No emie C. Combe MPI MiS Wednesday 24/03 at 17:00 · 70’s: algebraic topology; homotopy theory. 90’s: Algebra (Koszul duality), Geometry (moduli spaces of curves), Mathematical
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The realm of operads
Noemie C. CombeMPI MiS
Wednesday 24/03 at 17:00
Noemie C. Combe MPI MiS
The realm of operads
Where do operads appear?
I Situations in which you have an operation with multipleentries.
Figure: Higher operads, higher categories,T. Leinster
Noemie C. Combe MPI MiS
The realm of operads
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
The realm of operads
Algebra
Topology & Geometry Maths-Phys
Operads
Noemie C. Combe MPI MiS
The realm of 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
Noemie C. Combe MPI MiS
The realm of operads
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)
Noemie C. Combe MPI MiS
The realm of operads
Shrink the black box to a point, you obtain this graph \:
Noemie C. Combe MPI MiS
The realm of operads
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)
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
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 ....
Noemie C. Combe MPI MiS
The realm of operads
Operads behave a bit like a DAW
A digital audio workstation (DAW) is an electronic device usedfor recording, editing and producing audio files.
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
Micros
Audio files
Mixing console
Figure:
Noemie C. Combe MPI MiS
The realm of operads
Operads are everywhere
Operads can be applied everywhere ...... as long as you have a
symmetric monoidal category.
Noemie C. Combe MPI MiS
The realm of operads
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 )
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
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 .
Noemie C. Combe MPI MiS
The realm of operads
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 .
Noemie C. Combe MPI MiS
The realm of operads
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 .
Noemie C. Combe MPI MiS
The realm of operads
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 .
Noemie C. Combe MPI MiS
The realm of operads
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 .
Noemie C. Combe MPI MiS
The realm of operads
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 .
Noemie C. Combe MPI MiS
The realm of operads
Baking cake example
Figure: Spivak, 7sketches
Noemie C. Combe MPI MiS
The realm of operads
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
Noemie C. Combe MPI MiS
The realm of operads
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).
Noemie C. Combe MPI MiS
The realm of operads
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).
Noemie C. Combe MPI MiS
The realm of operads
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).
Noemie C. Combe MPI MiS
The realm of operads
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).
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
Panorama: types of operads
• Algebraic operads, ruling associative algebras (Wednesday)
• Topological operads, (little discs operad), loop space
• Geometric operads, coding Gromov–Witten invariants.
Noemie C. Combe MPI MiS
The realm of operads
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).
Noemie C. Combe MPI MiS
The realm of operads
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).
Noemie C. Combe MPI MiS
The realm of operads
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 realm of operads
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}.
Noemie C. Combe MPI MiS
The realm of operads
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}.
Noemie C. Combe MPI MiS
The realm of operads
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
Noemie C. Combe MPI MiS
The realm of operads
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).
Noemie C. Combe MPI MiS
The realm of operads
Compactification
Figure: Devadoss, Tesselations of the moduli spaces and mosaicoperad
Noemie C. Combe MPI MiS
The realm of operads
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).
Noemie C. Combe MPI MiS
The realm of operads
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
Noemie C. Combe MPI MiS
The realm of operads
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)!
Noemie C. Combe MPI MiS
The realm of operads
For more see my preprint Gauss Skizze-Operad andmonodromy on semisimple Frobenius manifolds, N.C. CombeMPIM 45-19 preprints.
Noemie C. Combe MPI MiS
The realm of operads
M0,n
Deligne–Mumford
Fulton--MacPherson
Axelrod–Singer
Kapranov
Getzler–Jones
Noemie C. Combe MPI MiS
The realm of operads
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
Noemie C. Combe MPI MiS
The realm of operads
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
Noemie C. Combe MPI MiS
The realm of operads
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
Noemie C. Combe MPI MiS
The realm of operads
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
Noemie C. Combe MPI MiS
The realm of operads
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
Noemie C. Combe MPI MiS
The realm of operads
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)
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
...
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
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.
Noemie C. Combe MPI MiS
The realm of operads
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