n-linear maps V1 × . . . × Vn → V.

• 2. The little n-cube operad En is a Top-enriched operad with
• ne object ∗ and MulEn(∗1, . . . , ∗k; ∗) = the toplogical space
• f configurations of k-many n-cubes within the unit n-cube

in Rn.

• 3. The operad Lie is a Vectk-eniched operad.

6

algebras

For operads O, P, the category of O-algebra in P is given by Fun(O, P). Lie algebras Lie algebras = AlgLie Vectk . Loop spaces May’s Recognition Theorem: For X path connected: X is equivalent to an n-fold loop space

nY

X AlgEn Top .

7

algebras

For operads O, P, the category of O-algebra in P is given by Fun(O, P). Lie algebras Lie algebras = AlgLie(Vectk). Loop spaces May’s Recognition Theorem: For X path connected: X is equivalent to an n-fold loop space

nY

X AlgEn Top .

7

algebras

For operads O, P, the category of O-algebra in P is given by Fun(O, P). Lie algebras Lie algebras = AlgLie(Vectk). Loop spaces May’s Recognition Theorem: For X path connected: X is equivalent to an n-fold loop space ΩnY ⇔ X ∈ AlgEn(Top).

7

topological operads

In the case of En the enrichment in spaces is important to 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

topological operads

In the case of En the enrichment in spaces is important to 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

topological operads

In the case of En the enrichment in spaces is important to 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

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

• perads are invented.

9

topological operads

But topological operads are often too rigid: ∙ Weakly equivalent topological operads O and P do not need to induce equivalent homotopy theories of O-algebras and P-algebras. ∙ Homotopy-invariant constructions such as homotopy (co)limits are difficult to set up. Therefore many different approaches to homotopy-coherent

• perads are invented.

9

topological operads

But topological operads are often too rigid: ∙ Weakly equivalent topological operads O and P do not need to induce equivalent homotopy theories of O-algebras and P-algebras. ∙ Homotopy-invariant constructions such as homotopy (co)limits are difficult to set up. Therefore many different approaches to homotopy-coherent

• perads are invented.

9

topological operads

But topological operads are often too rigid: ∙ Weakly equivalent topological operads O and P do not need to induce equivalent homotopy theories of O-algebras and P-algebras. ∙ Homotopy-invariant constructions such as homotopy (co)limits are difficult to set up. Therefore many different approaches to homotopy-coherent

• perads are invented.

9

Barwick’s Segal presheaves

The category ∆F Let F be the category of finite sets, define ∆F as follows: ∙ Ob( ): sequence of finite sets I0 I1 Im . Objects in are “levelwise” trees.

I0 1 2 3 I1 1 2 I2 10

Barwick’s Segal presheaves

The category ∆F Let F be the category of finite sets, define ∆F as follows: ∙ Ob(∆F): sequence of finite sets I0 → I1 → ... → Im = ∗. Objects in are “levelwise” trees.

I0 1 2 3 I1 1 2 I2 10

Barwick’s Segal presheaves

The category ∆F Let F be the category of finite sets, define ∆F as follows: ∙ Ob(∆F): sequence of finite sets I0 → I1 → ... → Im = ∗. Objects in ∆F are “levelwise” trees.

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥

I0={1,2,3}

.

❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

.

⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥

I1={1,2}

.

I2=∗ 10

Barwick’s Segal presheaves

∙ A morphism in ∆F (I0 → ... → Im = ∗) → (J0 → ... → Jn = ∗) is given by a map f: [m] → [n] in ∆ together with levelwise inclusions of edges which respects the tree structure.

11

Barwick’s Segal presheaves

Definition ∙ We write cn for the tree ({1, . . . , n} → ∗) and we call it a corolla. ∙ Let denote the

• category of spaces. A Segal presheaf F

is a presheaf

• p

which satisfies the Segal condition, i.e. F k0 km is a limit of the canonical diagram of corollas and edges in k0 km . ∙ We write Seg for the

• category of Segal presheaves.

12

Barwick’s Segal presheaves

Definition ∙ We write cn for the tree ({1, . . . , n} → ∗) and we call it a corolla. ∙ 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(k0 → ... → km) is a limit of the canonical diagram of corollas and edges in (k0 → ... → km). ∙ We write Seg for the

• category of Segal presheaves.

12

Barwick’s Segal presheaves

Definition ∙ We write cn for the tree ({1, . . . , n} → ∗) and we call it a corolla. ∙ 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(k0 → ... → km) is a limit of the canonical diagram of corollas and edges in (k0 → ... → km). ∙ We write SegF for the ∞-category of Segal presheaves.

