No emie C. Combe MPI MiS Wednesday 24/03 at 17:00 · 70’s: algebraic topology; homotopy theory....

The realm of operads

Noemie C. CombeMPI MiS

Wednesday 24/03 at 17:00

Noemie C. Combe MPI MiS


Where do operads appear?

I Situations in which you have an operation with multipleentries.

Figure: Higher operads, higher categories,T. Leinster



I Operads allow to describe and understand operations withmultiple entries (inputs), acting on certain algebras,topological spaces, or geometric manifolds.

I Define higher invariants in geometry and topology.




Algebra

Topology & Geometry Maths-Phys

Operads



An operad P consists of:

a collection {P(n)}n≥1 of k-vector spaces (or abstract n- ary

operations for each n),

+composition rule



We can think of an n-ary operation as a little black box with nwires coming in and one wire coming out:

Inputs

Output(s)



Shrink the black box to a point, you obtain this graph \:



About trees

Tree T - non-empty, connected oriented graph without loops(oriented or not).Property: For each vertex, there is at least one incomingedge and exactly one outgoing edge.

External edges: edges of the tree, bounded by a vertex at oneend only.

Internal edges: All other edges (i.e. those bounded by verticesat both ends)



Any tree has:- a unique outgoing external edge, called the output (orthe root) of the tree,- several ingoing external edges, called inputs or leaves ofthe tree.

Similarly, the edges going in and out of a vertex v of a tree willbe referred to as inputs and outputs at v.



Roughly speaking, an operad is a kind of super powerfulalgebra (a “higher structure algebra”).

Being a higher structure, allows to classify and organise wellknown algebras such as Lie algebras, commutative algebras, ortopological spaces, etc ....



Operads behave a bit like a DAW

A digital audio workstation (DAW) is an electronic device usedfor recording, editing and producing audio files.



I Inputs:space P(i1) = micro 1,space P(i2) = micro 2,...space P(in)= micro n.

I composition operation = (Mixing console)

I output: audio file 1.



Micros

Audio files

Mixing console

Figure:



Operads are everywhere

Operads can be applied everywhere ...... as long as you have a

symmetric monoidal category.



Category

A category C consists of data that satisfy certain properties.

Category

Objects: x,y,z,...

Morphisms: f : x → y

Composition: (f : x → y , g : y → z)⇒ g ◦ f : x → z

Properties

Identity morphism: Id : x → x

Associativity: (h ◦ g) ◦ f = h ◦ (g ◦ f )



Example : category of sets Set

Objects: sets ∅, {1}, {2}, ..., {1, 2, ..., n}Morphisms of sets: {1, 2, ..., n} → {1, 2, ...,m}I epimorphisms in Set are the surjective maps,

I monomorphisms are the injective maps,

I isomorphisms are the bijective maps.



Symmetric monoidal category

Ingredients:

1. Category C,

2. a tensor product ⊗ : C × C → C,

3. a unit object 1 ∈ C,

4. natural isomorphisms (X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z )

5. coherence axioms,

6. and symmetry isomorphisms cX ,Y : X ⊗ Y → Y ⊗ X suchthat cX ,Y cY ,X = id .



Baking cake example

Figure: Spivak, 7sketches



IllustrationCategory C: ’cake ingredients’.Ob(C): X = sugar, Y = white, W = butter, V = lemon,K =yolk.

Symmetry: M = Meringue = (sugar ⊗ white)= (X ⊗ Y )

( X︸︷︷︸sugar

⊗ Y︸︷︷︸white

) = ( Y︸︷︷︸white

⊗ X︸︷︷︸sugar

) = M︸︷︷︸meringue

.

Associativity:

Lemon filling = L = (sugar ⊗ butter ⊗ lemon ⊗ yolk)

L = ( X︸︷︷︸sugar

⊗ W︸︷︷︸butter

)⊗ V︸︷︷︸lemon

⊗K = X︸︷︷︸sugar

⊗( W︸︷︷︸butter

⊗ V︸︷︷︸lemon

)⊗ K



Mathematically speakingI An operad P consists of a collection of objects {P(n)}n≥1

(in C), such that the symmetric group Sr acts on P(r)

I composition maps:

◦i : P(k)× P(l)→ P(k + l − 1),

I a unit morphism η : 1→ P(1)

I satisfying some axioms (equivariance, unit, associativity).



Historical point

• 70’s: algebraic topology; homotopy theory.• 90’s: Algebra (Koszul duality), Geometry (moduli spaces ofcurves), Mathematical Physics (TQFT), due to the impulse ofY. Manin and M. Kontsevitch.• Nowadays: operads apply to algebraic topology, differentialgeometry, non-commutative geometry, mathematical physics,probabilities, combinatorics, algebraic combinatorics, highercategories and logic.



Panorama: types of operads

• Algebraic operads, ruling associative algebras (Wednesday)

• Topological operads, (little discs operad), loop space

• Geometric operads, coding Gromov–Witten invariants.



