A Quick Introduction to Operads
Eva Belmont
October 20, 2013
These are notes for a talk I gave about the first half of May’s Geometry of Iterated Loop
Spaces.
1. Introducing operads
An operad C is a collection of spaces C(n) for n ≥ 0 that satisfy a long list of properties.
Instead of writing them out, I invite you to look them up in [2, Def. 1.1], and offer the
following intuition instead. The prototypical operad is the sequence of spaces EX (n) =
Maps(X n , X) for some X. The properties that operads are required to have should be seen
as natural generalizations of the following properties of Maps(X n , X):
• Given a function of n variables g : X n → X, and n other maps fi1 : X i1 → P
X, . . . ,
i
n
fin : X → X, you can substitute the fij ’s into g to obtain a function of ni=1 ij
variables. (The resulting function is notated by γ(g; fi1 , · · · , fin ).)
• The identity is an element of EX (1).
• The symmetric group Σk acts on EX (k): given a function of k variables and a permutation σk , make a new function of k variables by permuting the variables according
to σk .
If C satisfies the first two properties but doesn’t have an action of the symmetric group, then
it is called a non-Σ operad.
Construction 1.1. Given a non-Σ operad C, you can always turn it into an operad D by
defining D(j) = C(j) × Σj . Then Σj acts freely on the right.
True to the name (and prototypical example), we often want to think about operations of
operads on a space X: this is defined to be a morphism of operads C → EX (i.e. a map
preserving the axioms I didn’t state – see [2, before Def. 1.2]).
Example 1.2. The simplest example of a non-Σ operad A(j) is where A(j) = ∗ for all j. I
claim that
operations of A on X ⇐⇒ associative multiplications µ : X × X → X.
The multiplication µ corresponds to the image of A(2) = ∗ in Maps(X 2 , X). Associativity
is guaranteed because γ(µ; 1, µ) (i.e. the operation (a, b, c) 7→ µ(a, µ(b, c))), and γ(µ; µ, 1)
(i.e. the operation (a, b, c) 7→ µ(µ(a, b), c)) are both elements of the image of A(3) = ∗ in
Maps(X 3 , X), which is a point so they must be the same.
Example 1.3. In general, instead of associative multiplications µ we might want to consider
homotopy-associative multiplications: that is, operations µ such that the maps (a, b, c) 7→
µ(a, µ(b, c)) and (a, b, c) 7→ µ(µ(a, b), c) are homotopic, not necessarily equal. So for every
a, b, c we get paths between µ(µ(a, b), c) and µ(a, µ(b, c)). In higher dimensions, it turns out
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2
we don’t just want homotopies between various parenthesizings, but also homotopies between
homotopies, and so on. This leads one to consider operads C such that C(j) is contractible.
For a concrete example of such an operad, consider the Stasheff polytope K• , also known as
the associahedron: Kn is the polytope with one vertices for every way of parenthesizing the
n elements a1 , · · · , an ; there is an edge between two vertices if one expression can be reached
from the other by changing one pair of parentheses, a 2-cell relating expressions that can be
reached from each other by two re-parenthesizings, etc.
Figure 1. K2 , K3 , and K4
An operation of K• on a space X would be a map K2 → EX (2) (i.e. the choice of a multiplication µ : X × X → X) that is homotopy-commutative in the ways dictated by the higher
maps Kn → EX (n).
We have been describing non-Σ operads. To add an operad structure without imposing any
extra conditions, apply Construction 1.1, so define M(j) := A(j) × Σj ∼
= Σj . Now 1 ∈ M(2)
2
corresponds to an associative multiplication µ : X → X, the nontrivial permutation τ ∈ Σ2
corresponds to the map (a, b) 7→ µ(b, a), and similarly permutations of n simply correspond
to permutations of the variables in products X n → X.
We also get the homotopy-associative version analogous to Example 1.3: if C is an operad
where C(j) is contractible, consider operads D where D(j) = C(j) × Σj .
Definition 1.4 (A∞ operads).
• M is the operad such that M(j) = Σj .
• An A∞ operad is an operad D such that π0 D(j) = Σj and all the connected components are contractible. This comes with a natural morphism to M and a free action
of Σj .
• X is an A∞ space if there is some A∞ operad D that “acts” on X, i.e. via a morphism
D → EX .