12

Barwick’s Segal presheaves

Segal presheaves encode operadic operations

• 1. F(∗) is the space of objects.
• 2. F

n is the space of n-ary multimorphisms.

• 3. The fibre F

n x1 xn y of the evaluation map F n

F

n 1

• ver x1

xn y can be identified with the space of multimorphisms Mul x1 xn y

13

Barwick’s Segal presheaves

Segal presheaves encode operadic operations

• 1. F(∗) is the space of objects.
• 2. F(cn) is the space of n-ary multimorphisms.
• 3. The fibre F

n x1 xn y of the evaluation map F n

F

n 1

• ver x1

xn y can be identified with the space of multimorphisms Mul x1 xn y

13

Barwick’s Segal presheaves

Segal presheaves encode operadic operations

• 1. F(∗) is the space of objects.
• 2. F(cn) is the space of n-ary multimorphisms.
• 3. The fibre F(cn)x1,...,xn,y of the evaluation map F(cn) → F(∗)n+1
• ver {x1, . . . , xn, y} can be identified with the space of

multimorphisms Mul(x1, . . . , xn; y).

13

example

F(I0 → I1 → ∗)x1,...,xnm,y1,...ym,z F(I0 → ∗)x1,...,xnm,z

I0 x11

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

. . .

xn1

③ ③ ③ ③ ③ ③ ③ ③ ③

x1m

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

. . .

xnm

③ ③ ③ ③ ③ ③ ③ ③ ③

I1

.

y1

◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗

. . . .

ym

♠♠♠♠♠♠♠♠♠♠♠♠♠♠

x11

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

. . .

xnm

③ ③ ③ ③ ③ ③ ③ ③ ③

I0 ∗

.

z

• .

z ∗

By the Segal condition: ( ∏

j∈J

MulO({xi}i∈Ij; yj)) × MulO({yj}j∈J; z) → MulO({xi}i∈I; z).

14

summary

∙ The definition of Segal presheaves is a homotopy coherent (∞-categorical) interpretation of May’s definition of operads. ∙ We need trees of “height” n to describe compositions in a Segal presheaf which are not strictly associative.

15

summary

∙ The definition of Segal presheaves is a homotopy coherent (∞-categorical) interpretation of May’s definition of operads. ∙ We need trees of “height” n to describe compositions in a Segal presheaf which are not strictly associative.

15

• perads via partial composition

Markl’s definition for operads An operads O consists of a class of objects, multimorphisms and partial compositions: For objects xi i I yj j J z and j J,

j

Mul xi i I yj Mul yj j J z Mul xi i I yj j J

j

z satisfying the unitality and associativity conditions.

16

• perads via partial composition

Markl’s definition for operads An operads O consists of a class of objects, multimorphisms and partial compositions: For objects {xi}i∈I, {yj}j∈J, z and j′ ∈ J,

• j′ : MulO({xi}i∈I, yj′)×MulO({yj}j∈J, z) → MulO({xi}i∈I∪{yj}j∈J\{j′}, z)

satisfying the unitality and associativity conditions.

16

• perads via partial composition

(f, g) f ◦2 g

x1

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

x2

tttttttttt

y1

■ ■ ■ ■ ■ ■ ■ ■ ■ ■

.

y2 y3

✉✉✉✉✉✉✉✉✉✉

• y1

◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗

x1

• x2

✇✇✇✇✇✇✇✇✇✇

y3

♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠

.

z

.

z

not lw.

17

dendroidal Segal spaces

What is the ∞-categorical interpretation of Markl’s operads? Definition (Moerdijk, Weiss) We write for the dendroidal category whose objects are trees and whose morphisms are generated by face maps, degeneracy maps and isomorphisms which satisfy certain dendroidal version of simplicial identities. We can regard as a full subcategory of .

18

dendroidal Segal spaces

What is the ∞-categorical interpretation of Markl’s operads? Definition (Moerdijk, Weiss) We write Ω for the dendroidal category whose objects are trees and whose morphisms are generated by face maps, degeneracy maps and isomorphisms which satisfy certain dendroidal version of simplicial identities. We can regard as a full subcategory of .

18

dendroidal Segal spaces

What is the ∞-categorical interpretation of Markl’s operads? Definition (Moerdijk, Weiss) We write Ω for the dendroidal category whose objects are trees and whose morphisms are generated by face maps, degeneracy maps and isomorphisms which satisfy certain dendroidal version of simplicial identities. We can regard ∆ as a full subcategory of Ω.

