10/22/2025, 11:53 AM
@Natsugou
I've started to notice many analogies on \( {\rm ete} (X)\) and on some other other contructions that "mods out" by connectedness and the contruction of projective spaces.
I'm trying to make those analogies more precise and formal.
Connected components contruction is like giving the possible "direction" of rays (half lines) at positive infinity. Eternal generators is like modding out lines by the parallel direction, but in the direction of negative infinity.
So both constructions are spiritually connected with the concept of projective space as the space of 1-dimensional subspaces (lines) of a vector space modded out by the parallel (scaling) relation.
In other words, if vector spaces over k have to do with the geometry of the field k, with dynamical system NSets have to do witht the "geometry" of the natural numbers (as a monoid).
Injective funtions/NSet are free like Nsets, like free vector spaces/modules over a Ring.
Surjective functions means that you can go back to negative infinity, so being surjective mean being "completed" by adding the eternal generators.
Much more is going on here, and much more have to do with tensor product of dynamical systems (induction/coinduction of representationa) aka change of base functors. Very exciting.
I've started to notice many analogies on \( {\rm ete} (X)\) and on some other other contructions that "mods out" by connectedness and the contruction of projective spaces.
I'm trying to make those analogies more precise and formal.
Connected components contruction is like giving the possible "direction" of rays (half lines) at positive infinity. Eternal generators is like modding out lines by the parallel direction, but in the direction of negative infinity.
So both constructions are spiritually connected with the concept of projective space as the space of 1-dimensional subspaces (lines) of a vector space modded out by the parallel (scaling) relation.
In other words, if vector spaces over k have to do with the geometry of the field k, with dynamical system NSets have to do witht the "geometry" of the natural numbers (as a monoid).
Injective funtions/NSet are free like Nsets, like free vector spaces/modules over a Ring.
Surjective functions means that you can go back to negative infinity, so being surjective mean being "completed" by adding the eternal generators.
Much more is going on here, and much more have to do with tensor product of dynamical systems (induction/coinduction of representationa) aka change of base functors. Very exciting.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)