Recall that one $ E_ {n, m} $ Algebra is a $ A_m $ algebra in $ E_n $ algebra. Here I index my $ A_m $ algebras so that's one $ A_1 $ algebra is pointed, a $ A_2 $ Algebra has a unitary multiplication, $ A_3 $ is homotopic associative, etc. In particular, a $ E_n $ Algebra is on the one hand a $ E_ {n, 1} $ algebra and on the other hand a $ E_ {n-1, infty} $ algebra.

Question: Suppose $ X $ is a homotopy k $-type and a $ E_ {n, m} $Algebra when done

he has a canonical $ E_ {n + 1} $ structure? In other words, how

connected is the opera map of the $ E_ {n + 1} $-operated

$ E_ {n, m} $-operad? And what would happen if $ m = $ 2?

Let me also briefly explain where this question comes from and why I am particularly interested in $ m = $ 2 Case.

A well-known theorem about monoidal categories says:

Theorem: a monoidal category $ mathcal {C} $ is braided if there is a monoidal duplication of the forgetful canonical map of the center of Drinfeld $ Z ( mathcal {C}) rightarrow mathcal {C} $.

I wondered what is the appropriate generalization of this statement in general $ E_n $ algebra. This is for whom k $-types make $ E_ {n + 1} $ structures on $ E_n $ the algebras correspond to the homotopic divisions of the forgetful map of the $ E_ {n + 1} $ center $ Z (A) rightarrow A $. If you think through the $ E_0 $ In this case, it is not difficult to see that a homotopic retraction of a $ E_1 $ Algebra is only one $ A_2 $ algebra. Apply this observation to $ E_0 $ algebras in $ E_n $ algebras we should not expect to $ A $ as above to be a $ E_ {n, 2} $ algebra. So, the above theorem corresponds to the assertion that a homotopy $ 1 $-type that is a $ E_ {1,2} $ Algebra is automatically a $ E_2 $ algebra. Basically, it's because of the Eckman-Hilton argument, which says that the two multiplications agree and that the second multiplication is $ A_ infty $ and not only $ A_2 $.