18

dendroidal Segal spaces

Definition (Cisinski, Moerdijk) A dendroidal Segal space is a functor F: Ωop → S which satisfies the Segal condition, i.e. F(T) is a limit of the canonical diagram of corollas and edges in T. We write Seg for the

• category.

The inclusions induces restriction functors Seg Seg Seg where Seg denotes the

• category of Segal spaces.

19

dendroidal Segal spaces

Definition (Cisinski, Moerdijk) A dendroidal Segal space is a functor F: Ωop → S which satisfies the Segal condition, i.e. F(T) is a limit of the canonical diagram of corollas and edges in T. We write SegΩ for the ∞-category. The inclusions induces restriction functors Seg Seg Seg where Seg denotes the

• category of Segal spaces.

19

dendroidal Segal spaces

Definition (Cisinski, Moerdijk) A dendroidal Segal space is a functor F: Ωop → S which satisfies the Segal condition, i.e. F(T) is a limit of the canonical diagram of corollas and edges in T. We write SegΩ for the ∞-category. The inclusions ∆F ← ֓ ∆ ֒ → Ω induces restriction functors SegF ↠ Seg ↞ SegΩ, where Seg denotes the ∞-category of Segal spaces.

19

complete objects

Definition (Rezk) A Segal space F is called complete, if its space of equivalences (a subspace of F([1])) is equivalent to F([0]). Rezk’s completion theorem The

• category CSeg of complete Segal spaces is the
• category of small
• categories.

We call a Segal presheaf or a dendroidal Segal space is complete if its underlying Segal space is complete.

20

complete objects

Definition (Rezk) A Segal space F is called complete, if its space of equivalences (a subspace of F([1])) is equivalent to F([0]). Rezk’s completion theorem The ∞-category CSeg of complete Segal spaces is the ∞-category of small ∞-categories. We call a Segal presheaf or a dendroidal Segal space is complete if its underlying Segal space is complete.

20

complete objects

Definition (Rezk) A Segal space F is called complete, if its space of equivalences (a subspace of F([1])) is equivalent to F([0]). Rezk’s completion theorem The ∞-category CSeg of complete Segal spaces is the ∞-category of small ∞-categories. ⇒ We call a Segal presheaf or a dendroidal Segal space is complete if its underlying Segal space is complete.

20

different models for ∞-operads

Dendroidal Sets Simplicial Operads Dendroidal complete Segal Spaces Lurie’s ∞-Operads Complete Segal Presheaves Cisinski Moerdijk Cisinski Moerdijk Barwick Heuts, Hinich, Moerdijk ?

21

different models for ∞-operads

Dendroidal Sets Simplicial Operads Dendroidal complete Segal Spaces Lurie’s ∞-Operads Complete Segal Presheaves Cisinski Moerdijk Cisinski Moerdijk Barwick Heuts, Hinich, Moerdijk ?

21

different models for ∞-operads

Dendroidal Sets Simplicial Operads Dendroidal complete Segal Spaces Lurie’s ∞-Operads Complete Segal Presheaves Cisinski Moerdijk Cisinski Moerdijk Barwick Heuts, Hinich, Moerdijk ?

21

different models for ∞-operads

Dendroidal Sets Simplicial Operads Dendroidal complete Segal Spaces Lurie’s ∞-Operads Complete Segal Presheaves Cisinski Moerdijk Cisinski Moerdijk Barwick Heuts, Hinich, Moerdijk ?

21

different models for ∞-operads

Dendroidal Sets Simplicial Operads Dendroidal complete Segal Spaces Lurie’s ∞-Operads Complete Segal Presheaves Cisinski Moerdijk Cisinski Moerdijk Barwick Heuts, Hinich, Moerdijk ?

21

how to prove the equivalence?

• 1. For every symmetric monoidal ∞-category V⊗, we define two

different models of V-enriched ∞-operads (∆F and Ω).

• 2. We reprove Rezk’s completion theorem in the operadic

setting.

• 3. We verify the equivalence of these two approaches.
• 4. We show that we recover dendroidal Segal spaces and Segal

presheaves by chosing .

22

how to prove the equivalence?

• 1. For every symmetric monoidal ∞-category V⊗, we define two

different models of V-enriched ∞-operads (∆F and Ω).

• 2. We reprove Rezk’s completion theorem in the operadic

setting.

• 3. We verify the equivalence of these two approaches.
• 4. We show that we recover dendroidal Segal spaces and Segal

presheaves by chosing .

22

how to prove the equivalence?

