(05/14/2021, 09:27 PM)JmsNxn Wrote: Secondly, equivariant seems to be a concept on flows; and this symbol seems independent of flows. The definition of equivariant seems reserved for flows. It kind of throws a stick in the mud when we just want a term for monoids.
I still don't get what your trying to express. In my understanding: equivariance is more general than flows (if with flows you restrict yourself to continuous solutions to differential eqs.). This symbol is more general than flows in the strict sense but it is at the right level of generality when talking of monoids (it is a special case of the concept of natural trasformation).
Some instances of equivariance are \( f(\lambda x)=\lambda f(x) \), \( \chi(f^{\circ n}(x))=g^{\circ n}(\chi(x)) \) or \( e^{\lambda z}=(e^z)^\lambda \).
The real problem with equivariance is... how you call the sets [f,g]? Equivariant class? Class of equivariance? This doesn't sound right. Maybe set of equivariances of f and g.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)