2. E∞ operads
Now suppose we want to model commutative operations. Let τ be the nontrivial permutation
θ
in Σ2 , and choose an operation D → EX , where D is an A∞ operad, and a multiplication
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µ ∈ D(2). Then µ and τ µ are in different components of D(2), and they probably don’t map
to the same element of EX .
Construction 2.1 (A strictly commutative analogue of M). Impose strict commutativity
the same way we imposed strict associativity – by taking M and stipulating that M(j) ×Σj
a point
is a point. So we consider the operad N where N (j) = ∗ with the trivial action of Σ.
Construction 2.2 (E∞ operads are homotopy-commutative analogues of N ). As with A∞
operads, instead of asking for C(j) (with the Σ-action) to be a point, we just ask it to be
contractible. We also ask Σ to act freely on C; this is a technical condition (and, annoyingly,
means N isn’t an E∞ operad).
Example 2.3 (The Little Cubes Operad). Let I = [0, 1] ⊂ R; specifying a linear map I → I
is equivalent to just specifying the range [a, b] ⊂ [0, 1]. Specifying n of these maps is the same
as specifying a “little cube” [a1 , b1 ] × · · · × [an , bn ] ⊂ I n ; this is a (particularly constrained)
linear map cn : I n → I n .
Definition 2.4. An element of Cn (j) is a map I n t · · · t I n → I n that specifies j little cubes
j
in I n that do not overlap.
Figure 2. An element of C3 (2)
One reason to care about Cn is that it acts on Ωn X. Ωn X is the space of maps (S n , ∗) →
(X, ∗), but we can think of it as the space of maps (I n , ∂I n ) → (X, ∗). Given j little n-cubes
hc1 , · · · , cj i and j maps fi : (I n /∂I n ) → (X ∗ ) ∈ Ωn X, create another map f : (I n , ∂I n ) →
(X, ∗) as follows:
(
fi (c−1
i (t)) if t ∈ im ci
f : t 7→
0
otherwise.
Figure 3. The map (f : (I n , ∂I n ) → (X, ∗)) ∈ Ωn X
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If n = 1, this is just the ordinary composition of j loops.
Figure 4. The map (f : I 1 → X) ∈ ΩX associated to an element of C1 (2)
3. Monads
Construction 3.1 (Monad associated to C). Given an operad C we can define a monad
(C, µ, η) where C is an endofunctor Top → Top:
.
G
CX =
C(j) × X j ∼
j≥0
where ∼ is described below for C = EX (∼ for arbitrary C is analogous):
• “including a basepoint works well”:
f :X n →X,
(a1 ,··· ,∗,··· ,an−1 )
∼
e
xj :=∗
f
f :X n−1 −→ X n −→X,
(a1 ,··· ,an−1 )
;
j
• permuting the variables in f is the same as permuting the order of the copies of X
in X j .
Example 3.2 .
(M X is the James reduced product (free monoid on X)). We defined M X =
F
j
∼, so it looks like there’s a copy of X j for every element of Σj , but the
j M(j) × X
second condition means that every element (σ, x) has a unique representative (1, x0 ). So
we can think of elements of M X as ordered strings x1 · · · xn , and the first condition just
guarantees that this plays nicely with inclusion of the basepoint (i.e. the monoid identity)
anywhere in the string.
Now let’s go back to the little cubes operad Cn ; recall we had an action of Cn on Ωn X for any
space X. If (Cn , µ, ηn ) is the monad corresponding to Cn , this induces a map θn : Cn Ωn X →
Ωn X.
Construction 3.3 (Morphism of monads Cn → Ωn S n ). There is an adjunction S n : Top →
Top : Ωn (where S n is the suspension functor) so Ωn S n is a monad. Then there is a map
Cn ηn
θ
n
αn : Cn X −→ Cn Ωn S n X −→
Ωn S n X.
Theorem 3.4 (“Approximation Theorem”). αn is a weak equivalence.
Proof. I said a little about this in the talk but here I’ll just leave this to [2, Ch. 6-7].
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References:
[1] Adams, J. F. Infinite Loop Spaces. Princeton University Press, 1978.
[2] May, Peter. The Geometry of Iterated Loop Spaces. Springer-Verlag, Lecture Notes in
Mathematics, 1972.
[3] Stasheff, Jim. “What is an operad?” http:www.ams.org/notices/200406/what-is.pdf.