• 1. For every symmetric monoidal ∞-category V⊗, we define two

different models of V-enriched ∞-operads (∆F and Ω).

• 2. We reprove Rezk’s completion theorem in the operadic

setting.

• 3. We verify the equivalence of these two approaches.
• 4. We show that we recover dendroidal Segal spaces and Segal

presheaves by chosing .

22

how to prove the equivalence?

• 1. For every symmetric monoidal ∞-category V⊗, we define two

different models of V-enriched ∞-operads (∆F and Ω).

• 2. We reprove Rezk’s completion theorem in the operadic

setting.

• 3. We verify the equivalence of these two approaches.
• 4. We show that we recover dendroidal Segal spaces and Segal

presheaves by chosing V = S.

22

enrichment

Definition (Segal) A symmetric monoidal ∞-category is a functor F∗ → Cat∞. Definition Let

• p be the fibration associated to a presentably

symmetric monoidal

• category

. Define by

Vert

• p

The functor Vert carries a tree to the set of its vertices.

23

enrichment

Definition (Segal) A symmetric monoidal ∞-category is a functor F∗ → Cat∞. Definition Let V⊗ → Fop

∗ be the fibration associated to a presentably

symmetric monoidal ∞-category V. Define by

Vert

• p

The functor Vert carries a tree to the set of its vertices.

23

enrichment

Definition (Segal) A symmetric monoidal ∞-category is a functor F∗ → Cat∞. Definition Let V⊗ → Fop

∗ be the fibration associated to a presentably

symmetric monoidal ∞-category V. Define ∆V

F by

∆V

F

• ❴✤

V⊗

• ∆F

Vert

Fop

∗ ,

The functor Vert carries a tree to the set of its vertices.

23

labeled trees

An object in ∆V

F is of the form ((I0 → ... → Im = ∗), (v1, . . . , vk)).

Objects in are labeled trees. v1 v2 v3

24

labeled trees

An object in ∆V

F is of the form ((I0 → ... → Im = ∗), (v1, . . . , vk)).

⇒ Objects in ∆V

F are labeled trees.

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ④④④④④④④④

v1

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

v2

⑥⑥⑥⑥⑥⑥⑥

v3

24

enrichment via Segal presheaves

Definition A V-enriched Segal presheaf is a presheaf F: ∆V,op

F

→ S satisfying

• 1. the Segal condition as above,
• 2. the Continous Segal condition. It implies that, for v

, the spaces F

n v x1 xn x determine an object

Mul x1 xn x . F is called complete, if its underlying Segal space is. We write CSeg for the corresponding

• category.

25

enrichment via Segal presheaves

Definition A V-enriched Segal presheaf is a presheaf F: ∆V,op

F

→ S satisfying

• 1. the Segal condition as above,
• 2. the Continous Segal condition. It implies that, for v

, the spaces F

n v x1 xn x determine an object

Mul x1 xn x . F is called complete, if its underlying Segal space is. We write CSeg for the corresponding

• category.

25

enrichment via Segal presheaves

Definition A V-enriched Segal presheaf is a presheaf F: ∆V,op

F

→ S satisfying

• 1. the Segal condition as above,
• 2. the Continous Segal condition.

It implies that, for v , the spaces F

n v x1 xn x determine an object

Mul x1 xn x . F is called complete, if its underlying Segal space is. We write CSeg for the corresponding

• category.

25

enrichment via Segal presheaves

Definition A V-enriched Segal presheaf is a presheaf F: ∆V,op

F

→ S satisfying

• 1. the Segal condition as above,
• 2. the Continous Segal condition. It implies that, for v ∈ V, the

spaces F(cn, v)x1,...,xn,x determine an object Mul(x1, . . . , xn; x) ∈ V. F is called complete, if its underlying Segal space is. We write CSeg for the corresponding

• category.

25

enrichment via Segal presheaves

Definition A V-enriched Segal presheaf is a presheaf F: ∆V,op

F

→ S satisfying

• 1. the Segal condition as above,
• 2. the Continous Segal condition. It implies that, for v ∈ V, the

spaces F(cn, v)x1,...,xn,x determine an object Mul(x1, . . . , xn; x) ∈ V. F is called complete, if its underlying Segal space is. We write CSegF(V) for the corresponding ∞-category.

25

Rezk’s completion theorem in the operadic setting

Definition A map F → G of Segal presheaves is ∙ fully faithful if F(cn, v)x1,...,xn,y ≃ G(cn, v)x1,...,xn,y. ∙ essentially surjective if it the induced functor of the underlying

• categories (complete Segal spaces) is.

