TITLE: Affine spaces as algebras for an operad?
QUESTION [10 upvotes]: (at.algebraic-topology because I don't know who else thinks about operads)
Let $A$ be an affine space, i.e. a torsor over (the abelian group underlying) a vector space $V$ over a field $K$. Then for any $a_1,\dotsc,a_n \in A$ and any $\lambda_1,\dotsc,\lambda_n\in K$ with $\sum \lambda_i = 1$, we can form the weighted average
$$ \lambda_1 a_1 + \dotsb + \lambda_n a_n \in A. $$
I'm trying to work out exactly what structure this exhibits on $A$.
Let's let $Pr_K(n) \subset K^n$ denote the subset of probability vectors, i.e. $n$-tuples which sum to 1. I believe this forms an operad: the map
$$ Pr_K(n)\times Pr_K(i_1) \times \dotsb \times Pr_K(i_n) \to Pr_K(i_1 + \dotsb + i_n)$$
sends
$$\langle \{\lambda_j\}_{j=1}^n; v_1,\dotsc, v_n\rangle \mapsto \lambda_1 v_1 \mathbin| \dotsb \mathbin| \lambda_n v_n$$
where by $|$ I mean "concatenate these tuples". Then I think $A$ should be an algebra over the operad $Pr_K$.
Is this correct? Is it written down somewhere? What's the standard name for this operad? Can you do anything with this perspective?

REPLY [5 votes]: For $K=\mathbb{R}$, the positive part of your operad (mentioned in Gabriel's comment), and its algebras have been discussed by Tom Leinster and others in connection to entropy. See, for example,
https://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html
http://www.maths.ed.ac.uk/~tl/b.pdf