I am reading the wiki page on operad theory and I am trying to figure out how exactly those "Little something" operads work which are mentioned there. Specifically, I am having a hard time, despite the verbal statements on the page, grasping what is really going on with the n-discs operads (you have an illustration on the right side of the page). Could anyone help me out?
(from the comments) What I do not really understand is: 1) the precise manner in which the inner discs are constructed and inserted 2) whether or not those smaller discs within the unity disc have to somehow cover it all, or what kind of constraints apply...
By the way, maybe I should point out at the relation to this other topic I have a bounty on: A question about Homotopy (Michael Harris's recent book)
I will concentrate on the little 2-discs operad $\newcommand{\DD}{\mathtt{D}_2}\DD$ (I think if you understand how it works, it shouldn't be a problem to understand the analog statements about the higher dimensional ones, or the little cubes operads).
Let $$D^2 = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 \le 1 \}$$ be the unit disk in $\mathbb{R}^2$. An operation $c \in \DD(r)$ in arity $r$ is a collection $(c_1, \dots, c_r)$, where each $c_i$ is an embedding $c_i : D^2 \hookrightarrow D^2$ of the form $$c_i(x,y) = (x_i + \lambda_i x, y_i + \lambda_i y)$$ (where of course $(x_i,y_i) \in D^2$, and $\lambda > 0$ is small enough such that $c_i(D^2) \subset D^2$). (Such an embedding is called a rectilinear embedding.) Furthermore, if $i \neq j$, we require that the image $\DeclareMathOperator{\itr}{int}\itr(D^2)$ of the interior of $D^2$ by $c_i$ and $c_j$ are disjoint; i.e. $$c_i(\itr(D^2)) \cap c_j(\itr(D^2)) = \varnothing;$$ this is equivalent to $d((x_i, y_i), (x_i, y_j)) \ge \lambda_i + \lambda_j$.
An element of $\DD(r)$ is advantageously represented like this:
Here, you see an element of $\DD(2)$. There are two embeddings $c_1, c_2 : D^2 \to D^2$, and I've represented their images in the unit disk. (You can recover the embedding from the image, because the embedding only depends on the center of the disk and its radius.)
Now, what about the operadic composition? Suppose you're given $c \in \DD(r)$, and $d^1 \in \DD(k_1)$, ..., $d^r \in \DD(k_r)$, and you want to compute $c(d^1, \dots, d^r)$. As before, write $c = (c_1, \dots, c_r)$, and $d^i = (d^i_1, \dots, d^i_{k_i})$ (where the $c_i$ and $d^i_j$ are rectilinear embeddings $D^2 \to D^2$ satisfying the conditions about disjointness of interiors).
Then what happens is that $c_i \circ d^i_j$ is again a rectilinear embedding $D^2 \to D^2$. Besides, since $d^i_j(\itr(D^2)) \cap d^i_k(\itr(D^2)) = \varnothing$ for $j \neq k$, then they're still disjoint when composed by $c_i$. And since $c_i(\itr(D^2)) \cap c_{i'}(\itr(D^2)) = \varnothing$ for $i \neq i'$, then the images of the interior of $c_i \circ d^i_j(\itr(D^2)) \cap c_{i'} \circ d^{i'}_{j'}(\itr(D^2)) = \varnothing$. So finally, you get a sequence of rectilinear embeddings $D^2 \to D^2$ $$(c_1 \circ d^1_1, \dots, c_1 \circ d^1_{k_1}, c_2 \circ d^2_1, \dots, \dots, c_r \circ d^r_{k_r})$$ that satisfy the conditions on the images of interior. In other words, this is an element of $\DD(k_1 + \dots + k_r)$, and we define $c(d^1, \dots, d^r)$ to be that element. And then one checks that the operad axioms are satisfied.
Now how does this look like pictorially (the above definition isn't very easy to understand at first)? I find it easier to think of operads in term of partial compositions. Consider the special element $1 = (\operatorname{id}) \in \DD(1)$ (this is the identity of $D^2$, which is a rectilinear embedding, so it defines an element of $\DD$). Then if $c \in \DD(n)$ and $d \in \DD(m)$, one defines $$c \circ_i d = c(1, \dots, 1, \underbrace{d}_{i}, 1, \dots, 1).$$ Basically, you "plug in" the operation $d$ in the $i$th position, and the identity in all the other positions. Then the composition in $\DD$ looks like this:
I hope all this helps you.
References