Completion Result There is an equivalence of

• categories

CSeg Seg FFES

1 26

Rezk’s completion theorem in the operadic setting

Definition A map F → G of Segal presheaves is ∙ fully faithful if F(cn, v)x1,...,xn,y ≃ G(cn, v)x1,...,xn,y. ∙ essentially surjective if it the induced functor of the underlying ∞-categories (complete Segal spaces) is. Completion Result There is an equivalence of

• categories

CSeg Seg FFES

1 26

Rezk’s completion theorem in the operadic setting

Definition A map F → G of Segal presheaves is ∙ fully faithful if F(cn, v)x1,...,xn,y ≃ G(cn, v)x1,...,xn,y. ∙ essentially surjective if it the induced functor of the underlying ∞-categories (complete Segal spaces) is. Completion Result There is an equivalence of ∞-categories CSegF(V) ≃ SegF(V)[FFES−1].

26

enriched dendroidal Segal spaces

Using the same arguments: ∙ Define ΩV by pullback, ∙ Define the

• category Seg
• f
• enriched dendroidal

Segal spaces using (continuous) Segal condition. ∙ Define FFES and prove CSeg Seg FFES

1 . 27

enriched dendroidal Segal spaces

Using the same arguments: ∙ Define ΩV by pullback, ∙ Define the ∞-category SegΩ(V) of V-enriched dendroidal Segal spaces using (continuous) Segal condition. ∙ Define FFES and prove CSeg Seg FFES

1 . 27

enriched dendroidal Segal spaces

Using the same arguments: ∙ Define ΩV by pullback, ∙ Define the ∞-category SegΩ(V) of V-enriched dendroidal Segal spaces using (continuous) Segal condition. ∙ Define FFES and prove CSegΩ(V) ≃ SegΩ(V)[FFES−1].

27

comparison

Theorem The ∞-categories CSegF(V) and CSegΩ(V) are equivalent. (Both are the

• category of algebras for certain monad.)

Theorem These

• categories are equivalent to the
• category of
• enriched operads coming from classical constructions using

model categories. (As long as such a model structure exists.)

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comparison

Theorem The ∞-categories CSegF(V) and CSegΩ(V) are equivalent. (Both are the ∞-category of algebras for certain monad.) Theorem These

• categories are equivalent to the
• category of
• enriched operads coming from classical constructions using

model categories. (As long as such a model structure exists.)

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comparison

Theorem The ∞-categories CSegF(V) and CSegΩ(V) are equivalent. (Both are the ∞-category of algebras for certain monad.) Theorem These ∞-categories are equivalent to the ∞-category of V-enriched operads coming from classical constructions using model categories. (As long as such a model structure exists.)

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applications

Theorem We have CSegF(S) ≃ CSegF and CSegΩ(S) ≃ CSegΩ. All different models for

• operads mentioned above are

equivalent. It provides a different proof for the fact that CSeg simplicial operads.

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applications

Theorem We have CSegF(S) ≃ CSegF and CSegΩ(S) ≃ CSegΩ. ⇒ All different models for ∞-operads mentioned above are equivalent. It provides a different proof for the fact that CSeg simplicial operads.

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applications

Theorem We have CSegF(S) ≃ CSegF and CSegΩ(S) ≃ CSegΩ. ⇒ All different models for ∞-operads mentioned above are equivalent. ⇒ It provides a different proof for the fact that CSegΩ ≃ simplicial operads.

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this is the starting point...

∙ Study Koszul duality of enriched ∞-operads (joint project with Haugseng). ∙ Define and study enriched

• properads or cyclic
• operads

(joint project with Hackney). For mor details please check: ∙ arxiv.org/abs/1606.03826 Joint work with Haugseng and Heuts. ∙ arxiv.org/abs/1707.08049 Joint work with Haugseng.

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this is the starting point...

∙ Study Koszul duality of enriched ∞-operads (joint project with Haugseng). ∙ Define and study enriched ∞-properads or cyclic ∞-operads (joint project with Hackney). For mor details please check: ∙ arxiv.org/abs/1606.03826 Joint work with Haugseng and Heuts. ∙ arxiv.org/abs/1707.08049 Joint work with Haugseng.

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this is the starting point...

∙ Study Koszul duality of enriched ∞-operads (joint project with Haugseng). ∙ Define and study enriched ∞-properads or cyclic ∞-operads (joint project with Hackney). For mor details please check: ∙ arxiv.org/abs/1606.03826 Joint work with Haugseng and Heuts. ∙ arxiv.org/abs/1707.08049 Joint work with Haugseng.

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