symmetric monoidal (∞,1)-category of spectra
A symmetric midpoint algebra that is also a 0-truncated symmetric 2-group.
A symmetric midpoint group or abelian midpoint group is a set $G$ with an element $0:G$, a function $-: G \to G$, a binary operation $(-)+(-):G \times G \to G$ and a binary operation $(-)\vert(-): G \times G \to G$ such that
$(G,+,0,-)$ is an abelian group
$(G,\vert,0,-)$ is a symmetric midpoint algebra
for all $a$ and $b$ in $G$, $a \vert b + a \vert b = a + b$
The rational numbers, real numbers, and the complex numbers with $a \vert b \coloneqq \frac{a + b}{2}$ are examples of symmetric midpoint groups.
The dyadic rational numbers are the free symmetric midpoint group on one generator.
The trivial group is a symmetric midpoint group, and is in fact a zero object in the category of symmetric midpoint groups.
Created on June 1, 2021 at 01:12:58. See the history of this page for a list of all contributions to it.