### Exercise 1 (Operadic ideal). Let (P,γ,η) be an operad, that is here a

```A LGEBRA , T OPOLOGY AND F JORDS
S UMMER S CHOOL , N ORDFJORDEID , N ORWAY
Let (P, γ, η) be an operad, that is here a monoid in the monoidal category (S-Mod, ◦, I) of S-modules. An
ideal of the operad P is a sub-S-module I ⊂ P such that
γ(µ, ν1 , . . . , νk ) ∈ I
when at least one of the µ, ν1 , . . . , νk lives in I.
(1) Show that the quotient S-module P/I is endowed with a canonical operad structure which satisfies
the classical property of quotients.
(2) Make explicit the ideal generated by
T(
?? 
?? 
?
−
?? ?? 
?? 
?
?? 
 ).
(3) Recover the nonsymmetric operad As.
Exercise 2 (Diassociative algebras).
By definition, a dimonoid is a set D equipped with two maps
a:D×D →D
and
`:D×D →D,
called the left operation and the right operation respectively, satisfying the following five relations


(x a y) a z = x a (y a z),






 (x a y) a z = x a (y ` z),
(x ` y) a z = x ` (y a z),




(x a y) ` z = x ` (y ` z),



 (x ` y) ` z = x ` (y ` z),
for any x, y, z ∈ D.
We denote by Di the set-theoretic nonsymmetric operad which encodes dimonoids.
(1) Make the free dimonoid on one element explicit.
(2) What is the cardinal of Din ?
(3) Describe the operad Di with each of the four equivalent definitions: monoidal, classical, partial
and combinatorial.
(4) Describe a functor
monoids → dimonoids
from the category of monoids to the category of dimonoids.
We consider now the associated linear notion of a diassociative algebra, which is a vector space A equipped
with two linear maps
a:A⊗A→A
and
1
`:A⊗A→A,
satisfying the same relations. We denote by Dias the nonsymmetric linear operad which encodes diassociative algebras. (Hence Diasn = KDin ).
(5) Give a presentation by generators E and relations R of the nonsymmetric operad Dias.
(6) Describe the ideal generated by R in the free operad T (E).
(7) Give an equivalent description of the induced functor
associative algebras → diassociative algebras
in terms of a morphism of nonsymmetric operads Dias → As.
Bonus: Find two different ways of defining this morphism.
(8) Show that the formula [x, y] := x a y − x ` y defines a functor
diassociative algebras → Leibniz algebras
from the category of diassociative algebras to the category of Leibniz algebras.
(9) Describe the symmetric operad Diass which encodes the category of diassociative algebras.
(10) In the same way as in (7), give an equivalent description of the functor
diassociative algebras → Leibniz algebras
in terms of a morphism of operads Leib → Diass.
Exercise 3 (Permutative algebras).
A permutative algebra is a vector space A equipped with a binary operation x ∗ y satisfying the following
relations:
(x ∗ y) ∗ z = x ∗ (y ∗ z) = (x ∗ z) ∗ y .
Answer the same kind of questions as in the previous exercise by replacing the operad Ass by Com and
replacing the operad Leib by Diass.
(1) Does the operad EndKs , where Ks is a one dimensional vector space concentrated in degree 1,
(2) Describe the category of algebras over the operad EndKs .
Exercise 5 (The operad of unital associative algebras).
(1) Describe the nonsymmetric (set-theoretical) operad uAs which encodes unital associative algebras
by making explicit its components uAsn , for any n ≥ 0, and its composition map uAs ◦ uAs →
uAs.
(2) Give a presentation of uAs by generators and relations.
(3) Can we apply the Koszul duality theory of operads to the operad uAs ?
Exercise 6 (The operad encoding Gerstenhaber algebras).
2
We work over the symmetric monoidal category of chain complexes.
(1) Describe the composite product P ◦ Q of two differential graded S-modules.
γ : P ◦P →P
& η : I→P.
(3) Recall that a dg operad is quasi-free is its underlying operad, i.e. after forgetting the differential,
is free. Show that the differential of a quasi-free operad is completely characterized by the image
of its generators.
Exercise 8 (Bar-cobar adjunction). Let C be a dg cooperad and let P be a dg operad.
(1) Prove the existence of a natural bijection
Homdg Op (ΩC, P) ∼
= Tw(C, P) .
(2) Find two universal twisting morphisms ι : C → ΩC and π : BP → P such that any twisting
morphism α : C → P factors through
= ΩC C
||
C gα
|
C
||
|
C!
||
α
/
C B
=P
{
{
B
{{
B
{{ π
B!
fα
{
{
BP.
(3) Are the two twisting morphisms ι and π Koszul morphisms ?
ι
(4) What can you say about the associated morphism (counit of adjunction) of dg operads ΩBP → P
?
Exercise 9 (A∞ -algebras).
¡
(1) Make the Koszul complex As ◦κ As of the nonsymmetric operad As explicit.
(2) Prove that the nonsymmetric operad As is Koszul.
¡
(3) Describe the nonsymmetric dg operad A∞ := ΩAs .
(4) Show that A∞ -algebras are the algebras over the nonsymmetric dg operad A∞ .