06/06/2021, 11:18 AM
Ok... I must take a pause...my brain is exploding and I think I'm going crazy. I literally think about this all day every minute. This is too big for me atm especially after I wrote down clearly the proof of functoriality and now that you remind me the thing about representations of functionals.
Under further investigation it turns out that the category I've defined, as long as we stick with groups, i.e. we keep invertibility in the time monoid, seems equivalent to a much simpler and fundamental one that has to do with "distance and outer composition in the group". That also links up heavily with the abstract nonsense pattern I found in the Jabotinsky/Compositional integral business of infinitesimal generators. I'm a bit surprised by this... I didn't expect to see the link with the subfunction operator this soon.
Also yes! it has something to do with adjoints in Hilbert spaces... and also metric spaces.... and also as "it just looks like complicated hilbert spaces": in the analogy functionals are to hilbert spaces what presheaves are to categories. And dual space is to the original space what the cateogory of presheaves is. I see something big. But I don't know what is.
I need to rework all of this from scratches. I need also to go back to basics of category theory.
I need several weeks of stop to tidy up my brain.
I'm sorry.
Under further investigation it turns out that the category I've defined, as long as we stick with groups, i.e. we keep invertibility in the time monoid, seems equivalent to a much simpler and fundamental one that has to do with "distance and outer composition in the group". That also links up heavily with the abstract nonsense pattern I found in the Jabotinsky/Compositional integral business of infinitesimal generators. I'm a bit surprised by this... I didn't expect to see the link with the subfunction operator this soon.
Also yes! it has something to do with adjoints in Hilbert spaces... and also metric spaces.... and also as "it just looks like complicated hilbert spaces": in the analogy functionals are to hilbert spaces what presheaves are to categories. And dual space is to the original space what the cateogory of presheaves is. I see something big. But I don't know what is.
I need to rework all of this from scratches. I need also to go back to basics of category theory.
I need several weeks of stop to tidy up my brain.
I'm sorry.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)