Example of operads:

little disc operad

Take (C,⊗): symmetric monoidal category of topologicalspaces, where ⊗ is the cartesian product.

I Let D(q,R) be a disc of center q and radius R < 1 in theEuclidean space Rn.

I D is the unit disc (q = (0, ..., 0) and R = 1).



Little disc operadThe little n-discs D(q,R) are contained in D i.e. there existsan embedding c : D→ D such that c(v) = Rv + q. So, wehave that D(q,R) = c(D).

Figure: Fresse, Little disc operad, graph complexesNoemie C. Combe MPI MiS


The Operad of little n-discs is a structure defined by:

I the collection of spaces Dn = {Dn(r), r ∈ N}, where

Dn(r) consists of r -tuples of little n-discs {c1, ..., cr} suchthat int(ci) ∩ int(cj) = ∅, for all pairs i 6= j .

I Composition operations:

◦i : Dn(k)× Dn(l)→ Dn(k + l − 1)

for all k , l ≥ 0 and i ∈ {1, 2, ..., k}.



Example

Let n = 2, and suppose we have D2(3) composed with D2(2)at i = 3:

◦3 : D2(3)× D2(2)→ D2(4)

You have:

Figure: Fresse, Little disc operad, graph complexes



Topological operads

Little disc operad ←→ configuration spaces ←→ Modulispaces of curves M0,n.

Configuration space for genus 0 Riemann surface:Conf (0, n) = {(x1, ..., xn) ∈ Pn|xi 6= xj}.

Intermediate step: Compactification i.e. Conf (0, n).



Compactification

Figure: Devadoss, Tesselations of the moduli spaces and mosaicoperad



Gauss - skizze operad

I Configuration space of npoints in C

I Conf (0, n)

I n-tuple (x1, ..., xn) ∈ Cn

I Space of degree npolynomials P in C

I {zn+an−2zn−2 +· · ·+a0}I roots of P

————————————————

Idea: Assign a graph to each n-tuple (x1, ..., xn) ∈ Cn, given byP−1(R ∪ ıR) (Gauss skizze).



Gauss - skizze operad

Consider Conf (0, 3). We want to compose with a Conf (0, 2)object at a given point i .

Figure: Degree 3 polynomial:z3 + 1.8 ∗ z + 1 + i

Figure: Degree 2 polynomials



Choose a given input of Conf (0, 3) (represented as a a root ofthe degree 3 polynomial). We compose with Conf (0, 2),(which can be one of the pictures above in blue column). Forexample:

• We need the singular part of Conf (0, n)!



For more see my preprint Gauss Skizze-Operad andmonodromy on semisimple Frobenius manifolds, N.C. CombeMPIM 45-19 preprints.



M0,n

Deligne–Mumford

Fulton--MacPherson

Axelrod–Singer

Kapranov

Getzler–Jones



Operadic zoo

Algebras:

I Operad

I cyclic operads

I k-modular

I dioperads

I properads

Graphs

I Rooted trees

I treesdirected connected genus0 graphs, all generagraphs,...

I connected + orientation+ on set of edges +genus marking

I connected directedgraphs w/o directedloops or parallel edges

I connected directedgraphs w/o directedloops



Operadic zoo:

What kind of operadic creatures can we find?

Modular operad. No distinction between inputs andoutputs.

EXAMPLE. The Deligne-Mumford moduli spaces of stablecurves of genus g with n + 1 points. The operadic compositemaps are defined by intersecting curves along their markedpoints.

(For more about the following objects, see reference : B.Vallette, Algebra + Homotopy = operad)



Operadic zoo

Properad. Several inputs and several outputs. But, incontrast to modular operads, where inputs and outputs areconfused, one keeps track of the inputs and the outputs.

EXAMPLE. Riemann surfaces, i.e. smooth compact complexcurves, with parametrized holomorphic holes form a properad.



Operadic zoo

Prop. Like a properad, but where one can also compose alongnon-necessarily connected graphs. This is the operadic notionwhich was introduced first, by Saunders MacLane as asymmetric monoidal category C.

EXAMPLE. The categories of cobordism, where the objectsare the d-dimensional manifolds and where the morphisms arethe (d + 1)-dimensional manifolds with d-dimension boundary,form a prop.



...



Conclusion

I Operads are universal and interfere in almost each domainof mathematics,

I appear in applied mathematics

I mathematical physics.

Operads are very flexible : many different ways of definingthem using the language which fits the most.



If you enjoyed this introduction to operads and want to knowmore:

Reading group:Tomorrow 25/03, at 17:00

Organisers: Noemie Combe & Joscha Diehl

Tomorrow: introduction to algebraic operads.



No emie C. Combe MPI MiS Wednesday 24/03 at 17:00 · 70’s: algebraic topology; homotopy theory....

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