2
votes
1answer
38 views

What is the meaning of “Homotopy of Little disc Operads”

I want to understand what means the homotopy of the little discs operad. I'm starting to research in this area and I have some questions. 1) I don't understand why little discs operad is a ...
1
vote
1answer
45 views

Spaces homotopy equivalent to $A_{\infty}$-spaces

I ask this question after reading Peter May's "Geometry of Iterated Loop Spaces", where the problem is definitely hinted at but I couldn't find a definite answer. Recall a symmetric operad ...
1
vote
1answer
65 views

Why is an $E_\infty$-operad a kind of ''strictification'' for a non-commutative operation?

A topological space $X$ which is an algebra over an $E_\infty$-operad $E$ consists of a sequence of maps $$ \mu_n':E(n)\times X^n\to X $$ with compatibility conditions. The spaces $E(n)$ are ...
2
votes
0answers
62 views

Γ-spaces and operads

I'm looking for a comprehensible reference that explains how $\Gamma$-spaces are related to $E_{\infty}$-operads. I've found some old publications but was hoping there are better references out there. ...
5
votes
1answer
234 views

Why is the recognition principle important?

The recognition principle basically states that (under some conditions) a topological space $X$ has the weak homotopy type of some $\Omega^k Y$ iff it is an $E_k$-algebra (ie. an algebra over the ...