Tag: automatically

Remember that one $ E_ {n, m} $ Algebra is one $ A_m $ Algebra in $ E_n $ Algebras. Here I indicate mine $ A_m $ Algebras, so that one $ A_1 $ Algebra is pointed $ A_2 $ Algebra has an integer multiplication, $ A_3 $ is homotopyassociative, etc. In particular a $ E_n $ Algebra is one $ E_ {n, 1} $ Algebra and on the other hand $ E_ {n-1, infty} $ Algebra.

Question: Accepted $ X $ is a homotopy $ k $Type and a $ E_ {n, m} $algebra, when does
it has a canonical $ E_ {n + 1} $ Structure? In other words, how
Connected is the map of the Operades from the $ E_ {n + 1} $-operad on the
$ E_ {n, m} $-operad? What if $ m = 2 $?

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

A well-known phrase about monoidal categories is:

Theorem: A monoidal category $ mathcal {C} $ is braided when the canonical forgetful map from the center of Drinfeld is divided monoidally $ Z ( mathcal {C}) rightarrow mathcal {C} $,

I was wondering what the proper generalization of this statement is too general $ E_n $ Algebras. That's for what $ k $Types do $ E_ {n + 1} $ Structures on $ E_n $ Algebras correspond to homotopy splits of the forgetful map from the $ E_ {n + 1} $ center $ Z (A) rightarrow A $, If you think about it $ E_0 $ Case, it is not hard to see that homotopy is being withdrawn $ E_1 $ Algebra is only one $ A_2 $ Algebra. Apply this observation $ E_0 $ Algebras in $ E_n $ We should only expect algebras $ A $ to be like above $ E_ {n, 2} $ Algebra. The above sentence thus corresponds to the statement that homotopy $ 1 $-Type that is one $ E_ {1,2} $ Algebra is automatically one $ E_2 $ Algebra. Roughly speaking, this is due to the Eckman-Hilton argument, which states that the two multiplies match and therefore the second multiplication is $ A_ infty $ and not only $ A_2 $,